diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/README.md b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/README.md index a36e30f80ff9eb1ca16a3398a29bf05c37e4be38..05cc014939435cccf995561250a39dd4935d7508 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/README.md +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/README.md @@ -1,20 +1,20 @@ -# [Incompressible Box Transport and Finite Computation](manuscript.pdf) - -**Author:** OpenAI -**Date:** October 6, 2026 - -## Citation - -```bibtex -@misc{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026, - author = {{OpenAI}}, - title = {{Incompressible Box Transport and Finite Computation}}, - howpublished = {OpenAI Math Release preprint - \href{https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf}{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026}}, - year = {2026} -} -``` - -## Version note - -This version supplies effective torus-projection and common-clock decay estimates, including the projected-force and pressure bounds; see the [previous version](../Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/manuscript.pdf). +# [Incompressible Box Transport and Finite Computation](manuscript.pdf) + +**Author:** OpenAI +**Date:** October 6, 2026 + +## Citation + +```bibtex +@misc{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026, + author = {{OpenAI}}, + title = {{Incompressible Box Transport and Finite Computation}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf}{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026}}, + year = {2026} +} +``` + +## Version note + +This version supplies effective torus-projection and common-clock decay estimates, including the projected-force and pressure bounds; see the [previous version](../Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/manuscript.pdf). diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/analysis.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/analysis.tex index 5f06ea1ea72f1449b748cfaa86da6221df515d99..d9262f206a70cb8bdd2268e52f244379a9b9fb41 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/analysis.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/analysis.tex @@ -1,282 +1,282 @@ -\section{Analytic classes and changes of physical coordinates}\label{sec:model} -The geometric constructions prescribe a smooth velocity first. This section -records exactly which other solutions are excluded by the energy argument. -Pressure assumptions are kept separate: continuity in a pressure norm, a -norm at each individual time, and an integrable pressure gradient are -different conditions. - -Write $\mathcal F_\nu[U]=U_t+(U\cdot\nabla)U-\nu\Delta U$. -All noncompact results below use $\mathbb R^3$; their velocities and forces -have one fixed compact spatial support unless the individual statement -says otherwise. This property concerns the constructed solution, not a -competing solution. On $[0,T]$, write $CH^k=C([0,T];H^k(\mathbb R^3))$ -and $C^1L^2=C^1([0,T];L^2(\mathbb R^3))$. Spatial constants in pressure -may depend on time. - -\begin{lemma}[The pressure term without a pressure norm]\label{lem:b-gradient} -If $w,g\in L^2(\mathbb R^3;\mathbb R^3)$ satisfy -$\operatorname{div}w=0$ and $\operatorname{curl}g=0$ in distributions, then -$\langle w,g\rangle_{L^2}=0$. In particular, any distributional gradient -$g=\nabla p\in L^2$ has zero pairing with $w$, without an assumption that -$p\in L^2$. -\end{lemma} -\begin{proof} -An $L^2$ field is a tempered distribution. The differential identities, -initially tested on compactly supported smooth functions, extend to -Schwartz tests by multiplying by cutoffs tending to one and using $L^2$ -convergence of the tests and their first derivatives. Fourier transformation -gives $\xi\cdot\widehat w=0$ and -$\xi_j\widehat g_k-\xi_k\widehat g_j=0$. These are identities of locally -integrable functions, so they hold almost everywhere. Away from $\xi=0$, -$\widehat g$ is parallel to $\xi$ and $\widehat w$ is perpendicular to it. -Their Hermitian product is zero. The exceptional point has measure zero; -Plancherel proves the assertion. -\end{proof} - -\begin{lemma}[Residual realization and separate comparison classes] -\label{lem:b-comparison}\label{lem:p2-realization} -Let $\nu>0$ and let $U$ be a prescribed smooth divergence-free velocity -on $[0,\infty)\times\mathbb R^3$, with $U(0)=0$. On each finite interval, -assume $U\in CH^2\cap C^1L^2$ and $U,\nabla U$ are bounded. Put -$f=\mathcal F_\nu[U]$. Then $(U,0)$ is a solution. Its velocity is unique -in each of the following separately stated comparison classes, with the -same force and initial data: -\begin{enumerate}[label=(\roman*),itemsep=2pt] -\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v$, and a -pressure representative $p\in CH^1$. -\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, -and an $H^1$ pressure representative at each time, with no required time -continuity in that pressure norm. -\item Strong solutions with $v\in CH^4\cap C^1H^2$ and distributional -$\nabla p\in CL^2$. -\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, -and a pressure representative $p\in CL^2$. -\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, -without an additional integrability or normalization condition on pressure. -\end{enumerate} -The pressure gradient is unique. An $L^2$ pressure representative, when -specified, is zero; otherwise pressure is determined up to a function of -time. The reference material flow is a smooth diffeomorphism at every -finite time and exists for every finite forward time. - -On a flat torus $\T_L^3$, the same conclusion holds in the classical class -where $v,v_t$, spatial derivatives through order two, and $p,\nabla p$ are -continuous on finite closed cylinders, with periodic mean-zero pressure. -No Sobolev conditions at infinity are involved in that assertion. -\end{lemma} -\begin{proof} -Substitution proves existence of the prescribed solution. Set $w=v-U$. -The difference equation is -\[ - w_t+(v\cdot\nabla)w+(w\cdot\nabla)U - =\nu\Delta w-\nabla p,\qquad \operatorname{div}w=0. -\] -The stated velocity regularity implies $w\in C^1L^2\cap CH^2$, so -$\frac d{dt}\|w\|_2^2=2\langle w,w_t\rangle$ at each time. In case -(iii), the stronger assumptions imply these inclusions. They also imply -bounded velocity and spatial derivatives through order two: for $j\le2$, -Cauchy--Schwarz in Fourier space uses the finite integral -$\int_{\mathbb R^3}|\xi|^{2j}(1+|\xi|^2)^{-4}\,d\xi$. - -For the transport term, insert a cutoff $\chi_R$ that equals one on the -ball of radius $R$, is supported in the ball of radius $2R$, and satisfies -$|\nabla\chi_R|\le C/R$. Its boundary error is at most -$CR^{-1}\|v\|_\infty\|w\|_2^2$, which tends to zero. Diffusion is -integrated by parts in $H^2$; alternatively its cutoff error is bounded by -$CR^{-1}\|w\|_2\|\nabla w\|_2$. The deformation term is bounded by -$\|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2$. - -We justify the pressure pairing according to the stated alternatives. -For (i) and (ii), at each fixed time approximate the chosen $H^1$ pressure -by compactly supported smooth functions; distributional incompressibility -and $w\in L^2$ give $\langle\nabla p,w\rangle=0$. Thus no continuity -of $p$ in time is used in (ii). For (iv), integration by parts on the -cutoff gives a pressure boundary term bounded by -$CR^{-1}\|p\|_2\|w\|_2$, also tending to zero. For (iii), apply -Lemma~\ref{lem:b-gradient} directly. Finally, in (v) the equation itself -gives at each time -\[ - \nabla p=f-v_t-(v\cdot\nabla)v+\nu\Delta v\in L^2, - \qquad \|(v\cdot\nabla)v\|_2\le\|v\|_\infty\|\nabla v\|_2. -\] -Here $f\in L^2$ follows from the identical estimate for $U$ and its stated -time and space regularity. The smooth pressure gradient is curl free, so -Lemma~\ref{lem:b-gradient} again cancels the term. This does not choose an -$L^2$ representative of the pressure. - -In every case the resulting pointwise energy inequality is -\[ - \frac12\frac d{dt}\|w\|_2^2+\nu\|\nabla w\|_2^2 - \le \|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2. -\] -The coefficient is bounded on each finite interval and $w(0)=0$, so an -integrating factor gives $w=0$. The pressure has already disappeared before -time integration; the proof imposes no unlisted temporal pressure norm. -Substitution then gives $\nabla p=0$. A spatially constant $L^2$ function -on $\mathbb R^3$ is zero, which proves the normalization assertion. - -On the torus every integration by parts is periodic and has no boundary -term. The specified classical regularity justifies norm differentiation, -diffusion, and pressure cancellation. The same energy inequality gives -uniqueness and the mean-zero normalization fixes pressure. In either domain, -the smooth reference velocity and its bounded spatial derivative give -unique material trajectories; bounded speed prevents finite-time escape. -Solving backwards on a finite interval gives the inverse flow, and smooth -ODE dependence gives its smoothness. -\end{proof} - -Cases (i)--(v) identify the precise hypotheses used by the applications; -their presence in a common lemma does not replace a theorem's declared -comparison class by a different one. In particular, a pressure norm is not -silently imposed in a gradient-only application. The proof is the classical -energy comparison for prescribed smooth solutions \cite[Section~18]{Leray1934}, -with its noncompact pressure pairings made explicit. - -For the flow $\Phi_t$, differentiation of its trajectory equation gives -\[ - U(t)=(\partial_t\Phi_t)\circ\Phi_t^{-1},\qquad - \partial_{tt}\Phi_t(a)= - (f-\nabla p+\nu\Delta U)(t,\Phi_t(a)). -\] -These identities describe the material form of the same constructed solution; -they do not add an independent existence claim. - -\begin{lemma}[Physical scaling]\label{lem:p2-chart} -For $a,b>0$, $x=x_0+ay$, and $s=bt$, let -$U(t,x)=abV(bt,(x-x_0)/a)$. Then trajectories are transformed by this change -of variables, incompressibility is preserved, and at the prescribed physical -viscosity $\nu$, -\[ - \mathcal F_\nu[U](t,x)=ab^2\left[ - V_s+(V\cdot\nabla_y)V-\frac{\nu}{a^2b}\Delta_yV - \right](bt,(x-x_0)/a). -\] -\end{lemma} -\begin{proof} -The trajectory equation gives $\dot x=abV$; differentiating in space gives -$\operatorname{div}_xU=b\operatorname{div}_yV$. The three residual terms -scale by $ab^2,ab^2$, and $b/a$, respectively, proving the formula. -\end{proof} -Thus a side-$10$ torus is kept as a physical side-$10$ torus. If a chart -is rescaled, the residual is recomputed with the fixed $\nu$; the equation -is not identified with a differently normalized viscosity. - -\begin{proposition}[Projection on a flat torus]\label{prop:projection-l2} -Let $g$ be a smooth force on $[0,\infty)\times\T_L^3$ with zero -spatial mean at each time. There is a unique smooth mean-zero potential -$\phi$ satisfying $\Delta\phi=\operatorname{div}g$. Put -$Qg=\nabla\phi$ and $Pg=g-Qg$. Then $Pg$ is solenoidal and mean zero, -and replacing $(g,p)$ by $(Pg,p-\phi)$ preserves the velocity and the -pressure normalization. - -Use normalized spatial measure, frequencies $\xi_k=2\pi k/L$, and norms -\[ - \|G\|_{H^s}^2=\sum_{k\in\Z^3}(1+|\xi_k|^2)^s|\widehat G(k)|^2, - \qquad - \|G\|_{C^r}=\max_{|\beta|\le r}\|\partial_x^\beta G\|_\infty. -\] -For all integers $h,r\ge0$, at each time, -\begin{equation}\label{eq:projection-estimate} - \begin{aligned} - \max\{\|\partial_t^h Pg\|_{C^r},\|\partial_t^h Qg\|_{C^r}\} - &\le S_L\|\partial_t^h g\|_{H^{r+2}},\\ - \|\partial_t^h\phi\|_{C^{r+1}} - &\le S_Ld_L\|\partial_t^h g\|_{H^{r+2}}, - \end{aligned} -\end{equation} -where $S_L=(1+52(L/(2\pi))^4)^{1/2}$ and -$d_L=(1+(L/(2\pi))^2)^{1/2}$. - -Uniform bounds, temporal $L^2$ bounds on spatial suprema, and qualitative -uniform decay of all mixed derivatives pass from $g$ to $Pg,Qg,\phi$. -A quantitative envelope for time order $h$ and output spatial order $r$ -passes to $Pg,Qg$ through order $r$ and to $\phi$ through order $r+1$ -when the input derivatives through spatial order $r+2$ satisfy that -envelope. The constants may depend on the derivative order. -Periodicity and stationarity, including either property on a time tail, -are preserved on their respective intervals. - -For computable $L>0$, effective mixed-derivative evaluation and effective -derivative bounds for $g$ on finite time slabs give effective evaluation -and such bounds for $Pg,Qg,\phi$. Corresponding supplied global input -bounds give effective global bounds in each assertion above. -\end{proposition} -\begin{proof} -For $k\ne0$ define -\[ - \widehat\phi(t,k)=-\frac{i\xi_k\cdot\widehat g(t,k)}{|\xi_k|^2}, - \qquad \widehat\phi(t,0)=0. -\] -The nonzero-frequency symbols of $Q$ and $P$ are complementary orthogonal -projections. Their $H^s$ operator norms are at most one, while the -multiplier for $\phi$ has norm at most $d_L$ from $H^s$ to $H^{s+1}$. -For $|\beta|\le r$, Cauchy--Schwarz bounds the absolute sum of the -Fourier series for $\partial_x^\beta G$ by -\[ - \left(\sum_{k\in\Z^3}(1+|\xi_k|^2)^{-2}\right)^{1/2} - \|G\|_{H^{r+2}}. -\] -The shell $|k|_\infty=n\ge1$ contains $24n^2+2\le26n^2$ points. -Since $\sum_{n\ge1}n^{-2}\le2$, the factor is at most $S_L$. -For the tail $|k|_\infty>N\ge1$, it is at most -\begin{equation}\label{eq:projection-tail} - \left(26(L/(2\pi))^4/N\right)^{1/2}. -\end{equation} -Applying these estimates to the multipliers proves -\eqref{eq:projection-estimate}, including the estimate for $\phi$ with -one additional spatial derivative. On compact time slabs these tails -converge uniformly for every mixed derivative. Thus the series define -smooth fields and may be differentiated term by term. The Fourier -identities give the Poisson equation, uniqueness of its mean-zero -solution, and $\operatorname{div}Pg=0$. Also -$-\nabla(p-\phi)+Pg=-\nabla p+g$. - -For every integer $n\ge0$, the multinomial expansion and Parseval give -\begin{equation}\label{eq:projection-derivative-bounds} - \|G\|_{H^n}^2 - \le\sum_{|\beta|\le n} - \frac{n!}{(n-|\beta|)!\,\beta!} - \|\partial_x^\beta G\|_\infty^2. -\end{equation} -Use $n=r+2$ and $G=\partial_t^h g$. This finite sum proves the uniform -bounds and qualitative decay assertions. If its input suprema are -bounded by $C_{h,\beta}a_h(t)$, it gives the same envelope $a_h$ for -the output, with the constants specified by -\eqref{eq:projection-estimate}--\eqref{eq:projection-derivative-bounds}. -Integrating the squared estimates proves the temporal $L^2$ assertion -from the corresponding input norms. The operators act at each time, -so they preserve the stated temporal symmetries. - -For effective evaluation, the supplied derivative bounds and -\eqref{eq:projection-tail} give a computable Fourier truncation on each -finite time slab. The finitely many coefficient integrals are computed -by Riemann sums. After scaling the torus to $[0,1]^3$, the integrand -for $\widehat G(k)$ has Lipschitz bound -\[ - L\sqrt3\max_j\|\partial_{x_j}G\|_\infty - +2\pi|k|\|G\|_\infty. -\] -This gives an effective quadrature error and hence evaluation of every -mixed derivative of all three fields. The displayed finite sums also -compute output bounds from the supplied input bounds. -\end{proof} -The estimates apply to the pressure shift $\phi$; in the constructions -with original pressure zero, the projected pressure is $-\phi$. -Projection generally changes pressure. It supplies no compact-support -conclusion on Euclidean space and is distinct from the direct zero-pressure -solenoidal shear construction. - -\paragraph{Effective flat profiles.} -For later cutoff formulas put -\begin{equation}\label{eq:p2-step} - \rho(s)=\begin{cases}e^{-1/s},&s>0,\\0,&s\le0,\end{cases} - \qquad \sigma(s)=\frac{\rho(s)}{\rho(s)+\rho(1-s)}. -\end{equation} -Positive-side derivatives of $\rho$ are polynomials in $1/s$ times -$e^{-1/s}$ and extend flatly at zero. The estimate -$v^me^{-v}\le(m+k)!v^{-k}$ gives effective error bounds near the seam, -and $\rho(s)+\rho(1-s)\ge e^{-2}$. Products, translations, rescalings, -and derivatives therefore yield effective cutoffs with all requested -derivative bounds. Periodization uses zero collars; at approximate real -arguments a finite superset of possible translates is evaluated. No real -equality test at a seam or execution of the encoded computation is used. +\section{Analytic classes and changes of physical coordinates}\label{sec:model} +The geometric constructions prescribe a smooth velocity first. This section +records exactly which other solutions are excluded by the energy argument. +Pressure assumptions are kept separate: continuity in a pressure norm, a +norm at each individual time, and an integrable pressure gradient are +different conditions. + +Write $\mathcal F_\nu[U]=U_t+(U\cdot\nabla)U-\nu\Delta U$. +All noncompact results below use $\mathbb R^3$; their velocities and forces +have one fixed compact spatial support unless the individual statement +says otherwise. This property concerns the constructed solution, not a +competing solution. On $[0,T]$, write $CH^k=C([0,T];H^k(\mathbb R^3))$ +and $C^1L^2=C^1([0,T];L^2(\mathbb R^3))$. Spatial constants in pressure +may depend on time. + +\begin{lemma}[The pressure term without a pressure norm]\label{lem:b-gradient} +If $w,g\in L^2(\mathbb R^3;\mathbb R^3)$ satisfy +$\operatorname{div}w=0$ and $\operatorname{curl}g=0$ in distributions, then +$\langle w,g\rangle_{L^2}=0$. In particular, any distributional gradient +$g=\nabla p\in L^2$ has zero pairing with $w$, without an assumption that +$p\in L^2$. +\end{lemma} +\begin{proof} +An $L^2$ field is a tempered distribution. The differential identities, +initially tested on compactly supported smooth functions, extend to +Schwartz tests by multiplying by cutoffs tending to one and using $L^2$ +convergence of the tests and their first derivatives. Fourier transformation +gives $\xi\cdot\widehat w=0$ and +$\xi_j\widehat g_k-\xi_k\widehat g_j=0$. These are identities of locally +integrable functions, so they hold almost everywhere. Away from $\xi=0$, +$\widehat g$ is parallel to $\xi$ and $\widehat w$ is perpendicular to it. +Their Hermitian product is zero. The exceptional point has measure zero; +Plancherel proves the assertion. +\end{proof} + +\begin{lemma}[Residual realization and separate comparison classes] +\label{lem:b-comparison}\label{lem:p2-realization} +Let $\nu>0$ and let $U$ be a prescribed smooth divergence-free velocity +on $[0,\infty)\times\mathbb R^3$, with $U(0)=0$. On each finite interval, +assume $U\in CH^2\cap C^1L^2$ and $U,\nabla U$ are bounded. Put +$f=\mathcal F_\nu[U]$. Then $(U,0)$ is a solution. Its velocity is unique +in each of the following separately stated comparison classes, with the +same force and initial data: +\begin{enumerate}[label=(\roman*),itemsep=2pt] +\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v$, and a +pressure representative $p\in CH^1$. +\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, +and an $H^1$ pressure representative at each time, with no required time +continuity in that pressure norm. +\item Strong solutions with $v\in CH^4\cap C^1H^2$ and distributional +$\nabla p\in CL^2$. +\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, +and a pressure representative $p\in CL^2$. +\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, +without an additional integrability or normalization condition on pressure. +\end{enumerate} +The pressure gradient is unique. An $L^2$ pressure representative, when +specified, is zero; otherwise pressure is determined up to a function of +time. The reference material flow is a smooth diffeomorphism at every +finite time and exists for every finite forward time. + +On a flat torus $\T_L^3$, the same conclusion holds in the classical class +where $v,v_t$, spatial derivatives through order two, and $p,\nabla p$ are +continuous on finite closed cylinders, with periodic mean-zero pressure. +No Sobolev conditions at infinity are involved in that assertion. +\end{lemma} +\begin{proof} +Substitution proves existence of the prescribed solution. Set $w=v-U$. +The difference equation is +\[ + w_t+(v\cdot\nabla)w+(w\cdot\nabla)U + =\nu\Delta w-\nabla p,\qquad \operatorname{div}w=0. +\] +The stated velocity regularity implies $w\in C^1L^2\cap CH^2$, so +$\frac d{dt}\|w\|_2^2=2\langle w,w_t\rangle$ at each time. In case +(iii), the stronger assumptions imply these inclusions. They also imply +bounded velocity and spatial derivatives through order two: for $j\le2$, +Cauchy--Schwarz in Fourier space uses the finite integral +$\int_{\mathbb R^3}|\xi|^{2j}(1+|\xi|^2)^{-4}\,d\xi$. + +For the transport term, insert a cutoff $\chi_R$ that equals one on the +ball of radius $R$, is supported in the ball of radius $2R$, and satisfies +$|\nabla\chi_R|\le C/R$. Its boundary error is at most +$CR^{-1}\|v\|_\infty\|w\|_2^2$, which tends to zero. Diffusion is +integrated by parts in $H^2$; alternatively its cutoff error is bounded by +$CR^{-1}\|w\|_2\|\nabla w\|_2$. The deformation term is bounded by +$\|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2$. + +We justify the pressure pairing according to the stated alternatives. +For (i) and (ii), at each fixed time approximate the chosen $H^1$ pressure +by compactly supported smooth functions; distributional incompressibility +and $w\in L^2$ give $\langle\nabla p,w\rangle=0$. Thus no continuity +of $p$ in time is used in (ii). For (iv), integration by parts on the +cutoff gives a pressure boundary term bounded by +$CR^{-1}\|p\|_2\|w\|_2$, also tending to zero. For (iii), apply +Lemma~\ref{lem:b-gradient} directly. Finally, in (v) the equation itself +gives at each time +\[ + \nabla p=f-v_t-(v\cdot\nabla)v+\nu\Delta v\in L^2, + \qquad \|(v\cdot\nabla)v\|_2\le\|v\|_\infty\|\nabla v\|_2. +\] +Here $f\in L^2$ follows from the identical estimate for $U$ and its stated +time and space regularity. The smooth pressure gradient is curl free, so +Lemma~\ref{lem:b-gradient} again cancels the term. This does not choose an +$L^2$ representative of the pressure. + +In every case the resulting pointwise energy inequality is +\[ + \frac12\frac d{dt}\|w\|_2^2+\nu\|\nabla w\|_2^2 + \le \|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2. +\] +The coefficient is bounded on each finite interval and $w(0)=0$, so an +integrating factor gives $w=0$. The pressure has already disappeared before +time integration; the proof imposes no unlisted temporal pressure norm. +Substitution then gives $\nabla p=0$. A spatially constant $L^2$ function +on $\mathbb R^3$ is zero, which proves the normalization assertion. + +On the torus every integration by parts is periodic and has no boundary +term. The specified classical regularity justifies norm differentiation, +diffusion, and pressure cancellation. The same energy inequality gives +uniqueness and the mean-zero normalization fixes pressure. In either domain, +the smooth reference velocity and its bounded spatial derivative give +unique material trajectories; bounded speed prevents finite-time escape. +Solving backwards on a finite interval gives the inverse flow, and smooth +ODE dependence gives its smoothness. +\end{proof} + +Cases (i)--(v) identify the precise hypotheses used by the applications; +their presence in a common lemma does not replace a theorem's declared +comparison class by a different one. In particular, a pressure norm is not +silently imposed in a gradient-only application. The proof is the classical +energy comparison for prescribed smooth solutions \cite[Section~18]{Leray1934}, +with its noncompact pressure pairings made explicit. + +For the flow $\Phi_t$, differentiation of its trajectory equation gives +\[ + U(t)=(\partial_t\Phi_t)\circ\Phi_t^{-1},\qquad + \partial_{tt}\Phi_t(a)= + (f-\nabla p+\nu\Delta U)(t,\Phi_t(a)). +\] +These identities describe the material form of the same constructed solution; +they do not add an independent existence claim. + +\begin{lemma}[Physical scaling]\label{lem:p2-chart} +For $a,b>0$, $x=x_0+ay$, and $s=bt$, let +$U(t,x)=abV(bt,(x-x_0)/a)$. Then trajectories are transformed by this change +of variables, incompressibility is preserved, and at the prescribed physical +viscosity $\nu$, +\[ + \mathcal F_\nu[U](t,x)=ab^2\left[ + V_s+(V\cdot\nabla_y)V-\frac{\nu}{a^2b}\Delta_yV + \right](bt,(x-x_0)/a). +\] +\end{lemma} +\begin{proof} +The trajectory equation gives $\dot x=abV$; differentiating in space gives +$\operatorname{div}_xU=b\operatorname{div}_yV$. The three residual terms +scale by $ab^2,ab^2$, and $b/a$, respectively, proving the formula. +\end{proof} +Thus a side-$10$ torus is kept as a physical side-$10$ torus. If a chart +is rescaled, the residual is recomputed with the fixed $\nu$; the equation +is not identified with a differently normalized viscosity. + +\begin{proposition}[Projection on a flat torus]\label{prop:projection-l2} +Let $g$ be a smooth force on $[0,\infty)\times\T_L^3$ with zero +spatial mean at each time. There is a unique smooth mean-zero potential +$\phi$ satisfying $\Delta\phi=\operatorname{div}g$. Put +$Qg=\nabla\phi$ and $Pg=g-Qg$. Then $Pg$ is solenoidal and mean zero, +and replacing $(g,p)$ by $(Pg,p-\phi)$ preserves the velocity and the +pressure normalization. + +Use normalized spatial measure, frequencies $\xi_k=2\pi k/L$, and norms +\[ + \|G\|_{H^s}^2=\sum_{k\in\Z^3}(1+|\xi_k|^2)^s|\widehat G(k)|^2, + \qquad + \|G\|_{C^r}=\max_{|\beta|\le r}\|\partial_x^\beta G\|_\infty. +\] +For all integers $h,r\ge0$, at each time, +\begin{equation}\label{eq:projection-estimate} + \begin{aligned} + \max\{\|\partial_t^h Pg\|_{C^r},\|\partial_t^h Qg\|_{C^r}\} + &\le S_L\|\partial_t^h g\|_{H^{r+2}},\\ + \|\partial_t^h\phi\|_{C^{r+1}} + &\le S_Ld_L\|\partial_t^h g\|_{H^{r+2}}, + \end{aligned} +\end{equation} +where $S_L=(1+52(L/(2\pi))^4)^{1/2}$ and +$d_L=(1+(L/(2\pi))^2)^{1/2}$. + +Uniform bounds, temporal $L^2$ bounds on spatial suprema, and qualitative +uniform decay of all mixed derivatives pass from $g$ to $Pg,Qg,\phi$. +A quantitative envelope for time order $h$ and output spatial order $r$ +passes to $Pg,Qg$ through order $r$ and to $\phi$ through order $r+1$ +when the input derivatives through spatial order $r+2$ satisfy that +envelope. The constants may depend on the derivative order. +Periodicity and stationarity, including either property on a time tail, +are preserved on their respective intervals. + +For computable $L>0$, effective mixed-derivative evaluation and effective +derivative bounds for $g$ on finite time slabs give effective evaluation +and such bounds for $Pg,Qg,\phi$. Corresponding supplied global input +bounds give effective global bounds in each assertion above. +\end{proposition} +\begin{proof} +For $k\ne0$ define +\[ + \widehat\phi(t,k)=-\frac{i\xi_k\cdot\widehat g(t,k)}{|\xi_k|^2}, + \qquad \widehat\phi(t,0)=0. +\] +The nonzero-frequency symbols of $Q$ and $P$ are complementary orthogonal +projections. Their $H^s$ operator norms are at most one, while the +multiplier for $\phi$ has norm at most $d_L$ from $H^s$ to $H^{s+1}$. +For $|\beta|\le r$, Cauchy--Schwarz bounds the absolute sum of the +Fourier series for $\partial_x^\beta G$ by +\[ + \left(\sum_{k\in\Z^3}(1+|\xi_k|^2)^{-2}\right)^{1/2} + \|G\|_{H^{r+2}}. +\] +The shell $|k|_\infty=n\ge1$ contains $24n^2+2\le26n^2$ points. +Since $\sum_{n\ge1}n^{-2}\le2$, the factor is at most $S_L$. +For the tail $|k|_\infty>N\ge1$, it is at most +\begin{equation}\label{eq:projection-tail} + \left(26(L/(2\pi))^4/N\right)^{1/2}. +\end{equation} +Applying these estimates to the multipliers proves +\eqref{eq:projection-estimate}, including the estimate for $\phi$ with +one additional spatial derivative. On compact time slabs these tails +converge uniformly for every mixed derivative. Thus the series define +smooth fields and may be differentiated term by term. The Fourier +identities give the Poisson equation, uniqueness of its mean-zero +solution, and $\operatorname{div}Pg=0$. Also +$-\nabla(p-\phi)+Pg=-\nabla p+g$. + +For every integer $n\ge0$, the multinomial expansion and Parseval give +\begin{equation}\label{eq:projection-derivative-bounds} + \|G\|_{H^n}^2 + \le\sum_{|\beta|\le n} + \frac{n!}{(n-|\beta|)!\,\beta!} + \|\partial_x^\beta G\|_\infty^2. +\end{equation} +Use $n=r+2$ and $G=\partial_t^h g$. This finite sum proves the uniform +bounds and qualitative decay assertions. If its input suprema are +bounded by $C_{h,\beta}a_h(t)$, it gives the same envelope $a_h$ for +the output, with the constants specified by +\eqref{eq:projection-estimate}--\eqref{eq:projection-derivative-bounds}. +Integrating the squared estimates proves the temporal $L^2$ assertion +from the corresponding input norms. The operators act at each time, +so they preserve the stated temporal symmetries. + +For effective evaluation, the supplied derivative bounds and +\eqref{eq:projection-tail} give a computable Fourier truncation on each +finite time slab. The finitely many coefficient integrals are computed +by Riemann sums. After scaling the torus to $[0,1]^3$, the integrand +for $\widehat G(k)$ has Lipschitz bound +\[ + L\sqrt3\max_j\|\partial_{x_j}G\|_\infty + +2\pi|k|\|G\|_\infty. +\] +This gives an effective quadrature error and hence evaluation of every +mixed derivative of all three fields. The displayed finite sums also +compute output bounds from the supplied input bounds. +\end{proof} +The estimates apply to the pressure shift $\phi$; in the constructions +with original pressure zero, the projected pressure is $-\phi$. +Projection generally changes pressure. It supplies no compact-support +conclusion on Euclidean space and is distinct from the direct zero-pressure +solenoidal shear construction. + +\paragraph{Effective flat profiles.} +For later cutoff formulas put +\begin{equation}\label{eq:p2-step} + \rho(s)=\begin{cases}e^{-1/s},&s>0,\\0,&s\le0,\end{cases} + \qquad \sigma(s)=\frac{\rho(s)}{\rho(s)+\rho(1-s)}. +\end{equation} +Positive-side derivatives of $\rho$ are polynomials in $1/s$ times +$e^{-1/s}$ and extend flatly at zero. The estimate +$v^me^{-v}\le(m+k)!v^{-k}$ gives effective error bounds near the seam, +and $\rho(s)+\rho(1-s)\ge e^{-2}$. Products, translations, rescalings, +and derivatives therefore yield effective cutoffs with all requested +derivative bounds. Periodization uses zero collars; at approximate real +arguments a finite superset of possible translates is evaluated. No real +equality test at a seam or execution of the encoded computation is used. diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/balanced.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/balanced.tex index b31ddf5c0d690a53b3b2f778a0f03c78c7e3d7b2..a7642b6f80d3a96f3b38bbbf1023e152b3d96f8f 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/balanced.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/balanced.tex @@ -1,166 +1,166 @@ -\section{A complete balanced three-stack realization}\label{sec:bb-balanced-three-stack} - -In a positional code with base $B$, replacing a prefix of length $r$ -by one of length $s$ gives an affine map with slope $B^{r-s}$. -Thus a three-stack instruction has determinant one when it preserves -the sum of the three prefix lengths. A third stack supplies reversible -history under this constraint: we first move a fresh blank from beyond -a work-tape boundary onto the history stack, then replace that blank -by the record. - -\begin{theorem}[Balanced three-stack realization] -\label{thm:bb-balanced-three-stack} -For every machine and finite input, there are effective coefficients -$f_0,f_1$ such that $f=f_0+\nu f_1$ on $\R^3$ has an explicit -zero-data solution $(U,0)$. Force and velocity have one compact -spatial support and bounded mixed derivatives of every order, and -are one-periodic after $t=1$. At the fixed label $x_*=(4,0,0)$, -\[ - \exists t\ge0:\quad X(t;x_*)\in(-1,2)^3 - \quad\Longleftrightarrow\quad\text{the machine halts}. -\] -For a fixed machine its repeated field is independent of the input. -The comparison class is $u\in C H^2\cap C^1L^2$, bounded $u,\nabla u$ -on finite intervals, and pressure modulo constants in $C H^1$. -The formula is valid for every fixed real $\nu>0$, with effective -evaluation when $\nu$ is computable and relative evaluation otherwise. -\end{theorem} -\begin{proof} -Normalize the finite machine to one halting state $h$, directing missing -nonhalting instructions to it by unchanged-letter stay moves. This preserves -initial halting as well. Let $\Gamma$ be the work alphabet and -$b\in\Gamma$ its blank. Use three infinite top-first stacks $L,R,J$. A rule replaces three -finite prefixes, leaving their tails untouched. Introduce boundary -$\star$, marked work letters $\bar a$, and controls $A_q,S_q,T_q,D_q$; -only $A_h$ is terminal. For each $q\ne h$, the preparatory rules are -\begin{align} - (A_q;\varepsilon,a,\varepsilon)&\to(S_q;\bar a,\varepsilon,\varepsilon), - \label{eq:bb-balanced-entry}\\ - (S_q;\varepsilon,c,\varepsilon)&\to(S_q;c,\varepsilon,\varepsilon), - \label{eq:bb-balanced-out-sweep}\\ - (S_q;\varepsilon,\star b,\varepsilon)&\to(T_q;\varepsilon,\star,b), - \label{eq:bb-balanced-blank}\\ - (T_q;c,\varepsilon,\varepsilon)&\to(T_q;\varepsilon,c,\varepsilon), - \label{eq:bb-balanced-return-sweep}\\ - (T_q;\bar a,\varepsilon,\varepsilon)&\to(D_q;\varepsilon,a,\varepsilon). - \label{eq:bb-balanced-restore} -\end{align} -Here $a,c\in\Gamma$. For $\delta(q,a)=(q',d,\sigma)$ add -\begin{equation}\label{eq:bb-balanced-step-table} -\begin{array}{c|c|c} -\sigma&\text{source}&\text{target}\\\hline -0&(D_q;\varepsilon,a,b)&(A_{q'};\varepsilon,d,\kappa)\\ --1&(D_q;c,a,b)&(A_{q'};\varepsilon,cd,\kappa)\quad(c\in\Gamma)\\ -1&(D_q;\varepsilon,ac,b)&(A_{q'};d,c,\kappa)\quad(c\in\Gamma)\\ -1&(D_q;\varepsilon,a\star b,b)&(A_{q'};d,b\star,\kappa). -\end{array} -\end{equation} -Every individual row occurrence, including its choice of $c$, receives -a distinct record symbol $\kappa$. - -\emph{Full-domain separation and initialized behavior.} -\label{lem:bb-balanced-processor}\label{proof:bb-balanced-processor} -Every rule preserves total prefix length: it is one in the transfer -rows, two in the blank-supply row, and respectively two, three, three, -four in the final rows. Source branches at $A_q$ test the right top; -at $S_q$ the right top separates work letters from the boundary; -at $T_q$ the left top separates work and marked letters. At $D_q$ -the right top fixes the work instruction, with the next right symbol -or left top separating its remaining branches. Target branches at -$S_q$ have distinct marked or ordinary left tops; at $T_q$ distinct -boundary or ordinary right tops; at $D_q$ distinct restored $a$; -and at an $A$ control distinct history tops $\kappa$. -This proves separation for arbitrary independent tails. - -Initialize -\[ - (A_{q_0};b^\infty,v\star b^\infty,b^\infty),\qquad - v=\begin{cases}\omega,&\omega\ne\varepsilon,\\b,&\omega=\varepsilon. - \end{cases} -\] -At a checkpoint, $L$ lists cells strictly left of the head and -$R=a_1\cdots a_m\star b^\infty$, $m\ge1$, lists the current cell -and a finite right block; beyond it the tape is blank. $J$ contains -completed records, newest first. The first rule marks $a_1$ while -moving it left; $m-1$ forward transfers expose $\star$. The blank -row removes one filler blank beyond that boundary and pushes it on -$J$, without changing the infinite blank filler. The reverse transfers -and restoration of $a_1$ restore exactly the original two work stacks. -The final rule replaces the fresh history blank by its record and -performs the work instruction. A left move pops the next cell from -$L$; a right move with $m\ge2$ pops the current cell from $R$; -when $m=1$ the last row exposes a blank and retains a nonempty right -block. Thus the invariant holds and one work step costs -$1+(m-1)+1+(m-1)+1+1=2m+2$ rules. An initially halted input needs -none. The records recover the head displacements as well. - -\emph{From balanced words to solid boxes.} -Give the enlarged alphabet $\Sigma$ the odd digits -$1,3,\ldots,2|\Sigma|-1$ in base $B=2|\Sigma|+1$. -For a finite word $v$ put $e(v)=\sum_{j=1}^{|v|}d(v_j)B^{-j}$; -for an infinite stack $\xi$ put -$e(\xi)=\sum_{j\ge1}d(\xi_j)B^{-j}$. -The prefix interval is $I(v)=e(v)+[0,B^{-|v|}]$, with -$I(\varepsilon)=[0,1]$. Every infinite tail satisfies -\[ - \frac1{B-1}\le e(\xi)\le\frac{2|\Sigma|-1}{B-1}<1, -\] -so its code lies strictly inside each applicable prefix interval. -Distinct incompatible prefixes have positive gaps, since digits differ -by at least two while the remaining interval has width at most one unit -in the differing place. Place control $s$ at -$o_s=(4i_s,0,0)$, with $i_{A_h}=0$ and distinct positive integers -for the others. The code of $(s;L,R,J)$ is -$o_s+(e(L),e(R),e(J))$. -A rule with input prefixes $\alpha_1,\alpha_2,\alpha_3$ -and output prefixes $\alpha'_1,\alpha'_2,\alpha'_3$ gives boxes -\[ - Q_i=o_s+\prod_jI(\alpha_j),\qquad - Q_i'=o_{s'}+\prod_jI(\alpha'_j). -\] -Both families are separately disjoint by the preceding full-prefix -separation. In control-relative coordinates their exact maps are -\begin{equation}\label{eq:bb-balanced-affine-rule} - y_j=e(\alpha'_j)+B^{|\alpha_j|-|\alpha'_j|} - (x_j-e(\alpha_j)). -\end{equation} -The identity $e(\alpha\xi)=e(\alpha)+B^{-|\alpha|}e(\xi)$ -proves their action on every code. Initialized stacks are eventually -blank, with rational codes -$e(vb^\infty)=e(v)+B^{-|v|}d(b)/(B-1)$. -The affine determinant equals -$B^{\sum|\alpha_j|-\sum|\alpha'_j|}=1$, exactly by balance. -Apply Theorem~\ref{lem:bb-balanced-box-routing} to these full boxes. - -\emph{Every intermediate time.} -All half-widths are at most $1/2$, so that lemma uses $R=1$. -Sources have first coordinate at least four, since no rule leaves -$A_h$. Targets with nonhalt control have the same property. -For such a branch the initial expansion and final contraction preserve -first coordinate at least four. During deformation its center has -first coordinate at least $4\Lambda$. Every scaled halt target -center has first coordinate at most $\Lambda$, whereas the endpoints -of each nonhalt parking path have first coordinate at least -$4\Lambda$. Since $\Lambda+4<4\Lambda$, the original segment -cannot meet an obstacle cube around a halt target. Any encountered -obstacle has first coordinate at least $4\Lambda$, so its boundary -detour remains at least $4\Lambda-4$. Subtracting a half-width at -most one still leaves first coordinate greater than two. Hence -\begin{equation}\label{eq:bb-balanced-no-premature-event} - \Phi_\tau(Q_i)\subset\{x_1>2\}\quad(0\le\tau\le1) - \quad\text{for every branch with nonhalt target}. -\end{equation} -This includes delivery after halt targets have already been filled. - -Finally load $x_*=(4,0,0)$ along the straight segment to the rational -initial code using the translation part of Lemma~\ref{lem:bb-curl}. -It avoids $(-1,2)^3$ for a nonhalt initial control. Repeat the box -field from time one. Induction at integer times applies the exact -full-box maps; the finite compiler cycles reach a code in $(0,1)^3$ -precisely upon halting. Equation~\eqref{eq:bb-balanced-no-premature-event} -excludes every intermediate false visit. The residual proposition -gives the force and uniqueness; compact support gives all finite-slab -Sobolev conditions and uniformly bounded kinetic energy. Every choice -was finite and effective, including the obstacle detours. The input -enters only the first loading path, completing the proof. -\end{proof} +\section{A complete balanced three-stack realization}\label{sec:bb-balanced-three-stack} + +In a positional code with base $B$, replacing a prefix of length $r$ +by one of length $s$ gives an affine map with slope $B^{r-s}$. +Thus a three-stack instruction has determinant one when it preserves +the sum of the three prefix lengths. A third stack supplies reversible +history under this constraint: we first move a fresh blank from beyond +a work-tape boundary onto the history stack, then replace that blank +by the record. + +\begin{theorem}[Balanced three-stack realization] +\label{thm:bb-balanced-three-stack} +For every machine and finite input, there are effective coefficients +$f_0,f_1$ such that $f=f_0+\nu f_1$ on $\R^3$ has an explicit +zero-data solution $(U,0)$. Force and velocity have one compact +spatial support and bounded mixed derivatives of every order, and +are one-periodic after $t=1$. At the fixed label $x_*=(4,0,0)$, +\[ + \exists t\ge0:\quad X(t;x_*)\in(-1,2)^3 + \quad\Longleftrightarrow\quad\text{the machine halts}. +\] +For a fixed machine its repeated field is independent of the input. +The comparison class is $u\in C H^2\cap C^1L^2$, bounded $u,\nabla u$ +on finite intervals, and pressure modulo constants in $C H^1$. +The formula is valid for every fixed real $\nu>0$, with effective +evaluation when $\nu$ is computable and relative evaluation otherwise. +\end{theorem} +\begin{proof} +Normalize the finite machine to one halting state $h$, directing missing +nonhalting instructions to it by unchanged-letter stay moves. This preserves +initial halting as well. Let $\Gamma$ be the work alphabet and +$b\in\Gamma$ its blank. Use three infinite top-first stacks $L,R,J$. A rule replaces three +finite prefixes, leaving their tails untouched. Introduce boundary +$\star$, marked work letters $\bar a$, and controls $A_q,S_q,T_q,D_q$; +only $A_h$ is terminal. For each $q\ne h$, the preparatory rules are +\begin{align} + (A_q;\varepsilon,a,\varepsilon)&\to(S_q;\bar a,\varepsilon,\varepsilon), + \label{eq:bb-balanced-entry}\\ + (S_q;\varepsilon,c,\varepsilon)&\to(S_q;c,\varepsilon,\varepsilon), + \label{eq:bb-balanced-out-sweep}\\ + (S_q;\varepsilon,\star b,\varepsilon)&\to(T_q;\varepsilon,\star,b), + \label{eq:bb-balanced-blank}\\ + (T_q;c,\varepsilon,\varepsilon)&\to(T_q;\varepsilon,c,\varepsilon), + \label{eq:bb-balanced-return-sweep}\\ + (T_q;\bar a,\varepsilon,\varepsilon)&\to(D_q;\varepsilon,a,\varepsilon). + \label{eq:bb-balanced-restore} +\end{align} +Here $a,c\in\Gamma$. For $\delta(q,a)=(q',d,\sigma)$ add +\begin{equation}\label{eq:bb-balanced-step-table} +\begin{array}{c|c|c} +\sigma&\text{source}&\text{target}\\\hline +0&(D_q;\varepsilon,a,b)&(A_{q'};\varepsilon,d,\kappa)\\ +-1&(D_q;c,a,b)&(A_{q'};\varepsilon,cd,\kappa)\quad(c\in\Gamma)\\ +1&(D_q;\varepsilon,ac,b)&(A_{q'};d,c,\kappa)\quad(c\in\Gamma)\\ +1&(D_q;\varepsilon,a\star b,b)&(A_{q'};d,b\star,\kappa). +\end{array} +\end{equation} +Every individual row occurrence, including its choice of $c$, receives +a distinct record symbol $\kappa$. + +\emph{Full-domain separation and initialized behavior.} +\label{lem:bb-balanced-processor}\label{proof:bb-balanced-processor} +Every rule preserves total prefix length: it is one in the transfer +rows, two in the blank-supply row, and respectively two, three, three, +four in the final rows. Source branches at $A_q$ test the right top; +at $S_q$ the right top separates work letters from the boundary; +at $T_q$ the left top separates work and marked letters. At $D_q$ +the right top fixes the work instruction, with the next right symbol +or left top separating its remaining branches. Target branches at +$S_q$ have distinct marked or ordinary left tops; at $T_q$ distinct +boundary or ordinary right tops; at $D_q$ distinct restored $a$; +and at an $A$ control distinct history tops $\kappa$. +This proves separation for arbitrary independent tails. + +Initialize +\[ + (A_{q_0};b^\infty,v\star b^\infty,b^\infty),\qquad + v=\begin{cases}\omega,&\omega\ne\varepsilon,\\b,&\omega=\varepsilon. + \end{cases} +\] +At a checkpoint, $L$ lists cells strictly left of the head and +$R=a_1\cdots a_m\star b^\infty$, $m\ge1$, lists the current cell +and a finite right block; beyond it the tape is blank. $J$ contains +completed records, newest first. The first rule marks $a_1$ while +moving it left; $m-1$ forward transfers expose $\star$. The blank +row removes one filler blank beyond that boundary and pushes it on +$J$, without changing the infinite blank filler. The reverse transfers +and restoration of $a_1$ restore exactly the original two work stacks. +The final rule replaces the fresh history blank by its record and +performs the work instruction. A left move pops the next cell from +$L$; a right move with $m\ge2$ pops the current cell from $R$; +when $m=1$ the last row exposes a blank and retains a nonempty right +block. Thus the invariant holds and one work step costs +$1+(m-1)+1+(m-1)+1+1=2m+2$ rules. An initially halted input needs +none. The records recover the head displacements as well. + +\emph{From balanced words to solid boxes.} +Give the enlarged alphabet $\Sigma$ the odd digits +$1,3,\ldots,2|\Sigma|-1$ in base $B=2|\Sigma|+1$. +For a finite word $v$ put $e(v)=\sum_{j=1}^{|v|}d(v_j)B^{-j}$; +for an infinite stack $\xi$ put +$e(\xi)=\sum_{j\ge1}d(\xi_j)B^{-j}$. +The prefix interval is $I(v)=e(v)+[0,B^{-|v|}]$, with +$I(\varepsilon)=[0,1]$. Every infinite tail satisfies +\[ + \frac1{B-1}\le e(\xi)\le\frac{2|\Sigma|-1}{B-1}<1, +\] +so its code lies strictly inside each applicable prefix interval. +Distinct incompatible prefixes have positive gaps, since digits differ +by at least two while the remaining interval has width at most one unit +in the differing place. Place control $s$ at +$o_s=(4i_s,0,0)$, with $i_{A_h}=0$ and distinct positive integers +for the others. The code of $(s;L,R,J)$ is +$o_s+(e(L),e(R),e(J))$. +A rule with input prefixes $\alpha_1,\alpha_2,\alpha_3$ +and output prefixes $\alpha'_1,\alpha'_2,\alpha'_3$ gives boxes +\[ + Q_i=o_s+\prod_jI(\alpha_j),\qquad + Q_i'=o_{s'}+\prod_jI(\alpha'_j). +\] +Both families are separately disjoint by the preceding full-prefix +separation. In control-relative coordinates their exact maps are +\begin{equation}\label{eq:bb-balanced-affine-rule} + y_j=e(\alpha'_j)+B^{|\alpha_j|-|\alpha'_j|} + (x_j-e(\alpha_j)). +\end{equation} +The identity $e(\alpha\xi)=e(\alpha)+B^{-|\alpha|}e(\xi)$ +proves their action on every code. Initialized stacks are eventually +blank, with rational codes +$e(vb^\infty)=e(v)+B^{-|v|}d(b)/(B-1)$. +The affine determinant equals +$B^{\sum|\alpha_j|-\sum|\alpha'_j|}=1$, exactly by balance. +Apply Theorem~\ref{lem:bb-balanced-box-routing} to these full boxes. + +\emph{Every intermediate time.} +All half-widths are at most $1/2$, so that lemma uses $R=1$. +Sources have first coordinate at least four, since no rule leaves +$A_h$. Targets with nonhalt control have the same property. +For such a branch the initial expansion and final contraction preserve +first coordinate at least four. During deformation its center has +first coordinate at least $4\Lambda$. Every scaled halt target +center has first coordinate at most $\Lambda$, whereas the endpoints +of each nonhalt parking path have first coordinate at least +$4\Lambda$. Since $\Lambda+4<4\Lambda$, the original segment +cannot meet an obstacle cube around a halt target. Any encountered +obstacle has first coordinate at least $4\Lambda$, so its boundary +detour remains at least $4\Lambda-4$. Subtracting a half-width at +most one still leaves first coordinate greater than two. Hence +\begin{equation}\label{eq:bb-balanced-no-premature-event} + \Phi_\tau(Q_i)\subset\{x_1>2\}\quad(0\le\tau\le1) + \quad\text{for every branch with nonhalt target}. +\end{equation} +This includes delivery after halt targets have already been filled. + +Finally load $x_*=(4,0,0)$ along the straight segment to the rational +initial code using the translation part of Lemma~\ref{lem:bb-curl}. +It avoids $(-1,2)^3$ for a nonhalt initial control. Repeat the box +field from time one. Induction at integer times applies the exact +full-box maps; the finite compiler cycles reach a code in $(0,1)^3$ +precisely upon halting. Equation~\eqref{eq:bb-balanced-no-premature-event} +excludes every intermediate false visit. The residual proposition +gives the force and uniqueness; compact support gives all finite-slab +Sobolev conditions and uniformly bounded kinetic energy. Every choice +was finite and effective, including the obstacle detours. The input +enters only the first loading path, completing the proof. +\end{proof} diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/clocks.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/clocks.tex index 3e97258eabfa6007f448d8a895984258457bfcdd..1aa909f39fc3b9417de71af92a75e89262535518 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/clocks.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/clocks.tex @@ -1,140 +1,140 @@ -\section{Clock profiles for selected processors}\label{sec:clocks} - -Lemma~\ref{ba:log-clock} gives the common logarithmic change of time. -We now apply it to two Euclidean processors with different pressure -classes, and then use a companion's autonomous spatial clock to obtain -stationary or decaying forcing on the torus. In every case the clock -traverses every finite computational time; periodic, stationary and -decaying profiles remain separate choices. - -\subsection{Preserving the declared comparison classes} -\label{sec:bb-late-logclock} - -The two templates below use different whole-space pressure assumptions. -Slowing their trajectories preserves those declared comparison classes -because the new reference velocity retains the required regularity on -every finite physical-time interval. - -\begin{proposition}\label{prop:bb-late-logclock} -The template of Theorem~\ref{thm:bb-direction-shuttle} admits a force -in $L^2([0,\infty)\times\R^3)$ with the same material label and -event. The template of Theorem~\ref{thm:bb-guarded-history} admits -a force in $L^2([0,\infty);H^j(\R^3))$ for every integer $j\ge0$. -Both retain uniform compact support, zero initial velocity, bounded -mixed derivatives, their stated pressure and velocity comparison -classes, and finite effective prescriptions. -\end{proposition} -\begin{proof}\label{proof:bb-late-logclock} -Apply Lemma~\ref{ba:log-clock} to each template, including its loading -interval if present. Both are initially zero, have one compact support, -and have uniformly bounded mixed derivatives. The lemma gives the exact -residual and all mixed bounds; fixed support also gives -$\|f(t)\|_{H^j}\le C_j(1+t)^{-1}$ for every fixed $j$. - -On each finite interval the slowed reference field retains every Sobolev -regularity and bounded derivative required by its original comparison -class. Case (i) of Lemma~\ref{lem:b-comparison} therefore applies to the -direction shuttle's continuous $H^1$ pressure representative, and case -(iii) applies to the guarded-history processor's continuous $L^2$ -pressure gradient. Both constructed pressures are zero. - -The new path is $X(\log(1+t);a_0)$. The clock lemma preserves the event -and the effective prescription. In particular, loaded samples $1+k$ -occur at physical times $e^{1+k}-1$, whereas the aligned construction's -samples $k$ occur at $e^k-1$. These times have no finite accumulation. -The new forces have the stated temporal integrability; no periodicity -in physical time is asserted for them. -\end{proof} - -\subsection{The input-offset alternative}\label{bc:input-offset-section} -A fixed material label can also be retained by shifting the state charts -by the initial tape code. The complete planar routing and spatial-clock -construction for this choice belongs to -\cite[Theorem~4.1 and Section~5]{OpenAI379StationaryA2}. -The following application fixes its match with the present conventions. - -\begin{corollary}[Two profiles of the input-offset processor] -\label{bc:input-offset-event} -On $\T^3$, the fixed label $P=(1/4,1/4,0)$ and strip -$O=\{1/32h_n$. Mark the old head, scan right to $g$, -write the record and install the new frontier at $g+1$, then return to -the mark. All intervening history is nonfrontier, the new frontier -cell is empty, and there is only one mark. Hence the same rows apply -when $r_0=2$. There are $g-h_n-1$ continuing right moves, -$g-h_n$ continuing left moves, and four other instructions, giving -$2(g-h_n)+3$. The last row clears the mark and makes the work -movement; the invariant is restored with frontier $g+1$. This proves -finite completion, indefinite legality for a nonhalting run, and the -stated halt equivalence, including initial halting. -\end{proof} - -\subsection{From symbolic cylinders to closed rectangles} - -Radix encodings turn a push or pop of a tape letter into an affine map; -this is the generalized-shift viewpoint of -Moore~\cite{Moore1990,Moore1991}. The following calculation adds the -separation of full closed rectangles needed for smooth localization. - -\begin{lemma}[Separated rectangle codes]\label{ba:rectangles} -Let a deterministic partial tape table on an alphabet $\Gamma$ of size -$m$ have the two incoming properties of Lemma~\ref{lem:bb-onecell}. -Assign distinct odd digits $e(a)\in\{1,3,\ldots,2m-1\}$ and choose -an integer $B\ge2m+1$. Its local rules admit effective positive diagonal -affine maps of determinant one between finite families of closed planar -rectangles. Sources are pairwise separated, as are targets. Both the -minimal subdivision and the subdivision by both adjacent letters given -below realize every tape in their full symbolic cylinders. -\end{lemma} -\begin{proof}\label{ba:rectangles-proof} -Use Lemma~\ref{lem:bb-onecell} with digits $d(a)=e(a)$. Its digit -hypotheses hold for every $B\ge2m+1$. In the notation -\[ - C(a_1a_2\cdots)=\sum_{j\ge1}e(a_j)B^{-j},\quad - h_a(v)=\frac{e(a)+v}{B},\quad - I_{a_1\cdots a_l}=h_{a_1}\cdots h_{a_l}([0,1]), -\] -the left stream starts immediately left of the head and the right stream -starts at the head. Place state $s$ in $k_s+a[0,1]^2$ with a common -rational $a>0$. The minimal branch maps are exactly -\eqref{eq:bb-one-cell-branches}; scaling both charts by $a$ leaves -$\operatorname{diag}(B^{-d},B^d)$ unchanged. - -For the additional full subdivision, restrict the right- and stay-move -sources to $x\in I_\gamma$, for every full letter $\gamma$. -Their targets become $I_{\beta\gamma}\times[0,1]$ and -$I_\gamma\times I_\beta$, respectively, for written letter $\beta$. -The already split left-move targets remain -$[0,1]\times I_{\gamma\beta}$. At a fixed target state the -incoming direction is fixed, and the pair $(\beta,\gamma)$ -distinguishes the branches in every case. Thus the added subdivision -preserves separate target separation, while source separation follows -by restriction of the shared source rectangles. All these statements -concern the filled closed rectangles and arbitrary symbolic tails. - -For completeness the quantitative gaps remain valid for the entire -range $B\ge2m+1$: distinct first-digit intervals have gap at least -$B^{-1}$ and distinct second-digit intervals at least $B^{-2}$. -The source and target families therefore have within-state gaps at -least $aB^{-2}$; different state squares have their own chosen gaps. -Tail values lie between $1/(B-1)$ and $(2m-1)/(B-1)<1$. -An initial constant tail after $l$ digits contributes -$B^{-l}e(a_{\rm tail})/(B-1)$, so initialized codes are rational. -At checkpoints the history records, or the known absolute origin of -the record block, also recover the absolute work-head position. -\end{proof} - - -\subsection{Conventions for the remaining geometric realizations}\label{bc:section} -\label{bc:frontier-section} -Unless a different table is specified, subsequent routes use the move-first recorder of -Lemma~\ref{lem:bb-movefirst}, with initial frontier two. The controls -$(C_q,J_r,S_r,P_q,T_q)$ there are written $(W_q,P_r,B_r,C_q,D_q)$ -here, and its empty/frontier/record symbols $(e,F,r)$ are written -$(E,F,R_r)$. Permuting work, history and mark coordinates in every row -preserves every guard. This identifies the full partial table, its inverse -and the exact checkpoint count $2(2+n-h')+4$, including arbitrary allowed -tapes. A nonterminal branch means one with both endpoint controls -nonterminal; a final branch into a halt can enter the observer. - -The full branches are those of Lemma~\ref{lem:bb-onecell}. For an alphabet -of size $m$ we use one of the following explicitly indicated conventions: -\begin{equation}\label{bc:digits} -(B,\{d_a\})=(2m+2,\{1,3,\ldots,2m-1\}),\quad -(2m+1,\{1,3,\ldots,2m-1\}),\quad -(2m,\{0,2,\ldots,2m-2\}). -\end{equation} -All satisfy its digit bound and separation hypotheses. In the last -convention the full blank has digit zero; boundary codes remain in the -closed-rectangle and neighborhood conclusions. Write -$C(a_1a_2\cdots)=\sum_{j\ge1}d_{a_j}B^{-j}$ and -$I_v=\sum_{j=1}^{|v|}d_{v_j}B^{-j}+[0,B^{-|v|}]$. -A constant tail after $k$ positions contributes -$d_{\rm tail}B^{-k}/(B-1)$, so the initial codes are rational. -Guarded three-cell rules use Lemma~\ref{lem:bb-window-rectangles} with -$(a,b)=(1,2)$ after their own full-domain inverse has been proved. - -\subsection{Analytic conventions for the geometric variants}\label{bc:analytic-section} -The whole-space baseline in these variants means smooth -$u\in CH^2\cap C^1L^2$, bounded $u,\nabla u$ on finite intervals, -and a pressure representative in $CH^1$. When the competitor-gradient -condition is omitted explicitly, the other conditions remain unchanged. -Both cases fall under Lemma~\ref{lem:b-comparison}(i); separately stated -pressure classes are retained. The torus baseline is its classical torus -clause. Unless a projection is specified, the force is -\begin{equation}\label{bc:residual} -f_U=U_t+(U\cdot\nabla)U-\nu\Delta U,\qquad (u,p)=(U,0). -\end{equation} -Torus projection is the separate choice in -Proposition~\ref{prop:projection-l2}, with changed pressure. -Localized affine motion is supplied by Lemma~\ref{lem:bb-curl} on a -compact time interval, using effective derivative bounds, positive lower -scale bounds and the positive cutoff margins specified in each route. -The potential is \eqref{eq:bb-potential}; the resulting flow fixes every -region disjoint from its support. In particular exponential diagonal -scales and linear reciprocal scales have respectively generators -$\theta'\log D$ and $\diag(l'/l,-l'/l,0)$. - -\begin{lemma}[Finite forcing mechanisms]\label{bc:analytic} -A finite family of the preceding pulses, joined with flat time collars, -can be used once for initialization and repeated with period one thereafter. -It gives a smooth effective velocity and residual force with all mixed -derivatives uniformly bounded. In $\R^3$ both have one fixed compact -spatial support, and the velocity has uniformly bounded kinetic energy. -In a torus chart, -periodized curl or planar Hamiltonian pulses have zero spatial mean, as -does their residual force. Initial velocity and pressure are zero. -Uniqueness holds in the comparison classes of Section~\ref{sec:model}; -in the whole-space class the competitor's gradient bound can be omitted -if the remaining conditions are retained. -\end{lemma} -\begin{proof}\label{bc:analytic-proof} -Each pulse is a finite expression in rational data, effective smooth -switches, their derivatives, positive diagonal scales and logarithms. -Finite positive separation margins bound every cutoff derivative. A finite -maximum bounds a period and the loading interval, and repetitions preserve -that bound. To evaluate near a join, a coarse enclosure of time gives a -finite superset of possibly active pulses. Summing them avoids an exact -comparison of a computable time with an integer. The flat collars give -smoothness. Compact support gives every required spatial Sobolev bound. -Lemma~\ref{lem:p2-realization} gives the asserted solution and comparison; -on a torus the integral of each curl or Hamiltonian derivative vanishes, -and the convection term is a divergence. Its flow satisfies -$\det D\Phi_t=1$, because differentiating this determinant gives -$(\divg U)(t,\Phi_t)\det D\Phi_t=0$ and its initial value is one. -The velocity and acceleration identities in that lemma consequently apply -to these volume-preserving diffeomorphisms. -\end{proof} - -For later use, spatial chart changes are always made at the velocity level. -If $x=o+a y$, set $\widetilde U(t,x)=aU(t,(x-o)/a)$ and then compute -\eqref{bc:residual} at the prescribed physical $\nu$. The time and convection -terms scale by $a$, while the Laplacian term scales by $a^{-1}$. -Thus chart rescaling does not silently change the fixed physical viscosity. +\section{History conventions and branch subdivision}\label{ba:codes} + +The general box motions now let us choose a recorder and an observer +separately. We begin with the two frontier recorders in +Section~\ref{sec:bb-frontiers}, which retain history in the +sense of Bennett~\cite{Bennett1973}. Later we retain the distinct +neighboring-cell guards and persistent-mark tables needed by other +realizations. A full-domain inverse, not merely agreement along an +initialized run, is the criterion for sharing a compiler. + +For these applications let $A$ be the work alphabet with distinguished +blank, $Q$ the states, $q_0$ the initial state, and $H\subseteq Q$ +the halting states. Write $h_n$ for the work head after $n$ steps and +\[ + \delta(q,a)=(q^+(r),b(r),d(r)),\qquad + r=(q,a)\in\mathcal R=(Q\setminus H)\times A, + \quad d(r)\in\{-1,0,1\}. +\] +The initial head is zero and the tape has finitely specified nonblank +cells. Replace missing instructions by unchanged-letter stay moves to +a halt. If one halt square is needed, merge halting states and redirect +their incoming instructions; no outgoing rule is changed. This preserves +initial halting as well. An unreachable halt state can be added when +needed. These are the same normalizations as in +Section~\ref{sec:bb-frontiers}. + +\subsection{Frontier recorders and their notation} + +We use the move-first recorder of Lemma~\ref{lem:bb-movefirst} +with its original notation and complete partial domain. +The delayed-head recorder of Lemma~\ref{lem:bb-direction} records the +work displacement until its final return step. Its table can also be +initialized with the frontier one cell farther to the right. + +\begin{lemma}[Six-rule history recorder]\label{ba:six-compiler} +For $r_0=1$ or $r_0=2$, there is a partial table with the two incoming +properties of Lemma~\ref{lem:bb-onecell}, frontier $r_0+n$ at checkpoint +$n$, and checkpoint control recording both the work state and the +preceding work displacement. Its next checkpoint takes +$2(r_0+n-h_n)+3$ instructions. The work head moves only on the last +instruction. +\end{lemma} +\begin{proof}\label{ba:six-proof} +Specialize Lemma~\ref{lem:bb-direction} to a singleton passive track +$O$ and delete that coordinate. Rename its controls +$(M(q,e),G(\tau),A(q,d),B(q,d))$ as +$(M_{q,e},R_\tau,F_{q,d},L_{q,d})$, and its frontier $F$ as $P$. +The full alphabet is +$A\times(\{\bot,P\}\sqcup\mathcal T)\times\{0,1\}$, +where $\mathcal T=(Q\setminus H)\times\{-1,0,1\}\times A$. +Every row and guard is the same under this bijection, so both incoming +properties hold on the entire partial domain, including unused controls. +Only $M_{q,e}$ with $q\in H$ are halting controls. + +Initialize $M_{q_0,0}$ with all marks zero, frontier at $r_0$, and +empty history elsewhere. The case $r_0=1$ is the initialization already +proved in the shared lemma. For either allowed $r_0$, checkpoint $n$ +has frontier $g=r_0+n>h_n$. Mark the old head, scan right to $g$, +write the record and install the new frontier at $g+1$, then return to +the mark. All intervening history is nonfrontier, the new frontier +cell is empty, and there is only one mark. Hence the same rows apply +when $r_0=2$. There are $g-h_n-1$ continuing right moves, +$g-h_n$ continuing left moves, and four other instructions, giving +$2(g-h_n)+3$. The last row clears the mark and makes the work +movement; the invariant is restored with frontier $g+1$. This proves +finite completion, indefinite legality for a nonhalting run, and the +stated halt equivalence, including initial halting. +\end{proof} + +\subsection{From symbolic cylinders to closed rectangles} + +Radix encodings turn a push or pop of a tape letter into an affine map; +this is the generalized-shift viewpoint of +Moore~\cite{Moore1990,Moore1991}. The following calculation adds the +separation of full closed rectangles needed for smooth localization. + +\begin{lemma}[Separated rectangle codes]\label{ba:rectangles} +Let a deterministic partial tape table on an alphabet $\Gamma$ of size +$m$ have the two incoming properties of Lemma~\ref{lem:bb-onecell}. +Assign distinct odd digits $e(a)\in\{1,3,\ldots,2m-1\}$ and choose +an integer $B\ge2m+1$. Its local rules admit effective positive diagonal +affine maps of determinant one between finite families of closed planar +rectangles. Sources are pairwise separated, as are targets. Both the +minimal subdivision and the subdivision by both adjacent letters given +below realize every tape in their full symbolic cylinders. +\end{lemma} +\begin{proof}\label{ba:rectangles-proof} +Use Lemma~\ref{lem:bb-onecell} with digits $d(a)=e(a)$. Its digit +hypotheses hold for every $B\ge2m+1$. In the notation +\[ + C(a_1a_2\cdots)=\sum_{j\ge1}e(a_j)B^{-j},\quad + h_a(v)=\frac{e(a)+v}{B},\quad + I_{a_1\cdots a_l}=h_{a_1}\cdots h_{a_l}([0,1]), +\] +the left stream starts immediately left of the head and the right stream +starts at the head. Place state $s$ in $k_s+a[0,1]^2$ with a common +rational $a>0$. The minimal branch maps are exactly +\eqref{eq:bb-one-cell-branches}; scaling both charts by $a$ leaves +$\operatorname{diag}(B^{-d},B^d)$ unchanged. + +For the additional full subdivision, restrict the right- and stay-move +sources to $x\in I_\gamma$, for every full letter $\gamma$. +Their targets become $I_{\beta\gamma}\times[0,1]$ and +$I_\gamma\times I_\beta$, respectively, for written letter $\beta$. +The already split left-move targets remain +$[0,1]\times I_{\gamma\beta}$. At a fixed target state the +incoming direction is fixed, and the pair $(\beta,\gamma)$ +distinguishes the branches in every case. Thus the added subdivision +preserves separate target separation, while source separation follows +by restriction of the shared source rectangles. All these statements +concern the filled closed rectangles and arbitrary symbolic tails. + +For completeness the quantitative gaps remain valid for the entire +range $B\ge2m+1$: distinct first-digit intervals have gap at least +$B^{-1}$ and distinct second-digit intervals at least $B^{-2}$. +The source and target families therefore have within-state gaps at +least $aB^{-2}$; different state squares have their own chosen gaps. +Tail values lie between $1/(B-1)$ and $(2m-1)/(B-1)<1$. +An initial constant tail after $l$ digits contributes +$B^{-l}e(a_{\rm tail})/(B-1)$, so initialized codes are rational. +At checkpoints the history records, or the known absolute origin of +the record block, also recover the absolute work-head position. +\end{proof} + + +\subsection{Conventions for the remaining geometric realizations}\label{bc:section} +\label{bc:frontier-section} +Unless a different table is specified, subsequent routes use the move-first recorder of +Lemma~\ref{lem:bb-movefirst}, with initial frontier two. The controls +$(C_q,J_r,S_r,P_q,T_q)$ there are written $(W_q,P_r,B_r,C_q,D_q)$ +here, and its empty/frontier/record symbols $(e,F,r)$ are written +$(E,F,R_r)$. Permuting work, history and mark coordinates in every row +preserves every guard. This identifies the full partial table, its inverse +and the exact checkpoint count $2(2+n-h')+4$, including arbitrary allowed +tapes. A nonterminal branch means one with both endpoint controls +nonterminal; a final branch into a halt can enter the observer. + +The full branches are those of Lemma~\ref{lem:bb-onecell}. For an alphabet +of size $m$ we use one of the following explicitly indicated conventions: +\begin{equation}\label{bc:digits} +(B,\{d_a\})=(2m+2,\{1,3,\ldots,2m-1\}),\quad +(2m+1,\{1,3,\ldots,2m-1\}),\quad +(2m,\{0,2,\ldots,2m-2\}). +\end{equation} +All satisfy its digit bound and separation hypotheses. In the last +convention the full blank has digit zero; boundary codes remain in the +closed-rectangle and neighborhood conclusions. Write +$C(a_1a_2\cdots)=\sum_{j\ge1}d_{a_j}B^{-j}$ and +$I_v=\sum_{j=1}^{|v|}d_{v_j}B^{-j}+[0,B^{-|v|}]$. +A constant tail after $k$ positions contributes +$d_{\rm tail}B^{-k}/(B-1)$, so the initial codes are rational. +Guarded three-cell rules use Lemma~\ref{lem:bb-window-rectangles} with +$(a,b)=(1,2)$ after their own full-domain inverse has been proved. + +\subsection{Analytic conventions for the geometric variants}\label{bc:analytic-section} +The whole-space baseline in these variants means smooth +$u\in CH^2\cap C^1L^2$, bounded $u,\nabla u$ on finite intervals, +and a pressure representative in $CH^1$. When the competitor-gradient +condition is omitted explicitly, the other conditions remain unchanged. +Both cases fall under Lemma~\ref{lem:b-comparison}(i); separately stated +pressure classes are retained. The torus baseline is its classical torus +clause. Unless a projection is specified, the force is +\begin{equation}\label{bc:residual} +f_U=U_t+(U\cdot\nabla)U-\nu\Delta U,\qquad (u,p)=(U,0). +\end{equation} +Torus projection is the separate choice in +Proposition~\ref{prop:projection-l2}, with changed pressure. +Localized affine motion is supplied by Lemma~\ref{lem:bb-curl} on a +compact time interval, using effective derivative bounds, positive lower +scale bounds and the positive cutoff margins specified in each route. +The potential is \eqref{eq:bb-potential}; the resulting flow fixes every +region disjoint from its support. In particular exponential diagonal +scales and linear reciprocal scales have respectively generators +$\theta'\log D$ and $\diag(l'/l,-l'/l,0)$. + +\begin{lemma}[Finite forcing mechanisms]\label{bc:analytic} +A finite family of the preceding pulses, joined with flat time collars, +can be used once for initialization and repeated with period one thereafter. +It gives a smooth effective velocity and residual force with all mixed +derivatives uniformly bounded. In $\R^3$ both have one fixed compact +spatial support, and the velocity has uniformly bounded kinetic energy. +In a torus chart, +periodized curl or planar Hamiltonian pulses have zero spatial mean, as +does their residual force. Initial velocity and pressure are zero. +Uniqueness holds in the comparison classes of Section~\ref{sec:model}; +in the whole-space class the competitor's gradient bound can be omitted +if the remaining conditions are retained. +\end{lemma} +\begin{proof}\label{bc:analytic-proof} +Each pulse is a finite expression in rational data, effective smooth +switches, their derivatives, positive diagonal scales and logarithms. +Finite positive separation margins bound every cutoff derivative. A finite +maximum bounds a period and the loading interval, and repetitions preserve +that bound. To evaluate near a join, a coarse enclosure of time gives a +finite superset of possibly active pulses. Summing them avoids an exact +comparison of a computable time with an integer. The flat collars give +smoothness. Compact support gives every required spatial Sobolev bound. +Lemma~\ref{lem:p2-realization} gives the asserted solution and comparison; +on a torus the integral of each curl or Hamiltonian derivative vanishes, +and the convection term is a divergence. Its flow satisfies +$\det D\Phi_t=1$, because differentiating this determinant gives +$(\divg U)(t,\Phi_t)\det D\Phi_t=0$ and its initial value is one. +The velocity and acceleration identities in that lemma consequently apply +to these volume-preserving diffeomorphisms. +\end{proof} + +For later use, spatial chart changes are always made at the velocity level. +If $x=o+a y$, set $\widetilde U(t,x)=aU(t,(x-o)/a)$ and then compute +\eqref{bc:residual} at the prescribed physical $\nu$. The time and convection +terms scale by $a$, while the Laplacian term scales by $a^{-1}$. +Thus chart rescaling does not silently change the fixed physical viscosity. diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compact-processors.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compact-processors.tex index 0e63498b8d27d83fcaa2b836a66b89f769c3b726..a8c900d68ff8d0c87be12b942f2b0e6737acaf72 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compact-processors.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compact-processors.tex @@ -1,189 +1,189 @@ -\section{Compact support and onto slow clocks}\label{ba:euclidean} - -A whole-space processor can use arbitrarily separated finite lanes. -We shall construct a periodic cycle with fixed compact support and also -give a separate decaying force with the same material event. We first -establish the time change used for the latter choice. Its clock must tend -to infinity: traversing only a finite amount of machine time would not -preserve the halting event. - -\subsection{The logarithmic clock at every derivative order} - -\begin{lemma}[Logarithmic clock with full derivative bounds] -\label{ba:log-clock} -Let $W(s,x)$ be a smooth solenoidal field for $s\ge0$, initially zero, -with every mixed derivative bounded and support in one compact set $K$. -For fixed computable $\nu>0$, set -\[ - s(t)=\log(1+t),\quad \rho(t)=(1+t)^{-1},\quad - U(t,x)=\rho(t)W(s(t),x),\quad - F=U_t+(U\cdot\nabla)U-\nu\Delta U. -\] -Every derivative $\partial_t^k\partial_x^\alpha U$ and -$\partial_t^k\partial_x^\alpha F$ is uniformly -$O((1+t)^{-1-k})$ and belongs to space-time $L^2$. -Both supports lie in $K$. If $\Xi$ is the material flow of $W$, that -of $U$ is $\Xi(s(t),a)$, so every fixed spatial reachability event is -unchanged. Effective finite formulas for $W$ give effective formulas -for $U,F$ and all derivatives. -\end{lemma} -\begin{proof}\label{ba:log-clock-proof} -Since $s'=\rho$ and $\rho'=-\rho^2$, -\begin{equation}\label{ba:clock-rule} - \partial_t\{\rho^jY(s(t),x)\} - =\rho^{j+1}(\partial_s-j)Y(s(t),x). -\end{equation} -For $P_{k,j}(D)=\prod_{r=0}^{k-1}(D-j-r)$, with $P_{0,j}=1$, -induction yields -\begin{equation}\label{ba:clock-induction} - \partial_t^k\{\rho^jY(s(t),x)\} - =\rho^{j+k}P_{k,j}(\partial_s)Y(s(t),x). -\end{equation} -Spatial differentiation commutes with this identity. Taking $j=1$ -proves the velocity bound at every order. The residual is exactly -\begin{equation}\label{ba:clock-residual} - F=\rho^2\{W_s-W+(W\cdot\nabla)W\}(s(t),x) - -\nu\rho\Delta W(s(t),x). -\end{equation} -Every mixed derivative of the braces is bounded by the finite product -rule and the hypotheses. Apply \eqref{ba:clock-induction} with $j=2$ -and $j=1$ to obtain orders $\rho^{2+k}$ and $\rho^{1+k}$, -respectively. This gives the force bound, including every spatial -multi-index. In particular no temporal derivative of the convection -term has been omitted. - -Supports do not increase under differentiation or multiplication. -For either field $Z$, the squared integral of an indicated derivative -is at most -\[ - |K|C_{k,\alpha}^2\int_0^\infty(1+t)^{-2-2k}\,dt - =\frac{|K|C_{k,\alpha}^2}{1+2k}. -\] -The zero datum remains zero, so residual realization applies. -The chain rule shows that $\Xi(s(t),a)$ solves the $U$ trajectory -equation with initial label $a$; uniqueness identifies it with the -actual trajectory. The clock is increasing onto $[0,\infty)$ with -inverse $e^s-1$, so it loses no finite computational time. Finally -logarithm on positive arguments, the rational powers of $1+t$, and -the finite-index evaluation of the template and its derivatives are -effective. This proves all assertions. -\end{proof} - - -\subsection{Loaded Euclidean processors} - -\begin{theorem}[Two full-box processors on Euclidean space] -\label{ba:euclidean-boxes} -For every finite machine and input, and every fixed computable $\nu>0$, -either the move-first recorder of Lemma~\ref{lem:bb-movefirst} or the -six-rule recorder of Lemma~\ref{ba:six-compiler} yields effective smooth forcing on $\mathbb R^3$ with one -compact spatial support, all mixed derivatives bounded, and period one -for $t\ge1$, such that -\[ - \exists t\ge0:\ (\Phi_t(0))_1<-1 - \quad\Longleftrightarrow\quad\text{halting}. -\] -The zero-data solution is unique in Lemma~\ref{ba:realization}'s class, -has pressure zero and uniformly bounded energy. Separately, the six-rule -processor admits a force square-integrable on all space-time with the -same support and event. -\end{theorem} -\begin{proof}\label{ba:euclidean-boxes-proof} -Write $\Gamma$ for the full alphabet of the recorder being used. -For the six-rule table use initial frontier 2 and base $B=2|\Gamma|+2$. -Give nonhalting controls lanes $3i+[0,1]$ and halting controls lanes -$-3i+[0,1]$, enumerating each group from 1; the second coordinate is -$[0,1]$. The minimal subdivision has gap $B^{-2}$. Use full thickness -$[-1,1]$, heights $10i$, and the transported-cutoff construction in -Section~\ref{sec:ba-transported-localization}, with initial padding $1/(10B^3)$. -Target horizontal padding is at most $1/(10B^2)$, below half the gap; -vertical middle intervals are also separated. - -For the move-first table use a single halt lane $-3+[0,1]$, other lanes -$2+2j+[0,1]$, base $B=2|\Gamma|+1$, the minimal subdivision, thickness -$[-1,1]$, and heights $4i$. Use current-box padding $1/(10B^2)$. -The middle-phase separation is -$2(1+1/(10B^2))<4$; the planar gaps handle the other phases. -This localization has a fixed current padding, as distinct from the -transported initial padding in the six-rule version. - -The rational initial code $P_*$ is reached from zero along -$\theta(t)P_*$ by a localized translation during $[0,1]$. -Repeating the chosen cycle produces a template $W$ with one compact -support and bounded mixed derivatives. The code identity at times -$1+n$ follows by induction through the local rules; the respective -recorder invariant supplies every next checkpoint in finite positive -time. A halt lies below $-1$ in the first coordinate. In a nonhalting -run all branch endpoints have positive first coordinate, and -\eqref{eq:bb-convex-coordinate} keeps it positive throughout every cycle. -Loading is nonnegative. Lemma~\ref{ba:realization} completes the PDE -argument. Uniform support and boundedness give a uniform energy bound. -The separate square-integrable choice follows from -Lemma~\ref{ba:log-clock}, without claiming periodicity in physical -time for that choice. -\end{proof} - - -\subsection{Two realizations of a left frontier} - -\begin{proposition}[Separate columns or solid-box parking] -\label{prop:bb-mid-stream}\label{prop:bb-mid-parking} -Both left-frontier tables have compact whole-space realizations, -periodic after $t=1$, with fixed label zero and event $X_1(t;0)<-1/2$. -The stopping version can use distinct-height stream functions and -has uniqueness with pointwise-in-time $H^1$ comparison pressure. -The continuing version transports full solid boxes by evacuation, -reshaping and delivery, and represents every local step, including -post-halt idle computation. It has in particular uniqueness with -comparison pressure in $C([0,T];H^1)$. -\end{proposition} -\begin{proof}\label{proof:bb-mid-stream}\label{proof:bb-mid-parking} -Let $\Sigma$ be the full work--history--mark alphabet of the chosen -left-frontier table, including its record symbols and any halting idle -records in the continuing version. Use odd digits in base $B=2|\Sigma|+2$, offset $-2$ for $C_h$ and -$2,4,6,\ldots$ for other controls. The two head-relative fractions -and height zero give a rational initial code $Z_*$. The rightmost -nonempty history cell is the original cell $-2$, an absolute landmark. - -For the stopping table, apply the stream-function construction -\eqref{eq:bb-mid-stream-path} at heights $z_i=3i$. -Choose source and target column paddings no greater than $1/2$ and -one third of their respective family gaps. A middle layer padding -$1/2$ keeps distinct layers disjoint. The explicit thickening bound -following that construction proves the neighborhood action. -Write $Z_*=(x_*,y_*,0)$. With a flat switch $\theta$ from zero to one, -load the origin along $\theta(t)Z_*$ using the velocity -$\nabla\times(0,0,H)$, where -$H(t,x,y,z)=\theta'(t)\chi(x,y,z)(x_*y-y_*x)$. -Choose $\chi$ compactly supported in $\R^3$ and equal to one near the -entire segment. The gate $\theta'$ vanishes on the time collars. -All nonhalt source and target lower-left first coordinates are at -least two. Inequality~\eqref{eq:bb-mid-stream-safe} therefore -excludes the observer during each period; loading has nonnegative -first coordinate. Halt codes lie in $(-2,-1)$. - -For the continuing table retain its unrestricted first-row history -symbol and its idle work rules. Thicken every rectangle by $[-1,1]$. -Choose $L$ ten larger than the maximum of zero and every source or -target right endpoint. Reserve parking projections -$(L+4i,0)+[-1,1]^2$ and let $\delta$ be the minimum of one and the -gaps in each union of one rectangle family with these reservations. -Use padding $\delta/10$, travel height five, and $7N$ slots: evacuate -all sources by three translations each, reshape all parked boxes by -$\operatorname{diag}(1+\theta(\lambda_i-1),[1+\theta(\lambda_i-1)]^{-1},1)$, -and deliver by three translations each. Horizontal widths stay at -most one, vertical half-width is one, and the reservations miss both -original families. Lemma~\ref{lem:bb-lift-parking} proves exact action -on the whole boxes and their neighborhoods even with source--target -overlap. A first localized translation loads $Z_*$ from the origin. -The sampling identity is now valid for every $k\ge0$, not just until -halting: $\Phi_{1+k}(0)=Z(C_k)$ for the continued table. - -For a nonhalting run all used endpoint centers have first coordinate -at least two, parking centers are farther right, and half-widths are -at most $1/2$. Translation, parking strain, and waiting therefore -keep first coordinate positive. A halt checkpoint lies in $(-2,-1)$. -Proposition~\ref{prop:bb-realization} proves the analytic assertions -for both constructions. Their compact support also gives uniformly -bounded kinetic energy. -\end{proof} +\section{Compact support and onto slow clocks}\label{ba:euclidean} + +A whole-space processor can use arbitrarily separated finite lanes. +We shall construct a periodic cycle with fixed compact support and also +give a separate decaying force with the same material event. We first +establish the time change used for the latter choice. Its clock must tend +to infinity: traversing only a finite amount of machine time would not +preserve the halting event. + +\subsection{The logarithmic clock at every derivative order} + +\begin{lemma}[Logarithmic clock with full derivative bounds] +\label{ba:log-clock} +Let $W(s,x)$ be a smooth solenoidal field for $s\ge0$, initially zero, +with every mixed derivative bounded and support in one compact set $K$. +For fixed computable $\nu>0$, set +\[ + s(t)=\log(1+t),\quad \rho(t)=(1+t)^{-1},\quad + U(t,x)=\rho(t)W(s(t),x),\quad + F=U_t+(U\cdot\nabla)U-\nu\Delta U. +\] +Every derivative $\partial_t^k\partial_x^\alpha U$ and +$\partial_t^k\partial_x^\alpha F$ is uniformly +$O((1+t)^{-1-k})$ and belongs to space-time $L^2$. +Both supports lie in $K$. If $\Xi$ is the material flow of $W$, that +of $U$ is $\Xi(s(t),a)$, so every fixed spatial reachability event is +unchanged. Effective finite formulas for $W$ give effective formulas +for $U,F$ and all derivatives. +\end{lemma} +\begin{proof}\label{ba:log-clock-proof} +Since $s'=\rho$ and $\rho'=-\rho^2$, +\begin{equation}\label{ba:clock-rule} + \partial_t\{\rho^jY(s(t),x)\} + =\rho^{j+1}(\partial_s-j)Y(s(t),x). +\end{equation} +For $P_{k,j}(D)=\prod_{r=0}^{k-1}(D-j-r)$, with $P_{0,j}=1$, +induction yields +\begin{equation}\label{ba:clock-induction} + \partial_t^k\{\rho^jY(s(t),x)\} + =\rho^{j+k}P_{k,j}(\partial_s)Y(s(t),x). +\end{equation} +Spatial differentiation commutes with this identity. Taking $j=1$ +proves the velocity bound at every order. The residual is exactly +\begin{equation}\label{ba:clock-residual} + F=\rho^2\{W_s-W+(W\cdot\nabla)W\}(s(t),x) + -\nu\rho\Delta W(s(t),x). +\end{equation} +Every mixed derivative of the braces is bounded by the finite product +rule and the hypotheses. Apply \eqref{ba:clock-induction} with $j=2$ +and $j=1$ to obtain orders $\rho^{2+k}$ and $\rho^{1+k}$, +respectively. This gives the force bound, including every spatial +multi-index. In particular no temporal derivative of the convection +term has been omitted. + +Supports do not increase under differentiation or multiplication. +For either field $Z$, the squared integral of an indicated derivative +is at most +\[ + |K|C_{k,\alpha}^2\int_0^\infty(1+t)^{-2-2k}\,dt + =\frac{|K|C_{k,\alpha}^2}{1+2k}. +\] +The zero datum remains zero, so residual realization applies. +The chain rule shows that $\Xi(s(t),a)$ solves the $U$ trajectory +equation with initial label $a$; uniqueness identifies it with the +actual trajectory. The clock is increasing onto $[0,\infty)$ with +inverse $e^s-1$, so it loses no finite computational time. Finally +logarithm on positive arguments, the rational powers of $1+t$, and +the finite-index evaluation of the template and its derivatives are +effective. This proves all assertions. +\end{proof} + + +\subsection{Loaded Euclidean processors} + +\begin{theorem}[Two full-box processors on Euclidean space] +\label{ba:euclidean-boxes} +For every finite machine and input, and every fixed computable $\nu>0$, +either the move-first recorder of Lemma~\ref{lem:bb-movefirst} or the +six-rule recorder of Lemma~\ref{ba:six-compiler} yields effective smooth forcing on $\mathbb R^3$ with one +compact spatial support, all mixed derivatives bounded, and period one +for $t\ge1$, such that +\[ + \exists t\ge0:\ (\Phi_t(0))_1<-1 + \quad\Longleftrightarrow\quad\text{halting}. +\] +The zero-data solution is unique in Lemma~\ref{ba:realization}'s class, +has pressure zero and uniformly bounded energy. Separately, the six-rule +processor admits a force square-integrable on all space-time with the +same support and event. +\end{theorem} +\begin{proof}\label{ba:euclidean-boxes-proof} +Write $\Gamma$ for the full alphabet of the recorder being used. +For the six-rule table use initial frontier 2 and base $B=2|\Gamma|+2$. +Give nonhalting controls lanes $3i+[0,1]$ and halting controls lanes +$-3i+[0,1]$, enumerating each group from 1; the second coordinate is +$[0,1]$. The minimal subdivision has gap $B^{-2}$. Use full thickness +$[-1,1]$, heights $10i$, and the transported-cutoff construction in +Section~\ref{sec:ba-transported-localization}, with initial padding $1/(10B^3)$. +Target horizontal padding is at most $1/(10B^2)$, below half the gap; +vertical middle intervals are also separated. + +For the move-first table use a single halt lane $-3+[0,1]$, other lanes +$2+2j+[0,1]$, base $B=2|\Gamma|+1$, the minimal subdivision, thickness +$[-1,1]$, and heights $4i$. Use current-box padding $1/(10B^2)$. +The middle-phase separation is +$2(1+1/(10B^2))<4$; the planar gaps handle the other phases. +This localization has a fixed current padding, as distinct from the +transported initial padding in the six-rule version. + +The rational initial code $P_*$ is reached from zero along +$\theta(t)P_*$ by a localized translation during $[0,1]$. +Repeating the chosen cycle produces a template $W$ with one compact +support and bounded mixed derivatives. The code identity at times +$1+n$ follows by induction through the local rules; the respective +recorder invariant supplies every next checkpoint in finite positive +time. A halt lies below $-1$ in the first coordinate. In a nonhalting +run all branch endpoints have positive first coordinate, and +\eqref{eq:bb-convex-coordinate} keeps it positive throughout every cycle. +Loading is nonnegative. Lemma~\ref{ba:realization} completes the PDE +argument. Uniform support and boundedness give a uniform energy bound. +The separate square-integrable choice follows from +Lemma~\ref{ba:log-clock}, without claiming periodicity in physical +time for that choice. +\end{proof} + + +\subsection{Two realizations of a left frontier} + +\begin{proposition}[Separate columns or solid-box parking] +\label{prop:bb-mid-stream}\label{prop:bb-mid-parking} +Both left-frontier tables have compact whole-space realizations, +periodic after $t=1$, with fixed label zero and event $X_1(t;0)<-1/2$. +The stopping version can use distinct-height stream functions and +has uniqueness with pointwise-in-time $H^1$ comparison pressure. +The continuing version transports full solid boxes by evacuation, +reshaping and delivery, and represents every local step, including +post-halt idle computation. It has in particular uniqueness with +comparison pressure in $C([0,T];H^1)$. +\end{proposition} +\begin{proof}\label{proof:bb-mid-stream}\label{proof:bb-mid-parking} +Let $\Sigma$ be the full work--history--mark alphabet of the chosen +left-frontier table, including its record symbols and any halting idle +records in the continuing version. Use odd digits in base $B=2|\Sigma|+2$, offset $-2$ for $C_h$ and +$2,4,6,\ldots$ for other controls. The two head-relative fractions +and height zero give a rational initial code $Z_*$. The rightmost +nonempty history cell is the original cell $-2$, an absolute landmark. + +For the stopping table, apply the stream-function construction +\eqref{eq:bb-mid-stream-path} at heights $z_i=3i$. +Choose source and target column paddings no greater than $1/2$ and +one third of their respective family gaps. A middle layer padding +$1/2$ keeps distinct layers disjoint. The explicit thickening bound +following that construction proves the neighborhood action. +Write $Z_*=(x_*,y_*,0)$. With a flat switch $\theta$ from zero to one, +load the origin along $\theta(t)Z_*$ using the velocity +$\nabla\times(0,0,H)$, where +$H(t,x,y,z)=\theta'(t)\chi(x,y,z)(x_*y-y_*x)$. +Choose $\chi$ compactly supported in $\R^3$ and equal to one near the +entire segment. The gate $\theta'$ vanishes on the time collars. +All nonhalt source and target lower-left first coordinates are at +least two. Inequality~\eqref{eq:bb-mid-stream-safe} therefore +excludes the observer during each period; loading has nonnegative +first coordinate. Halt codes lie in $(-2,-1)$. + +For the continuing table retain its unrestricted first-row history +symbol and its idle work rules. Thicken every rectangle by $[-1,1]$. +Choose $L$ ten larger than the maximum of zero and every source or +target right endpoint. Reserve parking projections +$(L+4i,0)+[-1,1]^2$ and let $\delta$ be the minimum of one and the +gaps in each union of one rectangle family with these reservations. +Use padding $\delta/10$, travel height five, and $7N$ slots: evacuate +all sources by three translations each, reshape all parked boxes by +$\operatorname{diag}(1+\theta(\lambda_i-1),[1+\theta(\lambda_i-1)]^{-1},1)$, +and deliver by three translations each. Horizontal widths stay at +most one, vertical half-width is one, and the reservations miss both +original families. Lemma~\ref{lem:bb-lift-parking} proves exact action +on the whole boxes and their neighborhoods even with source--target +overlap. A first localized translation loads $Z_*$ from the origin. +The sampling identity is now valid for every $k\ge0$, not just until +halting: $\Phi_{1+k}(0)=Z(C_k)$ for the continued table. + +For a nonhalting run all used endpoint centers have first coordinate +at least two, parking centers are farther right, and half-widths are +at most $1/2$. Translation, parking strain, and waiting therefore +keep first coordinate positive. A halt checkpoint lies in $(-2,-1)$. +Proposition~\ref{prop:bb-realization} proves the analytic assertions +for both constructions. Their compact support also gives uniformly +bounded kinetic energy. +\end{proof} diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compilers.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compilers.tex index b486b111883b0caafb7fb923f3c81ac2ebebf3ab..f4d02850a1314be4336e7cd6c15104e70b7b47a6 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compilers.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compilers.tex @@ -1,307 +1,307 @@ -\section{History records and complete symbolic domains} -\label{sec:bb-frontiers} - -The balanced-stack theorem is complete. The remaining realizations -use alternative recorders with tape data in two coordinates. -A motion of boxes is invertible. To use it for an ordinary machine, -we must retain whatever information an instruction erases. We use -finite history records in the sense of reversible computation, but -prove the local inverses below: reversibility of the initialized -execution would not by itself separate the full boxes. - -A work instruction is $r=(q,a)\mapsto(p_r,b_r,d_r)$, with -$d_r\in\{-1,0,1\}$. The work tape is indexed by $\Z$, starts with -head zero, and has only finitely many prescribed nonblank cells. -Halting states have no outgoing work instruction. Missing instructions -can be replaced by write-preserving stationary steps to a fresh halt -state. Unless a construction explicitly adds a dummy initial step, -an initially halting state is already a halt checkpoint. When a construction -uses a single halting state $h$, merge the original halting states and -redirect their incoming transitions; this preserves initial halting. In -the recorder tables $Q$ is the work-state set and $\Gamma$ the work -alphabet. Radix bases are chosen from the full cell alphabet, including -all history and mark tracks. - -\subsection{A one-cell code and its exact inverse test} - -A partial one-cell table sends a control and a scanned letter to a -new control, a written letter, and a displacement. Its two incoming -conditions are: the new control determines the displacement, and the -new control together with the written letter determines the entire -old control and read letter. - -\begin{lemma}[From one-cell tables to closed rectangles] -\label{lem:bb-onecell}\label{lem:bb-mid-prefix} -Give an alphabet of size $m$ digits separated by at least two, between -$0$ and $B-2$. Put -\[ -E(A)=\sum_{j\ge1}d(A_j)B^{-j},\qquad -I(v)=\sum_{j=1}^{|v|}d(v_j)B^{-j}+[0,B^{-|v|}]. -\] -For a head-relative tape $(a_j)_{j\in\Z}$ use the coordinates -\[ - (x,y)=\big(E(a_{-1}a_{-2}\cdots),E(a_0a_1\cdots)\big). -\] -After writing at cell zero, a displacement $e$ makes the old cell $e$ -the new cell zero. -A deterministic partial one-cell table satisfying the incoming -conditions acts, after separate state translations, by a finite list -of reciprocal affine maps on full closed rectangles. Each source -family and each target family is separately disjoint. For a rule -reading $a$, writing $b$, and moving by $e$, the branches are -\begin{equation}\label{eq:bb-one-cell-branches} -\begin{array}{c|c|c|c} -e&\text{source}&\text{target}&(x',y')\\\hline -1 &[0,1]\times I(a)&I(b)\times[0,1] - &((d(b)+x)/B,By-d(a))\\ -0 &[0,1]\times I(a)&[0,1]\times I(b) - &(x,y+(d(b)-d(a))/B)\\ --1&I(c)\times I(a)&[0,1]\times(d(c)+I(b))/B - &(Bx-d(c),[d(c)+(d(b)+By-d(a))/B]/B). -\end{array} -\end{equation} -The last row is split over every letter $c$. Each linear part is -$\operatorname{diag}(B^{-e},B^e)$. With odd digits -$1,3,\ldots,2m-1$ and $B\in\{2m+1,2m+2\}$, every code lies in -the interior of its applicable rectangle, and unscaled within-family -gaps are at least $B^{-2}$. -\end{lemma} -\begin{proof} -The identity $E(aA)=(d(a)+E(A))/B$ gives all three formulas and their -action on the independent tails. At the first differing digit, -incompatible prefix intervals have a positive gap: the digit difference -is at least two while the child interval has length one in that scale. -Thus the source state, read letter, and any left split separate sources. -At one target state the displacement is fixed. For a right or stationary -move the written letter separates the relevant target coordinate; -for a left move the pair $(c,b)$ separates the two-letter prefix. -The incoming inverse condition rules out duplicates. These arguments -concern the complete intervals, including their endpoints. -For odd digits the infinite tail lies between $1/(B-1)$ and -$(2m-1)/(B-1)<1$, proving interiority. The longest target prefix has -length two, giving the asserted gap. Uniformly scaling state squares -by $s$ scales gaps by $s$ and leaves the linear parts unchanged. -\end{proof} - -\subsection{Moving the work head before recording} - -We first place a frontier to the right of every possible work head. -A temporary mark lets the history excursion return to the correct -cell even though its length grows with the computation. - -\begin{lemma}[Move-first recorder] -\label{lem:bb-movefirst}\label{lem:bb-mid-right-compiler} -There is an effective one-cell table satisfying the incoming conditions -whose checkpoint configurations simulate the work machine. After -$n$ work steps its frontier is at $n+2$, its records occupy -$2,\ldots,n+1$, and its temporary marks are all zero. Each nonhalting -checkpoint has a finite positive continuation to the next one, with -no intervening checkpoint. -\end{lemma} -\begin{proof} -Use letters $(c,\ell,j)$ with work letter $c$, history -$\ell\in\{e,F\}\sqcup\{r\}$ and mark $j\in\{0,1\}$. -The controls are $C_q,J_r,S_r,P_p,T_p$. The complete partial table is -\[ -\begin{array}{lll} - C_q:(a,\ell,0)&\mapsto J_r:(b_r,\ell,0),d_r&\ell\ne F,\\ - J_r:(c,\ell,0)&\mapsto S_r:(c,\ell,1),1&\ell\ne F,\\ - S_r:(c,\ell,0)&\mapsto S_r:(c,\ell,0),1&\ell\ne F,\\ - S_r:(c,F,0)&\mapsto P_{p_r}:(c,r,0),1,&\\ - P_p:(c,e,0)&\mapsto T_p:(c,F,0),-1,&\\ - T_p:(c,\ell,0)&\mapsto T_p:(c,\ell,0),-1&\ell\ne F,\\ - T_p:(c,\ell,1)&\mapsto C_p:(c,\ell,0),0&\ell\ne F. -\end{array} -\] -Initialize the frontier at two, all other history empty, and all marks -zero. At checkpoint $n$ the head satisfies $|h_n|\le n$. -The first row updates the work tape and moves to $h'\le n+1$, strictly -left of the frontier $f=n+2$. The second row marks $h'$. The forward -scan writes $r$ at $f$, puts a new frontier at $f+1$, and the backward -scan stops exactly at the mark, which it clears. Its length is -$2(f-h')+4$, so it is finite and positive. This proves the invariant -inductively, including the exact work update and halt equivalence. - -For the full-domain inverse, $r$ recovers the old work pair into $J_r$; -mark one distinguishes entry into $S_r$ from its mark-zero loop; -a written record determines entry into $P_p$; a written $F$ -distinguishes entry into $T_p$ from its loop; and entry into $C_p$ -uniquely restores mark one. Every destination has the single incoming -displacement displayed in the table. No invariant on remote cells -was used. -\end{proof} - -\subsection{Recording before the work-head displacement} - -The next table leaves its mark at the old work head. Its final step -performs the work displacement; storing that displacement in the -checkpoint control keeps incoming moves unambiguous. - -\begin{lemma}[Direction-recording scan] -\label{lem:bb-direction}\label{lem:bb-mid-mark-compiler} -Let $D=\{-1,0,1\}$ and let $O$ be either a singleton or $\{0,1\}$. -An effective one-cell table on -$\Gamma\times(\{\bot,F\}\sqcup\mathcal R)\times\{0,1\}\times O$, -where $\mathcal R=\{(q,e,a):q\text{ nonhalting},e\in D,a\in\Gamma\}$, -satisfies the incoming conditions. It preserves the $O$ track in -its physical cells. Starting with frontier at one, its checkpoint -at work step $n$ has frontier $n+1$, records at $1,\ldots,n$, zero -marks, and the correct work configuration. A step from head $i$ -takes $5+2(n-i)$ local rules. -\end{lemma} -\begin{proof}\label{proof:bb-mid-mark-compiler} -Use main controls $M(q,e)$, forward controls $G(r)$, and $A(p,d),B(p,d)$ -for every work state $p$ and $d\in D$, including pairs not produced by -a work instruction. Copy the $O$ component in each row. In the first -three rows use every $r=(q,e,a)\in\mathcal R$ with -$\delta(q,a)=(p,c,d)$. The final three rows apply independently to -every $(p,d)\in Q\times D$, including pairs that are not outputs of -any work instruction. The variables $v\in\Gamma$ and -$h\in\{\bot,F\}\sqcup\mathcal R$ range over all letters subject to -the displayed restrictions. The whole table is -\[ -\begin{array}{lll} - M(q,e):(a,h,0)&\mapsto G(r):(c,h,1),1&h\ne F,\\ - G(r):(v,h,0)&\mapsto G(r):(v,h,0),1&h\ne F,\\ - G(r):(v,F,0)&\mapsto A(p,d):(v,r,0),1,&\\ - A(p,d):(v,\bot,0)&\mapsto B(p,d):(v,F,0),-1,&\\ - B(p,d):(v,h,0)&\mapsto B(p,d):(v,h,0),-1&h\ne F,\\ - B(p,d):(v,h,1)&\mapsto M(p,d):(v,h,0),d&h\ne F. -\end{array} -\] -Starting at $M(q_0,0)$, the first row writes and marks the old head. -The scan reaches frontier $j=n+1$, stores its record, creates the -next frontier, returns to the unique mark and makes the work move. -The count is $2(j-i)+3=5+2(n-i)$. Since $i\le nh_n$, -the forward scan reaches the frontier, moves it one cell right, and -the return scan reaches the mark. Its neighboring-cell guard holds -because the new frontier is $p_n+1$. The last rule clears the mark -and performs the work displacement. Counting the two traversals and -endpoint actions gives $2(p_n-h_n)+1$; induction gives $|h_n|\le n$. - -For injectivity on all relative tapes, into Go the old cursor is -output offset $-1$: a mark there distinguishes entry from the loop, -and $r$ restores the overwritten work letter and old Ready control. -Into Back the old cursor is offset $+1$. An output pointer at $+2$ -identifies a log insertion, with record at $+1$ recovering $r$ and -old pair $(P,E)$. A nonpointer at $+2$ identifies the guarded loop. -Into Ready the corresponding Back control uniquely restores the -cleared mark. These are all possibilities, proving the full inverse. -\end{proof} - -\begin{lemma}[Closed rectangles for a finite guarded window] -\label{lem:bb-window-rectangles} -Suppose a finite partial injective local tape map reads and writes only -indices $[-a,b-1]$, then shifts the cursor by $e$, where -$-a\le e\le b$. Split its domain by every complete word in that -window. With gapped radix coding the input prefix lengths are -$(a,b)$ and output lengths $(a+e,b-e)$; the affine branch has linear -part $\operatorname{diag}(B^{-e},B^e)$. Both full rectangle families -are separately disjoint. -\end{lemma} -\begin{proof} -After the writes the new left prefix consists of old indices -$e-1,e-2,\ldots,-a$, and the right prefix of $e,e+1,\ldots,b-1$. -The two remaining tails are free and unchanged, so the image is the -whole output cylinder. Distinct source cylinders have disjoint -images by injectivity. If both pairs of output prefixes were -compatible, a common extension would belong to both images; hence -at least one coordinate has incompatible prefixes. The gapped -interval argument then separates the complete closed rectangles. -The prefix lengths give the two reciprocal scales directly. -\end{proof} +\section{History records and complete symbolic domains} +\label{sec:bb-frontiers} + +The balanced-stack theorem is complete. The remaining realizations +use alternative recorders with tape data in two coordinates. +A motion of boxes is invertible. To use it for an ordinary machine, +we must retain whatever information an instruction erases. We use +finite history records in the sense of reversible computation, but +prove the local inverses below: reversibility of the initialized +execution would not by itself separate the full boxes. + +A work instruction is $r=(q,a)\mapsto(p_r,b_r,d_r)$, with +$d_r\in\{-1,0,1\}$. The work tape is indexed by $\Z$, starts with +head zero, and has only finitely many prescribed nonblank cells. +Halting states have no outgoing work instruction. Missing instructions +can be replaced by write-preserving stationary steps to a fresh halt +state. Unless a construction explicitly adds a dummy initial step, +an initially halting state is already a halt checkpoint. When a construction +uses a single halting state $h$, merge the original halting states and +redirect their incoming transitions; this preserves initial halting. In +the recorder tables $Q$ is the work-state set and $\Gamma$ the work +alphabet. Radix bases are chosen from the full cell alphabet, including +all history and mark tracks. + +\subsection{A one-cell code and its exact inverse test} + +A partial one-cell table sends a control and a scanned letter to a +new control, a written letter, and a displacement. Its two incoming +conditions are: the new control determines the displacement, and the +new control together with the written letter determines the entire +old control and read letter. + +\begin{lemma}[From one-cell tables to closed rectangles] +\label{lem:bb-onecell}\label{lem:bb-mid-prefix} +Give an alphabet of size $m$ digits separated by at least two, between +$0$ and $B-2$. Put +\[ +E(A)=\sum_{j\ge1}d(A_j)B^{-j},\qquad +I(v)=\sum_{j=1}^{|v|}d(v_j)B^{-j}+[0,B^{-|v|}]. +\] +For a head-relative tape $(a_j)_{j\in\Z}$ use the coordinates +\[ + (x,y)=\big(E(a_{-1}a_{-2}\cdots),E(a_0a_1\cdots)\big). +\] +After writing at cell zero, a displacement $e$ makes the old cell $e$ +the new cell zero. +A deterministic partial one-cell table satisfying the incoming +conditions acts, after separate state translations, by a finite list +of reciprocal affine maps on full closed rectangles. Each source +family and each target family is separately disjoint. For a rule +reading $a$, writing $b$, and moving by $e$, the branches are +\begin{equation}\label{eq:bb-one-cell-branches} +\begin{array}{c|c|c|c} +e&\text{source}&\text{target}&(x',y')\\\hline +1 &[0,1]\times I(a)&I(b)\times[0,1] + &((d(b)+x)/B,By-d(a))\\ +0 &[0,1]\times I(a)&[0,1]\times I(b) + &(x,y+(d(b)-d(a))/B)\\ +-1&I(c)\times I(a)&[0,1]\times(d(c)+I(b))/B + &(Bx-d(c),[d(c)+(d(b)+By-d(a))/B]/B). +\end{array} +\end{equation} +The last row is split over every letter $c$. Each linear part is +$\operatorname{diag}(B^{-e},B^e)$. With odd digits +$1,3,\ldots,2m-1$ and $B\in\{2m+1,2m+2\}$, every code lies in +the interior of its applicable rectangle, and unscaled within-family +gaps are at least $B^{-2}$. +\end{lemma} +\begin{proof} +The identity $E(aA)=(d(a)+E(A))/B$ gives all three formulas and their +action on the independent tails. At the first differing digit, +incompatible prefix intervals have a positive gap: the digit difference +is at least two while the child interval has length one in that scale. +Thus the source state, read letter, and any left split separate sources. +At one target state the displacement is fixed. For a right or stationary +move the written letter separates the relevant target coordinate; +for a left move the pair $(c,b)$ separates the two-letter prefix. +The incoming inverse condition rules out duplicates. These arguments +concern the complete intervals, including their endpoints. +For odd digits the infinite tail lies between $1/(B-1)$ and +$(2m-1)/(B-1)<1$, proving interiority. The longest target prefix has +length two, giving the asserted gap. Uniformly scaling state squares +by $s$ scales gaps by $s$ and leaves the linear parts unchanged. +\end{proof} + +\subsection{Moving the work head before recording} + +We first place a frontier to the right of every possible work head. +A temporary mark lets the history excursion return to the correct +cell even though its length grows with the computation. + +\begin{lemma}[Move-first recorder] +\label{lem:bb-movefirst}\label{lem:bb-mid-right-compiler} +There is an effective one-cell table satisfying the incoming conditions +whose checkpoint configurations simulate the work machine. After +$n$ work steps its frontier is at $n+2$, its records occupy +$2,\ldots,n+1$, and its temporary marks are all zero. Each nonhalting +checkpoint has a finite positive continuation to the next one, with +no intervening checkpoint. +\end{lemma} +\begin{proof} +Use letters $(c,\ell,j)$ with work letter $c$, history +$\ell\in\{e,F\}\sqcup\{r\}$ and mark $j\in\{0,1\}$. +The controls are $C_q,J_r,S_r,P_p,T_p$. The complete partial table is +\[ +\begin{array}{lll} + C_q:(a,\ell,0)&\mapsto J_r:(b_r,\ell,0),d_r&\ell\ne F,\\ + J_r:(c,\ell,0)&\mapsto S_r:(c,\ell,1),1&\ell\ne F,\\ + S_r:(c,\ell,0)&\mapsto S_r:(c,\ell,0),1&\ell\ne F,\\ + S_r:(c,F,0)&\mapsto P_{p_r}:(c,r,0),1,&\\ + P_p:(c,e,0)&\mapsto T_p:(c,F,0),-1,&\\ + T_p:(c,\ell,0)&\mapsto T_p:(c,\ell,0),-1&\ell\ne F,\\ + T_p:(c,\ell,1)&\mapsto C_p:(c,\ell,0),0&\ell\ne F. +\end{array} +\] +Initialize the frontier at two, all other history empty, and all marks +zero. At checkpoint $n$ the head satisfies $|h_n|\le n$. +The first row updates the work tape and moves to $h'\le n+1$, strictly +left of the frontier $f=n+2$. The second row marks $h'$. The forward +scan writes $r$ at $f$, puts a new frontier at $f+1$, and the backward +scan stops exactly at the mark, which it clears. Its length is +$2(f-h')+4$, so it is finite and positive. This proves the invariant +inductively, including the exact work update and halt equivalence. + +For the full-domain inverse, $r$ recovers the old work pair into $J_r$; +mark one distinguishes entry into $S_r$ from its mark-zero loop; +a written record determines entry into $P_p$; a written $F$ +distinguishes entry into $T_p$ from its loop; and entry into $C_p$ +uniquely restores mark one. Every destination has the single incoming +displacement displayed in the table. No invariant on remote cells +was used. +\end{proof} + +\subsection{Recording before the work-head displacement} + +The next table leaves its mark at the old work head. Its final step +performs the work displacement; storing that displacement in the +checkpoint control keeps incoming moves unambiguous. + +\begin{lemma}[Direction-recording scan] +\label{lem:bb-direction}\label{lem:bb-mid-mark-compiler} +Let $D=\{-1,0,1\}$ and let $O$ be either a singleton or $\{0,1\}$. +An effective one-cell table on +$\Gamma\times(\{\bot,F\}\sqcup\mathcal R)\times\{0,1\}\times O$, +where $\mathcal R=\{(q,e,a):q\text{ nonhalting},e\in D,a\in\Gamma\}$, +satisfies the incoming conditions. It preserves the $O$ track in +its physical cells. Starting with frontier at one, its checkpoint +at work step $n$ has frontier $n+1$, records at $1,\ldots,n$, zero +marks, and the correct work configuration. A step from head $i$ +takes $5+2(n-i)$ local rules. +\end{lemma} +\begin{proof}\label{proof:bb-mid-mark-compiler} +Use main controls $M(q,e)$, forward controls $G(r)$, and $A(p,d),B(p,d)$ +for every work state $p$ and $d\in D$, including pairs not produced by +a work instruction. Copy the $O$ component in each row. In the first +three rows use every $r=(q,e,a)\in\mathcal R$ with +$\delta(q,a)=(p,c,d)$. The final three rows apply independently to +every $(p,d)\in Q\times D$, including pairs that are not outputs of +any work instruction. The variables $v\in\Gamma$ and +$h\in\{\bot,F\}\sqcup\mathcal R$ range over all letters subject to +the displayed restrictions. The whole table is +\[ +\begin{array}{lll} + M(q,e):(a,h,0)&\mapsto G(r):(c,h,1),1&h\ne F,\\ + G(r):(v,h,0)&\mapsto G(r):(v,h,0),1&h\ne F,\\ + G(r):(v,F,0)&\mapsto A(p,d):(v,r,0),1,&\\ + A(p,d):(v,\bot,0)&\mapsto B(p,d):(v,F,0),-1,&\\ + B(p,d):(v,h,0)&\mapsto B(p,d):(v,h,0),-1&h\ne F,\\ + B(p,d):(v,h,1)&\mapsto M(p,d):(v,h,0),d&h\ne F. +\end{array} +\] +Starting at $M(q_0,0)$, the first row writes and marks the old head. +The scan reaches frontier $j=n+1$, stores its record, creates the +next frontier, returns to the unique mark and makes the work move. +The count is $2(j-i)+3=5+2(n-i)$. Since $i\le nh_n$, +the forward scan reaches the frontier, moves it one cell right, and +the return scan reaches the mark. Its neighboring-cell guard holds +because the new frontier is $p_n+1$. The last rule clears the mark +and performs the work displacement. Counting the two traversals and +endpoint actions gives $2(p_n-h_n)+1$; induction gives $|h_n|\le n$. + +For injectivity on all relative tapes, into Go the old cursor is +output offset $-1$: a mark there distinguishes entry from the loop, +and $r$ restores the overwritten work letter and old Ready control. +Into Back the old cursor is offset $+1$. An output pointer at $+2$ +identifies a log insertion, with record at $+1$ recovering $r$ and +old pair $(P,E)$. A nonpointer at $+2$ identifies the guarded loop. +Into Ready the corresponding Back control uniquely restores the +cleared mark. These are all possibilities, proving the full inverse. +\end{proof} + +\begin{lemma}[Closed rectangles for a finite guarded window] +\label{lem:bb-window-rectangles} +Suppose a finite partial injective local tape map reads and writes only +indices $[-a,b-1]$, then shifts the cursor by $e$, where +$-a\le e\le b$. Split its domain by every complete word in that +window. With gapped radix coding the input prefix lengths are +$(a,b)$ and output lengths $(a+e,b-e)$; the affine branch has linear +part $\operatorname{diag}(B^{-e},B^e)$. Both full rectangle families +are separately disjoint. +\end{lemma} +\begin{proof} +After the writes the new left prefix consists of old indices +$e-1,e-2,\ldots,-a$, and the right prefix of $e,e+1,\ldots,b-1$. +The two remaining tails are free and unchanged, so the image is the +whole output cylinder. Distinct source cylinders have disjoint +images by injectivity. If both pairs of output prefixes were +compatible, a common extension would belong to both images; hence +at least one coordinate has incompatible prefixes. The gapped +interval argument then separates the complete closed rectangles. +The prefix lengths give the two reciprocal scales directly. +\end{proof} diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/geometry.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/geometry.tex index 15679a19e3b23d1eecc4a9870493408bd295f241..ab1e31ba19b988aca78d3a8deab154c8428d85ab 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/geometry.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/geometry.tex @@ -1,324 +1,324 @@ -\section{Transporting complete boxes} -\label{sec:bb-boxes} - -We allow an instruction to change all three side lengths while -preserving their product. The geometric question is concrete: -given finitely many disjoint closed boxes and prescribed determinant-one -affine maps onto another disjoint family, can one carry out every map by -a smooth incompressible motion? The two families may overlap each -other. Consequently a destination cannot be occupied until every source -that obstructs it has been removed. - -We first extend an explicitly separated motion to a velocity field. -We then give two ways to produce that motion. Separate horizontal -projections permit a simple lift and transfer. For general solid boxes -we use parking and detours around the other occupied boxes. -Figure~\ref{fig:bb-routing} contrasts the height separation with the -evacuation order that makes overlapping source and target families harmless. - -\subsection{An affine motion and its curl extension} - -Throughout, a smooth switch is obtained by translating and rescaling -$\sigma$ in \eqref{eq:p2-step}, with its transition strictly inside -its allotted interval. Thus successive motions are stationary on -neighborhoods of their joining times. - -\begin{lemma}[A velocity on a neighborhood of each moving box] -\label{lem:bb-curl} -Let $K_i(t)=c_i(t)+D_i(t)K_i^0$, $1\le i\le N$, where $K_i^0$ is -an axis-parallel box, possibly with zero side lengths, and $D_i(t)$ is -positive diagonal with determinant one. Suppose the centers and scales -are smooth and constant near the ends of a compact time interval. -For the effective conclusion, assume computable endpoints for the -initial boxes $K_i^0$, effective evaluation of the centers' and scales' -derivatives, bounds for those derivatives, and positive lower bounds -for every scale on that interval. -If the coordinatewise $2\eta$ enlargements of the moving boxes are -pairwise disjoint and bounded, for a known $\eta>0$, a -compactly supported smooth divergence-free field realizes these motions -on open neighborhoods of the initial boxes. It vanishes near the time -endpoints. When all enlarged boxes lie in an interior torus chart, -the field extends to a mean-zero field on that torus. -The field is effective under the effective-data assumptions above; -smooth existence itself does not require computable data. -\end{lemma} -\begin{proof} -Choose a smooth cutoff $\chi_i$ which is one on the $\eta$ -enlargement of $K_i(t)$ and supported in its $2\eta$ enlargement, -using products of the effective switches. Put $y=x-c_i$ and -$L_i=\dot D_iD_i^{-1}$; then $\operatorname{tr}L_i=0$. -Take the curl of -\begin{equation}\label{eq:bb-potential} - \chi_i\{\tfrac12\dot c_i\times y+\tfrac13(L_i y)\times y\}. -\end{equation} -The identity -$\nabla\times(G\times y)=2G-y\operatorname{div}G+(y\cdot\nabla)G$ -shows that this curl equals $\dot c_i+L_i y$ on the plateau. -Disjoint supports permit summation over $i$. The path -$c_i(t)+D_i(t)D_i(t_0)^{-1}(x_0-c_i(t_0))$ has exactly that -derivative and remains in its prescribed box. ODE uniqueness proves -the formula. On the compact time interval the normalized matrix norm -has a finite bound $M$, computable under the effective-data assumptions; -an initial perturbation less than -$\eta/(2M)$ remains on the plateau. This proves the neighborhood -claim, also for zero-width initial boxes. -The potential is zero near time endpoints because the centers and -scales are constant there. Its support is bounded. Within a torus -chart extend the potentials by zero; periodic curls have zero integral. -\end{proof} - -\begin{proposition}[Fluid realization and a pressure refinement] -\label{prop:bb-realization} -Fix $\nu>0$. A finite collection of the preceding motions, repeated -with period one, or preceded by one finite loading interval, yields a -smooth prescribed velocity $U$ with $U(0)=0$. Suppose its spatial -support is contained in one compact set in $\R^3$, or its potentials -are contained in interior torus charts. Then -\begin{equation}\label{eq:bb-residual} - f=U_t+(U\cdot\nabla)U-\nu\Delta U -\end{equation} -has the same support and time periodicity and bounded mixed derivatives -of every order. The solution is $(U,0)$, with the uniqueness and -material-flow conclusions of Lemma~\ref{lem:p2-realization}. -In $\R^3$ its uniqueness statement also holds when the competing -pressure has an $H^1$ representative at each time, without requiring -continuity into $H^1$, while the other velocity hypotheses of -Section~\ref{sec:model} are retained. On the torus $U$ and $f$ have -zero mean. Under the effective-data assumptions of Lemma~\ref{lem:bb-curl}, -the formula is $f_0+\nu f_1$ with effective coefficients; -evaluation is effective for computable $\nu$, and relative to $\nu$ -otherwise. -\end{proposition} -\begin{proof} -The residual identity and Lemma~\ref{lem:b-comparison} give existence, -uniqueness and the material flow. In the whole-space pressure refinement -use case (ii): the reference field has compact support and smooth bounded -mixed derivatives, hence belongs to $CH^2\cap C^1L^2$ with bounded -velocity and gradient; the competing velocity and pointwise $H^1$ -pressure are exactly that case's hypotheses. No pressure continuity -in time is added. Integrating the residual on a torus uses -$\int U=0$, $\int\Delta U=0$ and -$\int(U\cdot\nabla)U=\int\operatorname{div}(U\otimes U)=0$. - -Each derivative of $e^{-1/s}$ for $s>0$ is a polynomial in $1/s$ -times $e^{-1/s}$. The estimate -$v^m e^{-v}\le(m+k)!v^{-k}$ supplies effective endpoint error bounds, -and the denominator defining $\sigma$ is at least $e^{-2}$. -Under those effective-data assumptions all positive scales have effective -positive lower bounds. Finite -compositions, differentiation, and the logarithms of positive computable -scales are therefore effective. At an approximate time, a coarse bound -gives a finite superset of possible translates of the zero-extended -unit template. This evaluates the periodic field without deciding -whether time is an integer. None of these operations iterates the -machine execution: the program contains the entire finite instruction -list. -\end{proof} - -\subsection{Three geometric schedules} - -For later use we describe the exact hypotheses of the schedules. -First suppose $P_i,Q_i\subset\R^2$ are finite families of closed -rectangles, possibly with zero side lengths, separately -pairwise disjoint, with affine maps -\[ - F_i(x)=q_i+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(x-p_i), - \qquad \lambda_i>0, -\] -where $p_i,q_i$ are their centers, and require $F_i(P_i)=Q_i$. -Their side lengths have a common finite bound. Source and target rectangles -belonging to different families need not be disjoint. The smooth existence -statements allow arbitrary real rectangle coordinates and scales. For the -effective conclusions, assume rational rectangle endpoints and rational -scales, together with computable chosen heights and vertical half-widths; -then a rational side-length bound and positive gap margins can be computed. - -\begin{lemma}[Separated projections and parking] -\label{lem:bb-lift-parking} -These reciprocal maps have the following smooth compactly supported -unit-time realizations, stationary on time collars. They are effective -under the rational-data assumptions just stated. -\begin{enumerate} -\item For the products $P_i\times[z_0-h,z_0+h]$, with common $h\ge0$ -and $z_0\in\R$, lift all boxes with their source projections fixed, -transfer them at sufficiently separated private heights, and lower -them with their target projections fixed. The middle motion may use -$g_i(s)=\lambda_i^s$ or $g_i(s)=1+s(\lambda_i-1)$ and matrix -$\operatorname{diag}(g_i,g_i^{-1},1)$, with the center interpolated -linearly in $s$. Each horizontal width stays between its endpoint -widths. In the linear case every point satisfies -\begin{equation}\label{eq:bb-convex-coordinate} - X_1(s)=(1-s)X_1(0)+sX_1(1). -\end{equation} -\item Alternatively, choose disjoint parking rectangles away from both -families. Move every source to its parking site before delivering any -target, using an upward, horizontal, and downward translation. Reshape -at the parking sites, then deliver by the same three translations. -This uses $7N$ motions. The same schedule applies to assigned onto positive -diagonal determinant-one maps between rational solid boxes whose centers -lie in a common horizontal plane and whose source and target horizontal -projections are separately disjoint. Require both horizontal widths during -reshaping to stay inside their reserved parking rectangles and the travel -height above the common plane to exceed twice the largest vertical -half-width. -\end{enumerate} -Both constructions act on neighborhoods of the whole source boxes. -\end{lemma} -\begin{proof} -In the first construction choose padding smaller than a quarter of -the within-family horizontal gaps and choose height differences larger -than twice the padded vertical half-width. During lift and descent, -horizontal separation suffices. During transfer, vertical separation -suffices. The two choices of $g_i$ are positive and monotone between -$1$ and $\lambda_i$; their reciprocals are monotone as well. Thus the -width claim holds. The affine path is -$c_i(s)+\operatorname{diag}(g_i,g_i^{-1},1)(X_0-c_i(0))$. -Its first coordinate gives \eqref{eq:bb-convex-coordinate} exactly for -the linear choice. Lemma~\ref{lem:bb-curl} now applies. - -For the second construction choose padding smaller than the gaps in -each union of source projections with reserved parking rectangles, and -of target projections with those rectangles. In the evacuation pass, -vertical motion is separated from uncollected sources and parked boxes. -During delivery it is separated from already filled targets and boxes -still parked. At travel height, the moving box is above every resting -box. A shape change stays within its own reserved projection. -These are the occupied sets at each actual time; source--target -intersection causes no interference because all sources have left -before delivery starts. Lemma~\ref{lem:bb-curl} applies to each motion -with every other box fixed. Finite concatenation gives the required -neighborhood map. -\end{proof} - -\begin{figure}[ht] -\centering -\begin{tikzpicture}[x=.75cm,y=.65cm,>=stealth,font=\small] - \draw[gray] (0,0)--(5,0); - \draw[thick,blue!65!black,->] (.5,0)--(.5,2.4)--(4.2,2.4)--(4.2,.1); - \draw[thick,orange!80!black,->] (1.6,0)--(1.6,1.3)--(3.1,1.3)--(3.1,.1); - \node[below] at (1,0) {sources}; - \node[below] at (3.7,0) {targets}; - \node[above] at (2.5,2.6) {Separate height layers}; - \begin{scope}[xshift=5.1cm] - \node[draw,rounded corners,minimum width=2.6cm] (s) at (1.8,2.8) {all source boxes}; - \node[draw,rounded corners,minimum width=2.6cm] (p) at (1.8,1.1) {all boxes parked}; - \node[draw,rounded corners,minimum width=2.6cm] (t) at (1.8,-.6) {all target boxes}; - \draw[->,thick] (s)--(p) node[midway,right] {evacuate}; - \draw[->,thick] (p)--(t) node[midway,right] {deliver}; - \end{scope} -\end{tikzpicture} -\caption{The two separation mechanisms in -Lemma~\ref{lem:bb-lift-parking}. Left: horizontal projections separate -the lift and descent, and distinct heights separate the transfer; -the second horizontal coordinate is suppressed. Right: a temporal -schedule, not a spatial drawing. Every source is removed before any -target is occupied, so overlap between the two families is harmless.} -\label{fig:bb-routing} -\end{figure} - -There is also a useful stream-function implementation of the first -schedule. Suppose $a_i,a_i'$ are the lower left corners of $P_i,Q_i$. -At height $z_i$, put $d_i=a_i'-a_i$, $b_i=\log\lambda_i$ and -$g_i(s)=a_i+sd_i$. On a neighborhood of the swept rectangle use -\begin{equation}\label{eq:bb-mid-stream-path} - \xi_i(s)=g_i(s)+\operatorname{diag}(\lambda_i^s,\lambda_i^{-s})(\xi-a_i). -\end{equation} -The Hamiltonian -\[ - H_i=d_{i1}y-d_{i2}x+b_i(x-g_{i1})(y-g_{i2}) -\] -generates this motion. Put $\widehat H_i=\chi_iH_i$, where $\chi_i$ -is one near the whole sweep and supported in its reserved layer. Use -\[ - \dot s(\partial_y\widehat H_i,-\partial_x\widehat H_i,0). -\] -Disjoint height layers separate these cutoffs. Vertical phases use -$\nabla\times(0,x\chi_i,0)$, equal to $(0,0,1)$ on their column -plateaus. If $M=\max_i(1,\lambda_i,\lambda_i^{-1})$ and the column -plateau margins are $\delta,\delta'$, an initial thickening smaller than -$\min(\delta/(4M),\delta'/(4M),1/(8M))$ stays on every plateau. -Thus the affine formula also holds on open neighborhoods. For points of -the original rectangle it has the additional bound -\begin{equation}\label{eq:bb-mid-stream-safe} - X_1(s)\ge\min(a_{i1},a'_{i1}). -\end{equation} -The bound follows because the first relative coordinate in -\eqref{eq:bb-mid-stream-path} is nonnegative. - -\subsection{Solid boxes with overlapping projections} - -The preceding lift requires separate horizontal projections. We now -remove that restriction; this is needed when all three coordinates -encode independent stacks. - -\begin{theorem}[Expansion, obstacle detours, and delivery] -\label{lem:bb-balanced-box-routing} -Let $Q_i=a_i+\prod_{j=1}^3[-h_{ij},h_{ij}]$ and -$Q_i'=b_i+\prod_{j=1}^3[-k_{ij},k_{ij}]$ be rational solid boxes, -with positive half-widths, each family pairwise disjoint. Suppose -$s_{ij}=k_{ij}/h_{ij}$ and $\prod_js_{ij}=1$. An effective compactly -supported smooth solenoidal field, zero outside the time interval -$[1/4,3/4]$, has time-one map -\[ - a_i+y\longmapsto b_i+\operatorname{diag}(s_{i1},s_{i2},s_{i3})y -\] -on a neighborhood of each $Q_i$. -\end{theorem} -\begin{proof}\label{proof:bb-balanced-box-routing} -For an empty family take the zero field. Otherwise let -$R=\max(1,h_{ij},k_{ij})$. Choose a rational $\Lambda>10$ so that -centers in each of the lists $(\Lambda a_i)$ and $(\Lambda b_i)$ -are separated by more than $20R$ in maximum norm. This is possible -because distinct disjoint boxes have distinct centers. Let -\[ - K=\max(4\Lambda,|\Lambda a_{i1}|,|\Lambda b_{i1}|), - \qquad d_i=(K+30Ri,0,0). -\] -The $d_i$ are parking centers, separated from each other and both scaled -families by at least $30R$ in their first coordinate. - -First multiply all source centers by a common factor increasing from -$1$ to $\Lambda$, while keeping shapes fixed. Next change all shapes -at those separated centers by factors $s_{ij}^{\theta}$, -$0\le\theta\le1$. Each half-width is the geometric interpolation -$h_{ij}^{1-\theta}k_{ij}^{\theta}\le R$. -Move the boxes one at a time from $\Lambda a_i$ to $d_i$, then, -after all have been parked, from $d_i$ to $\Lambda b_i$. -Finally shrink all target centers by a common factor from $\Lambda$ -to $1$, with target shapes fixed. - -Here is a finite construction of every single-box path. Around each -stationary center place the closed cube of maximum-norm radius $4R$. -These cubes are disjoint since stationary centers are separated by -more than $20R$, and neither endpoint is in a cube. Start with the -straight segment between the endpoints. Its intersections with the -cubes are rational intervals, found by linear inequalities. Replace -each portion inside a cube by a polygonal path on its boundary: join -the entry point to a vertex of a containing face, use cube edges to a -face containing the exit point, and join to that exit point. The cube -is convex face by face, so these segments remain on its boundary; -no other cube meets them. A tangency needs no replacement. The -resulting rational polygonal path has distance at least $4R$ from -every stationary center. Traverse its segments with flat switches, -stopping to all orders at vertices. Its geometric image is unchanged, -so smoothing the time parameter does not reduce clearance. - -For a uniform localization margin, let $g_a$ be the minimum over -$i0$, a +compactly supported smooth divergence-free field realizes these motions +on open neighborhoods of the initial boxes. It vanishes near the time +endpoints. When all enlarged boxes lie in an interior torus chart, +the field extends to a mean-zero field on that torus. +The field is effective under the effective-data assumptions above; +smooth existence itself does not require computable data. +\end{lemma} +\begin{proof} +Choose a smooth cutoff $\chi_i$ which is one on the $\eta$ +enlargement of $K_i(t)$ and supported in its $2\eta$ enlargement, +using products of the effective switches. Put $y=x-c_i$ and +$L_i=\dot D_iD_i^{-1}$; then $\operatorname{tr}L_i=0$. +Take the curl of +\begin{equation}\label{eq:bb-potential} + \chi_i\{\tfrac12\dot c_i\times y+\tfrac13(L_i y)\times y\}. +\end{equation} +The identity +$\nabla\times(G\times y)=2G-y\operatorname{div}G+(y\cdot\nabla)G$ +shows that this curl equals $\dot c_i+L_i y$ on the plateau. +Disjoint supports permit summation over $i$. The path +$c_i(t)+D_i(t)D_i(t_0)^{-1}(x_0-c_i(t_0))$ has exactly that +derivative and remains in its prescribed box. ODE uniqueness proves +the formula. On the compact time interval the normalized matrix norm +has a finite bound $M$, computable under the effective-data assumptions; +an initial perturbation less than +$\eta/(2M)$ remains on the plateau. This proves the neighborhood +claim, also for zero-width initial boxes. +The potential is zero near time endpoints because the centers and +scales are constant there. Its support is bounded. Within a torus +chart extend the potentials by zero; periodic curls have zero integral. +\end{proof} + +\begin{proposition}[Fluid realization and a pressure refinement] +\label{prop:bb-realization} +Fix $\nu>0$. A finite collection of the preceding motions, repeated +with period one, or preceded by one finite loading interval, yields a +smooth prescribed velocity $U$ with $U(0)=0$. Suppose its spatial +support is contained in one compact set in $\R^3$, or its potentials +are contained in interior torus charts. Then +\begin{equation}\label{eq:bb-residual} + f=U_t+(U\cdot\nabla)U-\nu\Delta U +\end{equation} +has the same support and time periodicity and bounded mixed derivatives +of every order. The solution is $(U,0)$, with the uniqueness and +material-flow conclusions of Lemma~\ref{lem:p2-realization}. +In $\R^3$ its uniqueness statement also holds when the competing +pressure has an $H^1$ representative at each time, without requiring +continuity into $H^1$, while the other velocity hypotheses of +Section~\ref{sec:model} are retained. On the torus $U$ and $f$ have +zero mean. Under the effective-data assumptions of Lemma~\ref{lem:bb-curl}, +the formula is $f_0+\nu f_1$ with effective coefficients; +evaluation is effective for computable $\nu$, and relative to $\nu$ +otherwise. +\end{proposition} +\begin{proof} +The residual identity and Lemma~\ref{lem:b-comparison} give existence, +uniqueness and the material flow. In the whole-space pressure refinement +use case (ii): the reference field has compact support and smooth bounded +mixed derivatives, hence belongs to $CH^2\cap C^1L^2$ with bounded +velocity and gradient; the competing velocity and pointwise $H^1$ +pressure are exactly that case's hypotheses. No pressure continuity +in time is added. Integrating the residual on a torus uses +$\int U=0$, $\int\Delta U=0$ and +$\int(U\cdot\nabla)U=\int\operatorname{div}(U\otimes U)=0$. + +Each derivative of $e^{-1/s}$ for $s>0$ is a polynomial in $1/s$ +times $e^{-1/s}$. The estimate +$v^m e^{-v}\le(m+k)!v^{-k}$ supplies effective endpoint error bounds, +and the denominator defining $\sigma$ is at least $e^{-2}$. +Under those effective-data assumptions all positive scales have effective +positive lower bounds. Finite +compositions, differentiation, and the logarithms of positive computable +scales are therefore effective. At an approximate time, a coarse bound +gives a finite superset of possible translates of the zero-extended +unit template. This evaluates the periodic field without deciding +whether time is an integer. None of these operations iterates the +machine execution: the program contains the entire finite instruction +list. +\end{proof} + +\subsection{Three geometric schedules} + +For later use we describe the exact hypotheses of the schedules. +First suppose $P_i,Q_i\subset\R^2$ are finite families of closed +rectangles, possibly with zero side lengths, separately +pairwise disjoint, with affine maps +\[ + F_i(x)=q_i+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(x-p_i), + \qquad \lambda_i>0, +\] +where $p_i,q_i$ are their centers, and require $F_i(P_i)=Q_i$. +Their side lengths have a common finite bound. Source and target rectangles +belonging to different families need not be disjoint. The smooth existence +statements allow arbitrary real rectangle coordinates and scales. For the +effective conclusions, assume rational rectangle endpoints and rational +scales, together with computable chosen heights and vertical half-widths; +then a rational side-length bound and positive gap margins can be computed. + +\begin{lemma}[Separated projections and parking] +\label{lem:bb-lift-parking} +These reciprocal maps have the following smooth compactly supported +unit-time realizations, stationary on time collars. They are effective +under the rational-data assumptions just stated. +\begin{enumerate} +\item For the products $P_i\times[z_0-h,z_0+h]$, with common $h\ge0$ +and $z_0\in\R$, lift all boxes with their source projections fixed, +transfer them at sufficiently separated private heights, and lower +them with their target projections fixed. The middle motion may use +$g_i(s)=\lambda_i^s$ or $g_i(s)=1+s(\lambda_i-1)$ and matrix +$\operatorname{diag}(g_i,g_i^{-1},1)$, with the center interpolated +linearly in $s$. Each horizontal width stays between its endpoint +widths. In the linear case every point satisfies +\begin{equation}\label{eq:bb-convex-coordinate} + X_1(s)=(1-s)X_1(0)+sX_1(1). +\end{equation} +\item Alternatively, choose disjoint parking rectangles away from both +families. Move every source to its parking site before delivering any +target, using an upward, horizontal, and downward translation. Reshape +at the parking sites, then deliver by the same three translations. +This uses $7N$ motions. The same schedule applies to assigned onto positive +diagonal determinant-one maps between rational solid boxes whose centers +lie in a common horizontal plane and whose source and target horizontal +projections are separately disjoint. Require both horizontal widths during +reshaping to stay inside their reserved parking rectangles and the travel +height above the common plane to exceed twice the largest vertical +half-width. +\end{enumerate} +Both constructions act on neighborhoods of the whole source boxes. +\end{lemma} +\begin{proof} +In the first construction choose padding smaller than a quarter of +the within-family horizontal gaps and choose height differences larger +than twice the padded vertical half-width. During lift and descent, +horizontal separation suffices. During transfer, vertical separation +suffices. The two choices of $g_i$ are positive and monotone between +$1$ and $\lambda_i$; their reciprocals are monotone as well. Thus the +width claim holds. The affine path is +$c_i(s)+\operatorname{diag}(g_i,g_i^{-1},1)(X_0-c_i(0))$. +Its first coordinate gives \eqref{eq:bb-convex-coordinate} exactly for +the linear choice. Lemma~\ref{lem:bb-curl} now applies. + +For the second construction choose padding smaller than the gaps in +each union of source projections with reserved parking rectangles, and +of target projections with those rectangles. In the evacuation pass, +vertical motion is separated from uncollected sources and parked boxes. +During delivery it is separated from already filled targets and boxes +still parked. At travel height, the moving box is above every resting +box. A shape change stays within its own reserved projection. +These are the occupied sets at each actual time; source--target +intersection causes no interference because all sources have left +before delivery starts. Lemma~\ref{lem:bb-curl} applies to each motion +with every other box fixed. Finite concatenation gives the required +neighborhood map. +\end{proof} + +\begin{figure}[ht] +\centering +\begin{tikzpicture}[x=.75cm,y=.65cm,>=stealth,font=\small] + \draw[gray] (0,0)--(5,0); + \draw[thick,blue!65!black,->] (.5,0)--(.5,2.4)--(4.2,2.4)--(4.2,.1); + \draw[thick,orange!80!black,->] (1.6,0)--(1.6,1.3)--(3.1,1.3)--(3.1,.1); + \node[below] at (1,0) {sources}; + \node[below] at (3.7,0) {targets}; + \node[above] at (2.5,2.6) {Separate height layers}; + \begin{scope}[xshift=5.1cm] + \node[draw,rounded corners,minimum width=2.6cm] (s) at (1.8,2.8) {all source boxes}; + \node[draw,rounded corners,minimum width=2.6cm] (p) at (1.8,1.1) {all boxes parked}; + \node[draw,rounded corners,minimum width=2.6cm] (t) at (1.8,-.6) {all target boxes}; + \draw[->,thick] (s)--(p) node[midway,right] {evacuate}; + \draw[->,thick] (p)--(t) node[midway,right] {deliver}; + \end{scope} +\end{tikzpicture} +\caption{The two separation mechanisms in +Lemma~\ref{lem:bb-lift-parking}. Left: horizontal projections separate +the lift and descent, and distinct heights separate the transfer; +the second horizontal coordinate is suppressed. Right: a temporal +schedule, not a spatial drawing. Every source is removed before any +target is occupied, so overlap between the two families is harmless.} +\label{fig:bb-routing} +\end{figure} + +There is also a useful stream-function implementation of the first +schedule. Suppose $a_i,a_i'$ are the lower left corners of $P_i,Q_i$. +At height $z_i$, put $d_i=a_i'-a_i$, $b_i=\log\lambda_i$ and +$g_i(s)=a_i+sd_i$. On a neighborhood of the swept rectangle use +\begin{equation}\label{eq:bb-mid-stream-path} + \xi_i(s)=g_i(s)+\operatorname{diag}(\lambda_i^s,\lambda_i^{-s})(\xi-a_i). +\end{equation} +The Hamiltonian +\[ + H_i=d_{i1}y-d_{i2}x+b_i(x-g_{i1})(y-g_{i2}) +\] +generates this motion. Put $\widehat H_i=\chi_iH_i$, where $\chi_i$ +is one near the whole sweep and supported in its reserved layer. Use +\[ + \dot s(\partial_y\widehat H_i,-\partial_x\widehat H_i,0). +\] +Disjoint height layers separate these cutoffs. Vertical phases use +$\nabla\times(0,x\chi_i,0)$, equal to $(0,0,1)$ on their column +plateaus. If $M=\max_i(1,\lambda_i,\lambda_i^{-1})$ and the column +plateau margins are $\delta,\delta'$, an initial thickening smaller than +$\min(\delta/(4M),\delta'/(4M),1/(8M))$ stays on every plateau. +Thus the affine formula also holds on open neighborhoods. For points of +the original rectangle it has the additional bound +\begin{equation}\label{eq:bb-mid-stream-safe} + X_1(s)\ge\min(a_{i1},a'_{i1}). +\end{equation} +The bound follows because the first relative coordinate in +\eqref{eq:bb-mid-stream-path} is nonnegative. + +\subsection{Solid boxes with overlapping projections} + +The preceding lift requires separate horizontal projections. We now +remove that restriction; this is needed when all three coordinates +encode independent stacks. + +\begin{theorem}[Expansion, obstacle detours, and delivery] +\label{lem:bb-balanced-box-routing} +Let $Q_i=a_i+\prod_{j=1}^3[-h_{ij},h_{ij}]$ and +$Q_i'=b_i+\prod_{j=1}^3[-k_{ij},k_{ij}]$ be rational solid boxes, +with positive half-widths, each family pairwise disjoint. Suppose +$s_{ij}=k_{ij}/h_{ij}$ and $\prod_js_{ij}=1$. An effective compactly +supported smooth solenoidal field, zero outside the time interval +$[1/4,3/4]$, has time-one map +\[ + a_i+y\longmapsto b_i+\operatorname{diag}(s_{i1},s_{i2},s_{i3})y +\] +on a neighborhood of each $Q_i$. +\end{theorem} +\begin{proof}\label{proof:bb-balanced-box-routing} +For an empty family take the zero field. Otherwise let +$R=\max(1,h_{ij},k_{ij})$. Choose a rational $\Lambda>10$ so that +centers in each of the lists $(\Lambda a_i)$ and $(\Lambda b_i)$ +are separated by more than $20R$ in maximum norm. This is possible +because distinct disjoint boxes have distinct centers. Let +\[ + K=\max(4\Lambda,|\Lambda a_{i1}|,|\Lambda b_{i1}|), + \qquad d_i=(K+30Ri,0,0). +\] +The $d_i$ are parking centers, separated from each other and both scaled +families by at least $30R$ in their first coordinate. + +First multiply all source centers by a common factor increasing from +$1$ to $\Lambda$, while keeping shapes fixed. Next change all shapes +at those separated centers by factors $s_{ij}^{\theta}$, +$0\le\theta\le1$. Each half-width is the geometric interpolation +$h_{ij}^{1-\theta}k_{ij}^{\theta}\le R$. +Move the boxes one at a time from $\Lambda a_i$ to $d_i$, then, +after all have been parked, from $d_i$ to $\Lambda b_i$. +Finally shrink all target centers by a common factor from $\Lambda$ +to $1$, with target shapes fixed. + +Here is a finite construction of every single-box path. Around each +stationary center place the closed cube of maximum-norm radius $4R$. +These cubes are disjoint since stationary centers are separated by +more than $20R$, and neither endpoint is in a cube. Start with the +straight segment between the endpoints. Its intersections with the +cubes are rational intervals, found by linear inequalities. Replace +each portion inside a cube by a polygonal path on its boundary: join +the entry point to a vertex of a containing face, use cube edges to a +face containing the exit point, and join to that exit point. The cube +is convex face by face, so these segments remain on its boundary; +no other cube meets them. A tangency needs no replacement. The +resulting rational polygonal path has distance at least $4R$ from +every stationary center. Traverse its segments with flat switches, +stopping to all orders at vertices. Its geometric image is unchanged, +so smoothing the time parameter does not reduce clearance. + +For a uniform localization margin, let $g_a$ be the minimum over +$i1$, and terminal codes lie strictly in $(-2,-1)\times(0,1)$. -The infinite tails are in $[\delta,1-\delta]$, -$\delta=1/(D-1)$, so finite-prefix decoding of length $b$ is possible -from error less than $(\delta/2)D^{-b}$ without a boundary ambiguity. - -Split full triples $[-1,1]$. By -Lemma~\ref{lem:bb-window-rectangles}, source prefix lengths $(1,2)$ -become $(1+d,2-d)$, target cylinders are complete, and both rectangle -families have gaps at least $D^{-3}$. Thicken by $[-1/4,1/4]$, -lift to height $4i$, use linear middle scale -$\mu_i=1+(D^{-d_i}-1)\theta$, and descend. Padding smaller than -$D^{-3}/4$ separates all phases. Repeating the pulse, zero on -$[0,1/12]$ and $[11/12,1]$, gives periodicity from zero without -loading. At a nonhalting branch's lift and descent the first -coordinate is greater than one; during the middle transfer the -particle's third coordinate is exactly $4i$, outside $(-1,1)$. -Thus even a horizontal route passing over the observation box cannot -cause a false visit. The integer samples and the compiler invariant -prove halting equivalence. The record count and pointer site $n+3$ -recover absolute indexing at checkpoints. - -For uniqueness apply Lemma~\ref{lem:b-comparison}(iii). -The explicit compactly supported smooth field lies in $CH^4\cap C^1H^2$ -on every finite interval and has pressure zero. The competing class -in the theorem is precisely case (iii), including its distributional -$CL^2$ gradient. Lemma~\ref{lem:b-gradient} cancels that gradient -against the solenoidal velocity difference without introducing a pressure -$L^2$ norm. Thus the velocity and pressure gradient are unique, while -pressure itself remains free up to a time-dependent spatial constant. -Proposition~\ref{prop:bb-late-logclock} gives the stated -decaying alternative. -\end{proof} - -\subsection{A flagged head and two-pass parking} -\label{sec:bb-two-pass-parking} - -\begin{theorem}\label{thm:bb-two-pass-parking} -For any machine and finitely specified initial tape, an effective -smooth force on $\R^3$, periodic from time zero, with one compact -spatial support and bounded mixed derivatives, realizes the event -\[ - \exists t\ge0:\quad X(t;0)\in(-6,-4)\times(-1,1)\times(-1,1) - \quad\Longleftrightarrow\quad\text{halting}. -\] -The zero-data velocity is unique among classical smooth solutions -with bounded $u,\nabla u$, $u\in C H^2\cap C^1L^2$ and $p\in C L^2$ -on every finite interval. The prescribed solution has uniformly -bounded kinetic energy. -\end{theorem} -\begin{proof}\label{proof:bb-two-pass-parking} -Merge terminal states into $q_\dagger$, send missing instructions -there by a stay step, and add a fresh initial state $q_*$, with no -incoming instruction, which writes back and stays while entering -the old initial state. For each normalized nonterminal rule -$r=(q,a)\mapsto(p_r,b_r,d_r)$ use cell letters -\[ - (w,h,j)\in\Gamma:=\mathcal A\times - (\{\circ,\star\}\sqcup\{[r]:r\in\mathcal R\})\times\{0,1\}, -\] -where $\mathcal A$ is the work alphabet and $\mathcal R$ is the -finite set of normalized instruction labels. Thus $\Gamma$ includes -all three tracks. -The complete table, with $h\ne\star$ wherever $h$ occurs, is -\[ -\begin{array}{lll} - M_q:(a,h,1)&\mapsto A_r:(b_r,h,0),d_r,\\ - A_r:(w,h,0)&\mapsto R_r:(w,h,1),1,\\ - R_r:(w,h,0)&\mapsto R_r:(w,h,0),1,\\ - R_r:(w,\star,0)&\mapsto B_{p_r}:(w,[r],0),1,\\ - B_q:(w,\circ,0)&\mapsto L_q:(w,\star,0),-1,\\ - L_q:(w,h,0)&\mapsto L_q:(w,h,0),-1,\\ - L_q:(w,h,1)&\mapsto M_q:(w,h,1),0. -\end{array} -\] -The destination fixes the incoming displacement. The record index -recovers the first row's old work pair; a flag distinguishes the -entries into $R_r$; a record distinguishes the entries into $B_q$; -a written frontier distinguishes entry and loop into $L_q$; -and entry into $M_q$ is the unique cell-preserving last row. -This proves the full one-cell inverse conditions. - -Initialize the sole flag at zero and the sole frontier at three, -with empty history elsewhere. At ready step $k$, only the work -head $i$ is flagged, records occupy $3,\ldots,k+2$, and the frontier -is $e_k=k+3$. The work rule clears the flag and moves to -$j=i+d_r\le e_k-2$, which is unflagged even for $d_r=0$. -It is flagged anew; the scan logs at $e_k$, advances the pointer, -returns, and preserves the ready flag. The two loop lengths are -$e_k-j-1$ and $e_k-j$, with five other rules, for a total -$2(e_k-j)+4$. Thus all scans terminate and ready halting is exact. - -Use odd digits in base $K=2|\Gamma|+2$, initial fractions $x_*,y_*$, -and state offsets $a_{M_{q_*}}=0$, $a_{M_{q_\dagger}}=-5$, and -$4,8,12,\ldots$ for the others. The code -$(a_s-x_*+x,y-y_*,0)$ starts at zero. Lemma~\ref{lem:bb-onecell} -gives $N$ full rectangle maps with scales $\lambda_i\in\{K^{-1},1,K\}$; -thicken by $[-1/4,1/4]$. The entire terminal block is inside the -observation box because $0-1$. Eventually constant tails are rational even for an -arbitrary finitely specified initial tape, and the pointer and record -count recover its absolute head position at ready checkpoints. - -For the required two-pass geometry, let $g$ be the minimum of one -and all positive separating coordinate gaps in either planar family, -and set $\epsilon=g/20$. Choose -\[ - C=20+\max(\{0\}\cup\{\text{right endpoints of source and target boxes}\}), - \qquad S_i=(C+4i,0,0). -\] -In a first pass give each source four consecutive slots: lift by ten, -translate above $S_i$, descend, and reshape there by the linear -reciprocal scale. In a second pass give each parked box three slots: -lift by ten, translate above its target, and descend. The total is -$7N$ slots, and every source is gone before a target is filled. -Widths remain at most one. Source gaps protect the first lift, -parking spacing protects strain and parking moves, target gaps protect -delivery, and travel height separates every horizontal transfer from -resting boxes. These inequalities remain strict under $2\epsilon$ -padding. Applying Lemma~\ref{lem:bb-curl} in chronological order -fixes every stationary box and gives the exact full-box map to the -moving one. Thus overlapping source and target families cause no -interference. - -Repeat the flat-ended template from zero and take its residual. -The integer sampling induction gives the table execution. In a -nonhalting run every endpoint box lies in $x_1>-1$; translations -interpolate corresponding endpoint points, strain occurs far to the -right, and waiting leaves the point fixed. The entire two-pass path -therefore remains in $x_1>-1$. A terminal sample lies inside the -observer. This proves the event for all times. - -For the exact pressure class use Lemma~\ref{lem:b-comparison}(iv). -The prescribed compactly supported smooth velocity satisfies the -reference $CH^2\cap C^1L^2$ and bounded-gradient conditions. The -competing velocity has precisely these bounds and a $CL^2$ pressure -representative, as that case requires. Its cutoff argument uses the -pressure itself, with error $CR^{-1}\|p\|_2\|u-U\|_2$, and assumes -no extra integrability of its derivative. The lemma gives velocity -uniqueness; the $L^2$ pressure representative is then zero. -Compact support and -bounded derivatives of the prescribed velocity give its uniform -energy bound and its global material flow. -\end{proof} - - -\subsection{A two-cell append and its guarded inverse}\label{bc:two-cell-section} -Use Lemma~\ref{lem:bb-window-compiler} with initial frontier $r_0=1$, -renaming Ready, Go and Back as $M,R,L$ and $(E,P)$ as -$(\perp,\star)$. The track order is work, mark, history. -The Ready history coordinate and the marked Back row remain unrestricted; -the unmarked Back row tests precisely the right-neighbor history. -Thus the full table, not only its initialized execution, is identical. -At checkpoint $n$, its frontier is $p_n=1+n>h_n$, where $h_n$ is -the work-head position, and the exact cycle length is $2(1+n-h_n)+1$. -Lemma~\ref{lem:bb-window-rectangles} applies with -$(a,b)=(1,2)$: the complete allowed triple fixes both history writes, -and its full image has prefix lengths $(1+e,2-e)$. - -\begin{theorem}[Guarded cylinders in a torus chart]\label{bc:guarded-torus} -On the unit torus, the fixed label $(1/8,0,0)$ and the strip -$\{x:x_1\pmod1\in(-1/4,0)\}$ realize halting by a general mean-zero -force with zero initial velocity and pressure. It is periodic after time -one and all its mixed derivatives are bounded. The torus comparison -class gives uniqueness. -\end{theorem} -\begin{proof}\label{bc:guarded-torus-proof} -Let $a$ be the augmented alphabet size and $k$ the control count in -Lemma~\ref{lem:bb-window-compiler}. Use zero-blank base $B=2a$, and enumerate -nonterminal and terminal states with positive integer indices $j\ge1$ -and first-coordinate offsets respectively -$L_0+3j$ and $-L_0-3j$, where $L_0=4k+ka^3+10$. -Refinement by triples gives at most $N=ka^3$ branches. Their source -prefix lengths are $(1,2)$ and image lengths $(1+e,2-e)$, so the -smallest possible grid gap is $B^{-3}$. If $P_i,Q_i$ are lower corners, -the branch map is -$Q_i+\diag(B^{-e_i},B^{e_i})(y-P_i)$. - -Choose three flat ramps $\alpha,\beta,\gamma$, active respectively in -$(1/8,1/4)$, $(3/8,5/8)$ and $(3/4,7/8)$. Set -\[ - c_i(t)=((1-\beta)P_i+\beta Q_i,\ i(\alpha-\gamma)), - \quad l_i=1-\beta+\beta B^{-e_i}. -\] -The moving rectangle has side lengths $l_i B^{-1}$ and -$l_i^{-1}B^{-2}$. Use a physical padding -$\varepsilon=B^{-3}/4$ around each moving rectangle in three dimensions. -During ascent and descent the planar endpoint gaps separate these -neighborhoods; during the affine phase the unit height gaps separate them. -Use Lemma~\ref{lem:bb-curl} for the paths -$c_i+\diag(l_i,l_i^{-1},1)(y-(P_i,0))$ and sum the disjoint fields. -The first coordinate is an endpoint convex combination even though the -paths are based at lower corners rather than centers. All supports lie -in $[-2L_0,2L_0]^3$, by the definitions of $L_0$ and $N$. - -Scale by $\rho=1/(32L_0)$, so the repeated mechanism is supported in -$[-1/16,1/16]^3$. Load the fixed label $(1/8,0,0)$ to the scaled -rational input code by a moving translation with cutoff radius $1/32$; -its support stays in $(-1/4,1/4)^3$. For a nonterminal initial code, -the entire first coordinate during loading is positive, and subsequent -nonterminal branches have first coordinate greater than $1/32$. -Terminal codes have first coordinate less than $-1/32$. Hence the stated -strip detects exactly terminal entry, including initial halting, without -any wraparound artifact. Periodize the compact curl fields and recompute -the force at physical $\nu$. Mean zero, bounded derivatives and comparison -follow from Lemma~\ref{bc:analytic}. -\end{proof} - -\subsection{A filled half-line and mean-corrected vertical motion} -\label{bc:halfline-section} -The next recorder starts with an infinite constant history tail rather -than a single frontier symbol. Its guard and its mean-zero vertical -realization both have independent roles. - -\begin{lemma}[History at a filled half-line]\label{bc:halfline-table} -A finite guarded table has a full-domain inverse and simulates the original -machine from a tape with a separately constant tail in each direction. -At checkpoint $n$ its nonempty history cells are exactly the absolute -indices below $2+n$. -\end{lemma} -\begin{proof}\label{bc:halfline-table-proof} -Normalize to one halt state $h$. For $i=(q,a)\in I=(Q\setminus\{h\})\times A$, -write $\delta(q,a)=(q_i,b_i,e_i)$. History letters are -$\mathcal H=\{\perp,\#\}\sqcup I$, with -$\mathcal H_+=\{\#\}\sqcup I$. The tracks are work, mark and history; -controls are $\mathsf R_q,\mathsf B_q,\mathsf T_i,\mathsf S_i$. -Write $\ell_j$ for the history at relative cell $j$. -Here is the complete table, with unmentioned cells unchanged: -\begin{equation}\label{bc:halfline-rules} -\begin{array}{c|c|c|l} -\text{input}&\text{output}&\text{move}&\text{condition}\\\hline -\mathsf R_q(a,0,k)&\mathsf T_i(b_i,0,k)&e_i&i=(q,a),\ q\ne h\\ -\mathsf T_i(c,0,k)&\mathsf S_i(c,1,k)&1&k\in\mathcal H_+\\ -\mathsf S_i(c,0,k)&\mathsf S_i(c,0,k)&1&k\in\mathcal H_+\\ -\mathsf S_i(c,0,\perp)&\mathsf B_{q_i}(c,0,i)&0&\ell_{+1}=\perp\\ -\mathsf B_q(c,0,k)&\mathsf B_q(c,0,k)&-1&k\in\mathcal H_+\\ -\mathsf B_q(c,1,k)&\mathsf R_q(c,0,k)&0&k\in\mathcal H_+. -\end{array} -\end{equation} -Initialize $\mathsf R_{q_0}$ at head zero with marks zero, history $\#$ -at every index below two and $\perp$ elsewhere. At checkpoint $n$, the -work head $p_n$ has $|p_n|\le n$ and the boundary is $r_n=2+n$. -After the work move, $p'=p_n+e_i\le n+11$, and terminal codes lie strictly in $(-2,-1)\times(0,1)$. +The infinite tails are in $[\delta,1-\delta]$, +$\delta=1/(D-1)$, so finite-prefix decoding of length $b$ is possible +from error less than $(\delta/2)D^{-b}$ without a boundary ambiguity. + +Split full triples $[-1,1]$. By +Lemma~\ref{lem:bb-window-rectangles}, source prefix lengths $(1,2)$ +become $(1+d,2-d)$, target cylinders are complete, and both rectangle +families have gaps at least $D^{-3}$. Thicken by $[-1/4,1/4]$, +lift to height $4i$, use linear middle scale +$\mu_i=1+(D^{-d_i}-1)\theta$, and descend. Padding smaller than +$D^{-3}/4$ separates all phases. Repeating the pulse, zero on +$[0,1/12]$ and $[11/12,1]$, gives periodicity from zero without +loading. At a nonhalting branch's lift and descent the first +coordinate is greater than one; during the middle transfer the +particle's third coordinate is exactly $4i$, outside $(-1,1)$. +Thus even a horizontal route passing over the observation box cannot +cause a false visit. The integer samples and the compiler invariant +prove halting equivalence. The record count and pointer site $n+3$ +recover absolute indexing at checkpoints. + +For uniqueness apply Lemma~\ref{lem:b-comparison}(iii). +The explicit compactly supported smooth field lies in $CH^4\cap C^1H^2$ +on every finite interval and has pressure zero. The competing class +in the theorem is precisely case (iii), including its distributional +$CL^2$ gradient. Lemma~\ref{lem:b-gradient} cancels that gradient +against the solenoidal velocity difference without introducing a pressure +$L^2$ norm. Thus the velocity and pressure gradient are unique, while +pressure itself remains free up to a time-dependent spatial constant. +Proposition~\ref{prop:bb-late-logclock} gives the stated +decaying alternative. +\end{proof} + +\subsection{A flagged head and two-pass parking} +\label{sec:bb-two-pass-parking} + +\begin{theorem}\label{thm:bb-two-pass-parking} +For any machine and finitely specified initial tape, an effective +smooth force on $\R^3$, periodic from time zero, with one compact +spatial support and bounded mixed derivatives, realizes the event +\[ + \exists t\ge0:\quad X(t;0)\in(-6,-4)\times(-1,1)\times(-1,1) + \quad\Longleftrightarrow\quad\text{halting}. +\] +The zero-data velocity is unique among classical smooth solutions +with bounded $u,\nabla u$, $u\in C H^2\cap C^1L^2$ and $p\in C L^2$ +on every finite interval. The prescribed solution has uniformly +bounded kinetic energy. +\end{theorem} +\begin{proof}\label{proof:bb-two-pass-parking} +Merge terminal states into $q_\dagger$, send missing instructions +there by a stay step, and add a fresh initial state $q_*$, with no +incoming instruction, which writes back and stays while entering +the old initial state. For each normalized nonterminal rule +$r=(q,a)\mapsto(p_r,b_r,d_r)$ use cell letters +\[ + (w,h,j)\in\Gamma:=\mathcal A\times + (\{\circ,\star\}\sqcup\{[r]:r\in\mathcal R\})\times\{0,1\}, +\] +where $\mathcal A$ is the work alphabet and $\mathcal R$ is the +finite set of normalized instruction labels. Thus $\Gamma$ includes +all three tracks. +The complete table, with $h\ne\star$ wherever $h$ occurs, is +\[ +\begin{array}{lll} + M_q:(a,h,1)&\mapsto A_r:(b_r,h,0),d_r,\\ + A_r:(w,h,0)&\mapsto R_r:(w,h,1),1,\\ + R_r:(w,h,0)&\mapsto R_r:(w,h,0),1,\\ + R_r:(w,\star,0)&\mapsto B_{p_r}:(w,[r],0),1,\\ + B_q:(w,\circ,0)&\mapsto L_q:(w,\star,0),-1,\\ + L_q:(w,h,0)&\mapsto L_q:(w,h,0),-1,\\ + L_q:(w,h,1)&\mapsto M_q:(w,h,1),0. +\end{array} +\] +The destination fixes the incoming displacement. The record index +recovers the first row's old work pair; a flag distinguishes the +entries into $R_r$; a record distinguishes the entries into $B_q$; +a written frontier distinguishes entry and loop into $L_q$; +and entry into $M_q$ is the unique cell-preserving last row. +This proves the full one-cell inverse conditions. + +Initialize the sole flag at zero and the sole frontier at three, +with empty history elsewhere. At ready step $k$, only the work +head $i$ is flagged, records occupy $3,\ldots,k+2$, and the frontier +is $e_k=k+3$. The work rule clears the flag and moves to +$j=i+d_r\le e_k-2$, which is unflagged even for $d_r=0$. +It is flagged anew; the scan logs at $e_k$, advances the pointer, +returns, and preserves the ready flag. The two loop lengths are +$e_k-j-1$ and $e_k-j$, with five other rules, for a total +$2(e_k-j)+4$. Thus all scans terminate and ready halting is exact. + +Use odd digits in base $K=2|\Gamma|+2$, initial fractions $x_*,y_*$, +and state offsets $a_{M_{q_*}}=0$, $a_{M_{q_\dagger}}=-5$, and +$4,8,12,\ldots$ for the others. The code +$(a_s-x_*+x,y-y_*,0)$ starts at zero. Lemma~\ref{lem:bb-onecell} +gives $N$ full rectangle maps with scales $\lambda_i\in\{K^{-1},1,K\}$; +thicken by $[-1/4,1/4]$. The entire terminal block is inside the +observation box because $0-1$. Eventually constant tails are rational even for an +arbitrary finitely specified initial tape, and the pointer and record +count recover its absolute head position at ready checkpoints. + +For the required two-pass geometry, let $g$ be the minimum of one +and all positive separating coordinate gaps in either planar family, +and set $\epsilon=g/20$. Choose +\[ + C=20+\max(\{0\}\cup\{\text{right endpoints of source and target boxes}\}), + \qquad S_i=(C+4i,0,0). +\] +In a first pass give each source four consecutive slots: lift by ten, +translate above $S_i$, descend, and reshape there by the linear +reciprocal scale. In a second pass give each parked box three slots: +lift by ten, translate above its target, and descend. The total is +$7N$ slots, and every source is gone before a target is filled. +Widths remain at most one. Source gaps protect the first lift, +parking spacing protects strain and parking moves, target gaps protect +delivery, and travel height separates every horizontal transfer from +resting boxes. These inequalities remain strict under $2\epsilon$ +padding. Applying Lemma~\ref{lem:bb-curl} in chronological order +fixes every stationary box and gives the exact full-box map to the +moving one. Thus overlapping source and target families cause no +interference. + +Repeat the flat-ended template from zero and take its residual. +The integer sampling induction gives the table execution. In a +nonhalting run every endpoint box lies in $x_1>-1$; translations +interpolate corresponding endpoint points, strain occurs far to the +right, and waiting leaves the point fixed. The entire two-pass path +therefore remains in $x_1>-1$. A terminal sample lies inside the +observer. This proves the event for all times. + +For the exact pressure class use Lemma~\ref{lem:b-comparison}(iv). +The prescribed compactly supported smooth velocity satisfies the +reference $CH^2\cap C^1L^2$ and bounded-gradient conditions. The +competing velocity has precisely these bounds and a $CL^2$ pressure +representative, as that case requires. Its cutoff argument uses the +pressure itself, with error $CR^{-1}\|p\|_2\|u-U\|_2$, and assumes +no extra integrability of its derivative. The lemma gives velocity +uniqueness; the $L^2$ pressure representative is then zero. +Compact support and +bounded derivatives of the prescribed velocity give its uniform +energy bound and its global material flow. +\end{proof} + + +\subsection{A two-cell append and its guarded inverse}\label{bc:two-cell-section} +Use Lemma~\ref{lem:bb-window-compiler} with initial frontier $r_0=1$, +renaming Ready, Go and Back as $M,R,L$ and $(E,P)$ as +$(\perp,\star)$. The track order is work, mark, history. +The Ready history coordinate and the marked Back row remain unrestricted; +the unmarked Back row tests precisely the right-neighbor history. +Thus the full table, not only its initialized execution, is identical. +At checkpoint $n$, its frontier is $p_n=1+n>h_n$, where $h_n$ is +the work-head position, and the exact cycle length is $2(1+n-h_n)+1$. +Lemma~\ref{lem:bb-window-rectangles} applies with +$(a,b)=(1,2)$: the complete allowed triple fixes both history writes, +and its full image has prefix lengths $(1+e,2-e)$. + +\begin{theorem}[Guarded cylinders in a torus chart]\label{bc:guarded-torus} +On the unit torus, the fixed label $(1/8,0,0)$ and the strip +$\{x:x_1\pmod1\in(-1/4,0)\}$ realize halting by a general mean-zero +force with zero initial velocity and pressure. It is periodic after time +one and all its mixed derivatives are bounded. The torus comparison +class gives uniqueness. +\end{theorem} +\begin{proof}\label{bc:guarded-torus-proof} +Let $a$ be the augmented alphabet size and $k$ the control count in +Lemma~\ref{lem:bb-window-compiler}. Use zero-blank base $B=2a$, and enumerate +nonterminal and terminal states with positive integer indices $j\ge1$ +and first-coordinate offsets respectively +$L_0+3j$ and $-L_0-3j$, where $L_0=4k+ka^3+10$. +Refinement by triples gives at most $N=ka^3$ branches. Their source +prefix lengths are $(1,2)$ and image lengths $(1+e,2-e)$, so the +smallest possible grid gap is $B^{-3}$. If $P_i,Q_i$ are lower corners, +the branch map is +$Q_i+\diag(B^{-e_i},B^{e_i})(y-P_i)$. + +Choose three flat ramps $\alpha,\beta,\gamma$, active respectively in +$(1/8,1/4)$, $(3/8,5/8)$ and $(3/4,7/8)$. Set +\[ + c_i(t)=((1-\beta)P_i+\beta Q_i,\ i(\alpha-\gamma)), + \quad l_i=1-\beta+\beta B^{-e_i}. +\] +The moving rectangle has side lengths $l_i B^{-1}$ and +$l_i^{-1}B^{-2}$. Use a physical padding +$\varepsilon=B^{-3}/4$ around each moving rectangle in three dimensions. +During ascent and descent the planar endpoint gaps separate these +neighborhoods; during the affine phase the unit height gaps separate them. +Use Lemma~\ref{lem:bb-curl} for the paths +$c_i+\diag(l_i,l_i^{-1},1)(y-(P_i,0))$ and sum the disjoint fields. +The first coordinate is an endpoint convex combination even though the +paths are based at lower corners rather than centers. All supports lie +in $[-2L_0,2L_0]^3$, by the definitions of $L_0$ and $N$. + +Scale by $\rho=1/(32L_0)$, so the repeated mechanism is supported in +$[-1/16,1/16]^3$. Load the fixed label $(1/8,0,0)$ to the scaled +rational input code by a moving translation with cutoff radius $1/32$; +its support stays in $(-1/4,1/4)^3$. For a nonterminal initial code, +the entire first coordinate during loading is positive, and subsequent +nonterminal branches have first coordinate greater than $1/32$. +Terminal codes have first coordinate less than $-1/32$. Hence the stated +strip detects exactly terminal entry, including initial halting, without +any wraparound artifact. Periodize the compact curl fields and recompute +the force at physical $\nu$. Mean zero, bounded derivatives and comparison +follow from Lemma~\ref{bc:analytic}. +\end{proof} + +\subsection{A filled half-line and mean-corrected vertical motion} +\label{bc:halfline-section} +The next recorder starts with an infinite constant history tail rather +than a single frontier symbol. Its guard and its mean-zero vertical +realization both have independent roles. + +\begin{lemma}[History at a filled half-line]\label{bc:halfline-table} +A finite guarded table has a full-domain inverse and simulates the original +machine from a tape with a separately constant tail in each direction. +At checkpoint $n$ its nonempty history cells are exactly the absolute +indices below $2+n$. +\end{lemma} +\begin{proof}\label{bc:halfline-table-proof} +Normalize to one halt state $h$. For $i=(q,a)\in I=(Q\setminus\{h\})\times A$, +write $\delta(q,a)=(q_i,b_i,e_i)$. History letters are +$\mathcal H=\{\perp,\#\}\sqcup I$, with +$\mathcal H_+=\{\#\}\sqcup I$. The tracks are work, mark and history; +controls are $\mathsf R_q,\mathsf B_q,\mathsf T_i,\mathsf S_i$. +Write $\ell_j$ for the history at relative cell $j$. +Here is the complete table, with unmentioned cells unchanged: +\begin{equation}\label{bc:halfline-rules} +\begin{array}{c|c|c|l} +\text{input}&\text{output}&\text{move}&\text{condition}\\\hline +\mathsf R_q(a,0,k)&\mathsf T_i(b_i,0,k)&e_i&i=(q,a),\ q\ne h\\ +\mathsf T_i(c,0,k)&\mathsf S_i(c,1,k)&1&k\in\mathcal H_+\\ +\mathsf S_i(c,0,k)&\mathsf S_i(c,0,k)&1&k\in\mathcal H_+\\ +\mathsf S_i(c,0,\perp)&\mathsf B_{q_i}(c,0,i)&0&\ell_{+1}=\perp\\ +\mathsf B_q(c,0,k)&\mathsf B_q(c,0,k)&-1&k\in\mathcal H_+\\ +\mathsf B_q(c,1,k)&\mathsf R_q(c,0,k)&0&k\in\mathcal H_+. +\end{array} +\end{equation} +Initialize $\mathsf R_{q_0}$ at head zero with marks zero, history $\#$ +at every index below two and $\perp$ elsewhere. At checkpoint $n$, the +work head $p_n$ has $|p_n|\le n$ and the boundary is $r_n=2+n$. +After the work move, $p'=p_n+e_i\le n+1-1$, and halt codes $<-5$. Thicken the rectangles of -Lemma~\ref{lem:bb-onecell} by $[-1,1]$, give branch $i$ height $10i$, -and use the linear scale $g_i=1+\theta(\lambda_i-1)$ in the lift -schedule. A padding smaller than one tenth of the minimum of one -and all within-family planar gaps suffices. The middle strain has -potential $(0,0,(\dot g_i/g_i)(x_1-c_{i1})(x_2-c_{i2}))$. -Equation~\eqref{eq:bb-convex-coordinate} excludes every nonhalting -visit to $x_1<-5$. The table and routing are fixed for the machine; -only $x_*,y_*$ change with the input. The passive origin bit remains -at physical cell zero, so no absolute tape information is discarded. - -For the second case first merge all halting work states into a single -state $q_h$, redirecting incoming transitions and retaining initial halting. -Delete the passive track from the direction recorder, and now let -$\mathcal A$ denote that full alphabet. Use odd digits with -$B=2|\mathcal A|+2$ and -$\lambda=1/[16(J+1)]$, where $J$ counts the controls. Number them -from zero and set -\[ - S_s(x,y)=(A_s+\lambda x,1/4+\lambda y),\qquad - A_s=2\lambda\operatorname{index}(s)+ - \begin{cases}5/8,&s=M(q_h,e),\\1/4,&\text{otherwise}. - \end{cases} -\] -The nonhalt squares lie in $[1/4,3/8]\times[1/4,1/4+\lambda]$; -halt squares lie in $[5/8,3/4]\times[1/4,1/4+\lambda]$. -Write $m$ for the number of branches. Use coding height $1/4$, -branch heights $1/2+i/[4(m+1)]$, and -the same linear first scale. One hundredth of the minimum of -$1/16$, $1/[4(m+1)]$, and the source and target planar gaps is -valid padding; absent pair gaps are omitted. All supports lie inside -the cube, and all horizontal widths are at most $\lambda$. -A localized translation with padding $1/1000$ loads the rational -initial point from the fixed label in one unit interval. - -In a nonhalting period the center has first coordinate at most $3/8$ -and half-width at most $\lambda/2$, hence is below $1/2$; loading -has the same bound. Halting codes lie in the observer. The exact -sampling induction gives one table step per period, with checkpoint -cycle $2(k+1-h_k)+3$. The record block beginning at cell one -recovers absolute indexing at checkpoints. An initial halt is -observed at time one. -\end{proof} - - -\subsection{A one-time torus loader} - -The next placement uses a separate first interval to load the code. Only -the instruction cycle is then repeated, so periodicity begins at time one. - -\begin{theorem}[A separate one-unit loader on the torus] -\label{ba:loaded-torus} -For every machine and finite input and every fixed computable $\nu>0$, -there is an effective smooth -mean-zero force, one-periodic for $t\ge1$ with bounded mixed derivatives, -whose unique smooth zero-data solution with zero normalized pressure -satisfies -\[ - \exists t\ge0:\ \Phi_t(1/2,1/2,1/4) - \in(1/16,1/4)\times\mathbb T^2 - \quad\Longleftrightarrow\quad\text{halting}. -\] -Uniqueness holds in the classical class of Section~\ref{sec:model}. -\end{theorem} -\begin{proof}\label{ba:loaded-torus-proof} -Use the single-halt move-first recorder of -Lemma~\ref{lem:bb-movefirst}, without an entry state. Let -$\Gamma$ be its full alphabet and $N$ its number of controls. Put -$B=2|\Gamma|+1$, $a=1/(16N)$, the halt square at -$(1/8,1/4)+a[0,1]^2$, and the other squares at -$(1/2+2ja,1/4)+a[0,1]^2$, $0\le j\le N-2$. -Use the minimal subdivision, so right and stay moves are not split by -the left letter. Its gaps are at least $a/B^2$. -For $M$ branches choose -\[ - h=\frac1{64(M+1)},\quad Z_i=\frac12+\frac{i}{4(M+1)},\quad - \varepsilon=\min\{1/128,h/4,a/(4B^2)\}. -\] -Apply linear-first-scale full-box routing from height $z_0=1/4$. -Middle heights differ by $16h$; source and target gaps handle the -vertical phases. Centers lie in -$[1/8,5/8]\times[1/4,5/16]$, widths are at most $a$, and all padded -boxes lie in $(1/16,15/16)^3$. This is a construction on the specified -unit torus, with no change of physical viscosity. - -In the first unit interval translate a box of side $1/64$ from the -fixed label to the rational initial code, using Lemma~\ref{lem:bb-curl}. -The segment and a small padding lie inside the same chart. Repeat -only the instruction cycle afterward. Code induction proves the -checkpoint identities. A halting code has first coordinate in -$[1/8,3/16]$, including an initial halt reached by loading. For a -nonhalting run both centers of every used branch have first coordinate -at least $1/2$, with half-width at most $a/2$; loading has the same -lower center bound. Hence the particle always stays to the right of -$1/4$. Residual realization proves every analytic assertion. The -separate loading field accounts for the qualification $t\ge1$ in the -periodicity statement. -\end{proof} - - -\subsection{Repeated torus loading branches}\label{ba:torus} - -A loader can instead be one branch of every period. To keep its target -separate from all computational targets, place the initial code in a state -square with no incoming instruction. This condition concerns the entire -target family, not only the distinguished execution. We implement the -same startup principle first with full boxes, then with fixed columns. - -\begin{theorem}[A loader repeated as part of the processor] -\label{ba:periodic-torus} -For every finite machine and input, and every fixed computable $\nu>0$, -there is an effective smooth mean-zero force on the unit flat torus, -one-periodic from time zero, with all mixed derivatives bounded. Its -zero-data solution has pressure zero, is unique in the classical class -of Section~\ref{sec:model}, and satisfies -\[ - \exists t\ge0:\ \Phi_t(1/8,1/8,1/4) - \in\{5/81/(2B)$ from the endpoints -of $[0,1]$, so the new target lies strictly inside the entry square. -There is no other target in that square. Both enlarged branch lists -therefore remain separately separated, with gaps at least $a/B^2$. -The extra source is separated from all state squares. - -Number the branches, including loading, by $1\le i\le M$. Set -\[ - z_0=\tfrac14,\qquad h=\frac1{16(M+1)},\qquad - Z_i=\tfrac12+\frac{i}{4(M+1)},\qquad - \varepsilon=\frac1{10}\min(h,a/B^2). -\] -Thicken each planar source by $[z_0-h,z_0+h]$. Lift the resulting -full box to center height $Z_i$, move it horizontally, and lower it -to $z_0$. For an instruction with displacement $d_i$, use -$D_i=\operatorname{diag}(B^{-d_iw_2},B^{d_iw_2},1)$; -loading has $d_i=0$. Its center is the planar endpoint interpolation -plus $(Z_i-z_0)(w_1-w_3)e_3$. Adjacent middle heights are $4h$ apart, -while the padded half-thickness is $h+\varepsilon<2h$. -Source and target gaps handle the other phases. Horizontal half-widths -never exceed $a/2$, horizontal centers lie between $1/8$ and $3/4+a$, -and vertical centers lie between $1/4$ and $3/4$. The padded motions -are thus strictly inside the unit cube. Lemma~\ref{lem:bb-curl} gives a -chart field, extended periodically in space and time. Its zero time -collars give zero initial velocity. - -The first cycle loads the fixed particle. Each subsequent cycle maps -its exact code to the next local code until the first halt, or forever -in a nonhalting run. Lemma~\ref{lem:bb-movefirst} identifies the finite -checkpoint times with machine steps. The added entry state handles an -original initial halt. A halt code is in the target slab. For a -nonhalting execution every used source and target center, including the -loader, has first coordinate at most $3/8$. Exponential interpolation -keeps the center below that bound and the half-width below $a/2$; -the particle cannot reach $5/8$ at any intermediate time. The final -transition into a halt is not subject to this nonhalting-endpoint bound. -Lemma~\ref{ba:realization} gives the force, its mean, its effective -derivatives, and uniqueness. Repeating the loading field is harmless -to the initialized trajectory and makes the force periodic from zero. -\end{proof} - -\paragraph{A fixed-column implementation.}\label{bc:unit-periodic-section} -The same untargeted-entry condition also permits a repeated initializer -using fixed source and target columns. The next construction retains its -explicit column potentials and its fixed label; its middle motion uses -linear interpolation of the first planar scale. - -\begin{theorem}[A periodic initializer on the unit torus] -\label{bc:unit-periodic-event} -There is an effective halting realization on $\T^3$ with fixed label -$a_*=(3/8,1/4,1/2)$ and fixed strip $5/8R+2$. -Lemma~\ref{lem:bb-lift-parking} gives the full $7N$-motion schedule. - -Repeat this unit pulse from zero. Nonhalting endpoint boxes and all -parking boxes have nonnegative first coordinate. Translations preserve -that bound, and strain takes place in positive parking envelopes. -Therefore a nonhalting path never enters the negative observation box. -A halting code lies in $[-3,-2]\times[0,1]\times\{0\}$ inside it. -The residual proposition finishes the proof. Loading was a repeated -instruction, so no exceptional first-time force is required. -\end{proof} - - -\begin{remark}[A different observation regime] -For comparison, the distinct velocity-field detector of -\cite[Theorem~1.1(2) and Section~5]{OpenAI379EulerianC} uses -$\R^2\times\T$ and the event -$\int_{\{X_2>0\}\times\T}u_3>1/2$. Its force has globally bounded -mixed derivatives and horizontal support compact on each finite time -interval. The nonnegative scalar velocity has diffusion tails; after -loading its total mass is one. Its cutoff estimate gives loss -$\delta<1/24$, nonhalting mass at most $\delta$ in the observed half-space, -and halting mass above $3/4$. Those conclusions follow from the cited -advection--diffusion proof and have no uniform spatial support assertion. -They are a separate force and observation regime from the material test -proved here. -\end{remark} +\section{Choosing initialization and the observer} +\label{sec:bb-initialization} + +The preceding recorders supply complete instruction tables. We now +specify how their boxes are positioned and how the fixed material label +becomes the initial code. We compare one-time loading, a loading branch +repeated in every cycle, and chart translations that place the initial +code at the label itself. The latter two choices can preserve periodicity +from time zero. Each construction also needs a bound between integer +rule times; the endpoint coding alone does not give a material event. + +Unless stated otherwise, the force in this section is the residual +\eqref{eq:bb-residual}, with pressure zero. Its smoothness, all-order +mixed derivative bounds, finite effective prescription and uniqueness +are as in Proposition~\ref{prop:bb-realization}. Whole-space forces +have one compact spatial support; torus forces have zero spatial mean. +All statements use a fixed positive computable viscosity, with the +relative interpretation of that proposition for arbitrary fixed +positive viscosity. + +\subsection{Torus loading and direct chart placement} + +\begin{proposition}[A move-first material test]\label{thm:bb-torus33} +The move-first recorder has a unit-torus realization, periodic after +$t=1$, whose fixed label $(1/4,1/2,1/4)$ enters +$\{x_1\in(1/2,1)\pmod1\}$ exactly when the work machine halts. +For a fixed machine only the loading interval depends on the input. +\end{proposition} +\begin{proof}\label{proof:bb-torus33} +Use Lemma~\ref{lem:bb-movefirst}, let $\Sigma$ be its full cell alphabet, +and enumerate its $N$ controls by $i=1,\ldots,N$. +Set $s=1/[16(N+1)]$ and give the $i$th control a square +of side $s$ centered at +\[ + m_i=(o_i+i/[4(N+1)],1/2),\qquad + o_i=\begin{cases}5/8,&C_q\text{ with }q\text{ halting},\\1/8,&\text{otherwise}. + \end{cases} +\] +Use odd digits and $B=2|\Sigma|+2$. The code at height $z_0=1/4$ +is $(m_i+s(E(L)-1/2,E(R)-1/2),z_0)$. +It is rational initially. Lemma~\ref{lem:bb-onecell} gives $J$ +branches with separately separated rectangles, all side lengths at +most $s$ and endpoints on the grid of mesh +$[32(N+1)B^2]^{-1}$. For branch $j$ choose height +$z_j=1/2+j/[4(J+1)]$. Use the three-phase lift of +Lemma~\ref{lem:bb-lift-parking}, with exponential middle scale +$\lambda_j^\theta$ and center interpolation. A padding parameter +\[ + \epsilon=\min\{1/128,[320(N+1)B^2]^{-1},[40(J+1)]^{-1}\} +\] +separates the enlarged sheets in the source, height, and target phases +and keeps them inside the unit cube. Their thickness may then be +chosen positive by Lemma~\ref{lem:bb-curl}. For $J=0$ the repeated +field is zero. + +During $[0,1]$ a localized curl translation takes the fixed label +along the segment to the initial code. Both segment and a small +padding lie in the chart. Repeat the branch pulse thereafter. +At each subsequent integer time the particle is the next table code. +If both endpoint controls are nonhalting, the interpolated center and +half-width bounds give $x_1<3/8+s<1/2$ throughout the motion. +The loader also stays below $1/2$ in a nonhalting run. A halting +checkpoint lies in the upper half, reached in finitely many positive +local steps; an initial halt is reached at the end of loading. +This proves the event for every real time. +\end{proof} + +\begin{proposition}[A guarded-window material test]\label{thm:bb-window37} +The table of Lemma~\ref{lem:bb-window-compiler} has a unit-torus +realization, periodic after $t=1$, with fixed label $(1/4,1/4,1/4)$ +and event $x_1\in(1/2,7/8)\pmod1$, occurring exactly when the +work machine halts. At work step $n$ its next +checkpoint occurs after exactly $2(p_n-h_n)+1\le4n+5$ unit periods. +\end{proposition} +\begin{proof}\label{proof:bb-window37} +Choose $r_0=2$ in Lemma~\ref{lem:bb-window-compiler}: initially the +frontier is at cell two, all other history cells are empty, and all +marks are zero. Thus $p_n=n+2$, with records in cells $2,\ldots,n+1$. +Split the full three-cell windows $[-1,1]$ and apply +Lemma~\ref{lem:bb-window-rectangles} with $(a,b)=(1,2)$. +Let $\mathcal A$ be the full cell alphabet of this window table and use +odd digits in base $B=2|\mathcal A|+1$. +For $S$ controls let $\epsilon=1/[16(S+1)]$ and place their squares +at $o_s+\epsilon[0,1]^2$, where +\[ + o_s=(\xi_s,1/4+2\kappa(s)\epsilon),\quad + \kappa(s)\in\{1,\ldots,S\},\quad + \xi_s=\begin{cases}5/8,&s\text{ halting Ready},\\1/4,&\text{otherwise}. + \end{cases} +\] +Write $J$ for the number of branches and $e_j$ for branch $j$'s cursor +displacement. The full source and target rectangle gaps are at least +$\epsilon B^{-3}$. Put their common height at $1/4$, assign branch +$j$ height $1/2+j/[4(J+1)]$, and use the exponential middle scale +$\operatorname{diag}(B^{-e_j\theta},B^{e_j\theta},1)$. +Take one tenth of +$\min(1/32,\epsilon B^{-3},1/[4(J+1)])$ as padding. +Horizontal source and target gaps and the private heights separate +the three phases, while all widths remain at most $\epsilon$. +The strain can equivalently be generated by +$(0,0,-e_j\dot\theta\log B\,(x_1-c_{j1})(x_2-c_{j2}))$, +retaining the explicit diagonal-strain implementation. + +A first unit interval loads the label along the segment to the rational +initial code. Thereafter the exact rectangle action and the compiler +lemma give checkpoint times $t_0=1$ and +$t_{n+1}=t_n+2(p_n-h_n)+1$. A nonhalting center stays in +$[1/4,1/4+\epsilon]$, with half-width at most $\epsilon/2$; +thus every intermediate first coordinate remains below $1/2$. +The loader has the same exclusion. Halt codes are in +$[5/8,5/8+\epsilon]$, inside the observer, including an initial halt. +The history block starts at the known absolute cell two, so the +head-relative code also recovers absolute indexing at checkpoints. +\end{proof} + +\begin{proposition}[Two direction-recording placements] +\label{thm:bb-aligned38}\label{thm:bb-direction39} +The direction recorder gives both of the following realizations. +\begin{enumerate} +\item On $\R^3$, with its passive origin bit, there is a force periodic +from time zero and a fixed-origin event $X_1(t;0)<-5$. Input dependence +can be confined to two translations of the state charts. On every +implemented full box the first coordinate is a convex combination +of its endpoint first coordinates. +\item On the unit torus, without the passive track, a force periodic +after $t=1$ has fixed label $(1/8,1/8,1/4)$ and event +$x_1\in(1/2,7/8)\pmod1$. For a fixed machine its repeated part is +independent of the input. +\end{enumerate} +Both events are equivalent to halting. +\end{proposition} +\begin{proof}\label{proof:bb-aligned38}\label{proof:bb-direction39} +For the first case introduce a fresh terminal state $\dagger$ and +replace every missing instruction and every originally terminal +state--symbol pair by a write-preserving stationary transition into it. +The original initial control is therefore nonterminal in this normalized +table, even when the original machine is initially halted. Apply +Lemma~\ref{lem:bb-direction} with its passive origin track, and let +$\mathcal A$ be the resulting full cell alphabet. Use odd digits in +base $B=2|\mathcal A|+1$ +and initial fractions $x_*,y_*$. Index the initial main state by zero, +the main $\dagger$ states by distinct negative integers, and the rest +by distinct positive integers. State $s$ has placement +\[ + (x,y)\longmapsto(10k(s)+x-x_*,y-y_*). +\] +The initialized point is zero; nonhalt codes have first coordinate +$>-1$, and halt codes $<-5$. Thicken the rectangles of +Lemma~\ref{lem:bb-onecell} by $[-1,1]$, give branch $i$ height $10i$, +and use the linear scale $g_i=1+\theta(\lambda_i-1)$ in the lift +schedule. A padding smaller than one tenth of the minimum of one +and all within-family planar gaps suffices. The middle strain has +potential $(0,0,(\dot g_i/g_i)(x_1-c_{i1})(x_2-c_{i2}))$. +Equation~\eqref{eq:bb-convex-coordinate} excludes every nonhalting +visit to $x_1<-5$. The table and routing are fixed for the machine; +only $x_*,y_*$ change with the input. The passive origin bit remains +at physical cell zero, so no absolute tape information is discarded. + +For the second case first merge all halting work states into a single +state $q_h$, redirecting incoming transitions and retaining initial halting. +Delete the passive track from the direction recorder, and now let +$\mathcal A$ denote that full alphabet. Use odd digits with +$B=2|\mathcal A|+2$ and +$\lambda=1/[16(J+1)]$, where $J$ counts the controls. Number them +from zero and set +\[ + S_s(x,y)=(A_s+\lambda x,1/4+\lambda y),\qquad + A_s=2\lambda\operatorname{index}(s)+ + \begin{cases}5/8,&s=M(q_h,e),\\1/4,&\text{otherwise}. + \end{cases} +\] +The nonhalt squares lie in $[1/4,3/8]\times[1/4,1/4+\lambda]$; +halt squares lie in $[5/8,3/4]\times[1/4,1/4+\lambda]$. +Write $m$ for the number of branches. Use coding height $1/4$, +branch heights $1/2+i/[4(m+1)]$, and +the same linear first scale. One hundredth of the minimum of +$1/16$, $1/[4(m+1)]$, and the source and target planar gaps is +valid padding; absent pair gaps are omitted. All supports lie inside +the cube, and all horizontal widths are at most $\lambda$. +A localized translation with padding $1/1000$ loads the rational +initial point from the fixed label in one unit interval. + +In a nonhalting period the center has first coordinate at most $3/8$ +and half-width at most $\lambda/2$, hence is below $1/2$; loading +has the same bound. Halting codes lie in the observer. The exact +sampling induction gives one table step per period, with checkpoint +cycle $2(k+1-h_k)+3$. The record block beginning at cell one +recovers absolute indexing at checkpoints. An initial halt is +observed at time one. +\end{proof} + + +\subsection{A one-time torus loader} + +The next placement uses a separate first interval to load the code. Only +the instruction cycle is then repeated, so periodicity begins at time one. + +\begin{theorem}[A separate one-unit loader on the torus] +\label{ba:loaded-torus} +For every machine and finite input and every fixed computable $\nu>0$, +there is an effective smooth +mean-zero force, one-periodic for $t\ge1$ with bounded mixed derivatives, +whose unique smooth zero-data solution with zero normalized pressure +satisfies +\[ + \exists t\ge0:\ \Phi_t(1/2,1/2,1/4) + \in(1/16,1/4)\times\mathbb T^2 + \quad\Longleftrightarrow\quad\text{halting}. +\] +Uniqueness holds in the classical class of Section~\ref{sec:model}. +\end{theorem} +\begin{proof}\label{ba:loaded-torus-proof} +Use the single-halt move-first recorder of +Lemma~\ref{lem:bb-movefirst}, without an entry state. Let +$\Gamma$ be its full alphabet and $N$ its number of controls. Put +$B=2|\Gamma|+1$, $a=1/(16N)$, the halt square at +$(1/8,1/4)+a[0,1]^2$, and the other squares at +$(1/2+2ja,1/4)+a[0,1]^2$, $0\le j\le N-2$. +Use the minimal subdivision, so right and stay moves are not split by +the left letter. Its gaps are at least $a/B^2$. +For $M$ branches choose +\[ + h=\frac1{64(M+1)},\quad Z_i=\frac12+\frac{i}{4(M+1)},\quad + \varepsilon=\min\{1/128,h/4,a/(4B^2)\}. +\] +Apply linear-first-scale full-box routing from height $z_0=1/4$. +Middle heights differ by $16h$; source and target gaps handle the +vertical phases. Centers lie in +$[1/8,5/8]\times[1/4,5/16]$, widths are at most $a$, and all padded +boxes lie in $(1/16,15/16)^3$. This is a construction on the specified +unit torus, with no change of physical viscosity. + +In the first unit interval translate a box of side $1/64$ from the +fixed label to the rational initial code, using Lemma~\ref{lem:bb-curl}. +The segment and a small padding lie inside the same chart. Repeat +only the instruction cycle afterward. Code induction proves the +checkpoint identities. A halting code has first coordinate in +$[1/8,3/16]$, including an initial halt reached by loading. For a +nonhalting run both centers of every used branch have first coordinate +at least $1/2$, with half-width at most $a/2$; loading has the same +lower center bound. Hence the particle always stays to the right of +$1/4$. Residual realization proves every analytic assertion. The +separate loading field accounts for the qualification $t\ge1$ in the +periodicity statement. +\end{proof} + + +\subsection{Repeated torus loading branches}\label{ba:torus} + +A loader can instead be one branch of every period. To keep its target +separate from all computational targets, place the initial code in a state +square with no incoming instruction. This condition concerns the entire +target family, not only the distinguished execution. We implement the +same startup principle first with full boxes, then with fixed columns. + +\begin{theorem}[A loader repeated as part of the processor] +\label{ba:periodic-torus} +For every finite machine and input, and every fixed computable $\nu>0$, +there is an effective smooth mean-zero force on the unit flat torus, +one-periodic from time zero, with all mixed derivatives bounded. Its +zero-data solution has pressure zero, is unique in the classical class +of Section~\ref{sec:model}, and satisfies +\[ + \exists t\ge0:\ \Phi_t(1/8,1/8,1/4) + \in\{5/81/(2B)$ from the endpoints +of $[0,1]$, so the new target lies strictly inside the entry square. +There is no other target in that square. Both enlarged branch lists +therefore remain separately separated, with gaps at least $a/B^2$. +The extra source is separated from all state squares. + +Number the branches, including loading, by $1\le i\le M$. Set +\[ + z_0=\tfrac14,\qquad h=\frac1{16(M+1)},\qquad + Z_i=\tfrac12+\frac{i}{4(M+1)},\qquad + \varepsilon=\frac1{10}\min(h,a/B^2). +\] +Thicken each planar source by $[z_0-h,z_0+h]$. Lift the resulting +full box to center height $Z_i$, move it horizontally, and lower it +to $z_0$. For an instruction with displacement $d_i$, use +$D_i=\operatorname{diag}(B^{-d_iw_2},B^{d_iw_2},1)$; +loading has $d_i=0$. Its center is the planar endpoint interpolation +plus $(Z_i-z_0)(w_1-w_3)e_3$. Adjacent middle heights are $4h$ apart, +while the padded half-thickness is $h+\varepsilon<2h$. +Source and target gaps handle the other phases. Horizontal half-widths +never exceed $a/2$, horizontal centers lie between $1/8$ and $3/4+a$, +and vertical centers lie between $1/4$ and $3/4$. The padded motions +are thus strictly inside the unit cube. Lemma~\ref{lem:bb-curl} gives a +chart field, extended periodically in space and time. Its zero time +collars give zero initial velocity. + +The first cycle loads the fixed particle. Each subsequent cycle maps +its exact code to the next local code until the first halt, or forever +in a nonhalting run. Lemma~\ref{lem:bb-movefirst} identifies the finite +checkpoint times with machine steps. The added entry state handles an +original initial halt. A halt code is in the target slab. For a +nonhalting execution every used source and target center, including the +loader, has first coordinate at most $3/8$. Exponential interpolation +keeps the center below that bound and the half-width below $a/2$; +the particle cannot reach $5/8$ at any intermediate time. The final +transition into a halt is not subject to this nonhalting-endpoint bound. +Lemma~\ref{ba:realization} gives the force, its mean, its effective +derivatives, and uniqueness. Repeating the loading field is harmless +to the initialized trajectory and makes the force periodic from zero. +\end{proof} + +\paragraph{A fixed-column implementation.}\label{bc:unit-periodic-section} +The same untargeted-entry condition also permits a repeated initializer +using fixed source and target columns. The next construction retains its +explicit column potentials and its fixed label; its middle motion uses +linear interpolation of the first planar scale. + +\begin{theorem}[A periodic initializer on the unit torus] +\label{bc:unit-periodic-event} +There is an effective halting realization on $\T^3$ with fixed label +$a_*=(3/8,1/4,1/2)$ and fixed strip $5/8R+2$. +Lemma~\ref{lem:bb-lift-parking} gives the full $7N$-motion schedule. + +Repeat this unit pulse from zero. Nonhalting endpoint boxes and all +parking boxes have nonnegative first coordinate. Translations preserve +that bound, and strain takes place in positive parking envelopes. +Therefore a nonhalting path never enters the negative observation box. +A halting code lies in $[-3,-2]\times[0,1]\times\{0\}$ inside it. +The residual proposition finishes the proof. Loading was a repeated +instruction, so no exceptional first-time force is required. +\end{proof} + + +\begin{remark}[A different observation regime] +For comparison, the distinct velocity-field detector of +\cite[Theorem~1.1(2) and Section~5]{OpenAI379EulerianC} uses +$\R^2\times\T$ and the event +$\int_{\{X_2>0\}\times\T}u_3>1/2$. Its force has globally bounded +mixed derivatives and horizontal support compact on each finite time +interval. The nonnegative scalar velocity has diffusion tails; after +loading its total mass is one. Its cutoff estimate gives loss +$\delta<1/24$, nonhalting mass at most $\delta$ in the observed half-space, +and halting mass above $3/4$. Those conclusions follow from the cited +advection--diffusion proof and have no uniform spatial support assertion. +They are a separate force and observation regime from the material test +proved here. +\end{remark} diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/introduction.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/introduction.tex index 2f6645999c682c489b51f51d4b9075ba18a6c12f..d3d9117d1d341a0523590cb2d248e422ec6cbefa 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/introduction.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/introduction.tex @@ -1,173 +1,173 @@ -\section{Introduction}\label{sec:introduction} -A finite instruction can prescribe a volume-preserving affine map on a -box. A fluid realization must do more: it must move every point of that -box, preserve incompressibility outside it, and leave the other instructions -available. Source boxes and destination boxes can overlap. Moreover, a -particle used to report the answer must avoid the observation region -throughout every nonhalting transition. We study these requirements together. - -Our geometric result is Theorem~\ref{lem:bb-balanced-box-routing}. -It extends any finite collection of positive diagonal determinant-one maps -between separately disjoint rational solid boxes to the time-one map of -an effective smooth compactly supported incompressible velocity. The maps -hold on neighborhoods of the entire closed boxes. The proof separates -centers, changes shapes, evacuates the sources to storage, and delivers -boxes along paths avoiding everything still occupied. Positive margins make -the construction effective and permit additional observation guards. - -The first computational application, Theorem~\ref{thm:bb-balanced-three-stack}, -uses all three coordinates for symbolic data. Three stacks retain a tape -and its execution history. Every local instruction preserves the total -length of the prefixes it replaces, so its affine map has determinant one. -A fixed particle, initially in a fluid at rest, enters the fixed cube -$(-1,2)^3$ exactly when the encoded machine halts. The force is smooth, -compactly supported in space and periodic after a one-unit loading interval. -Its prescription uses the finite rule table, not the executed computation. -This application completes the main construction before we vary -initialization, history guards and observers. - -The passage from a symbolic instruction to a fluid observation begins -with retained history, which makes the instruction injective on its -entire domain of tapes, not just along the chosen computation. A rule -specifies finite prefixes and leaves their infinite tails free; gapped -radix expansions place the codes in separately disjoint closed source -and target boxes. Prefix replacement extends to an affine map of each -whole box, which smooth incompressible transport then realizes. -An additional geometric bound must exclude unintended visits to -the observer between successive rule times. Finally, the prescribed -velocity determines a body force, and an energy comparison identifies -the resulting solution in the declared class. Each later variant changes -one or more of these steps while retaining this separation of tasks. - -The remaining constructions answer more specific questions. Two tape -coordinates can be thickened into solid boxes, or their planar scale changes -can be compensated in the normal direction. A temporary mark, a persistent -mark, an explicit frontier and a boundary between occupied and unused -history give different partial instruction maps. We prove their inverses -on their full domains because this is what separates the target boxes. -A bounded observation box permits horizontal motion above it; a slab needs -a coordinate bound throughout that motion. Storage and obstacle detours -supply further alternatives. Loading once gives eventual periodicity, -whereas placing the initial code at the fixed label or repeating a loading -branch gives periodicity from time zero. Onto slow clocks give separate -decaying forces with derivative norms square-integrable in time. - -\subsection{Prior work and the role of forcing} -The computational starting point is Turing's finite machine model -\cite{Turing1936}. Landauer's discussion of retained intermediate information -\cite[Section~3]{Landauer1961} and Bennett's explicit history recording -\cite[Eqs.~(9)--(11)]{Bennett1973} explain how an irreversible computation -can be embedded in an invertible mechanism. We use that principle, but -prove the finite guarded tables and their complete inverses here. No -complexity bound or history-erasure procedure from those works is asserted -for these tables. - -Moore's generalized shifts turn symbolic updates into affine operations -on positional expansions \cite{Moore1990,Moore1991}. Our gapped expansions -serve the same purpose. Moore also realizes invertible generalized shifts -by smooth planar diffeomorphisms and three-dimensional flows -\cite[Section~6, Theorem~12 and its corollary]{Moore1991}. -Cardona, Miranda, Peralta-Salas and Presas extend the affine maps on -neighborhoods of separated planar blocks to an area-preserving disk -diffeomorphism, using contraction, transport, expansion and Moser's area -correction \cite[Proposition~5.1]{CardonaMirandaPeraltaSalasPresas2021}. -This is a geometric predecessor of the routing problem here. Our local -curl formulas implement positive diagonal determinant-one motions in -three dimensions directly; the proof supplies effective support and -neighborhood bounds and controls observation at intermediate times. - -Computational universality has been established for stationary Euler -flows with adapted geometry \cite{CardonaMirandaPeraltaSalasPresas2021}. -Cardona, Miranda and Peralta-Salas also construct universal Euclidean -Beltrami flows, with infinite energy, and robust simulations of -tape-bounded machines by unforced Navier--Stokes flows on a flat torus -\cite[arXiv version~3, Theorem~1 and Remark~29]{CardonaMirandaPeraltaSalas2023Beltrami}. -Dyhr, Gonz\'alez-Prieto, Miranda and Peralta-Salas obtain stationary -unforced Navier--Stokes universality using an adapted metric and the -Hodge Laplacian -\cite[Theorem~A]{DyhrGonzalezPrietoMirandaPeraltaSalas2026}. - -Here the ambient metric is fixed and flat, the viscosity is -prescribed, and the initial velocity is zero. The machine is placed in an -external body force. Given an explicit solenoidal velocity $U$, the identity -$f=U_t+(U\cdot\nabla)U-\nu\Delta U$ supplies that force. The mathematical -work is the effective construction of $U$ and its all-time observation, -followed by uniqueness in the stated comparison class. We do not assert a -regularity theorem for arbitrary fluid data. - -The companion \cite{OpenAI379Shear} gives an entry construction using -solenoidal shears, and \cite{OpenAI379Sheets} treats prefix instructions -and other planar or normal-compensated realizations. They explain adjacent -geometric choices; the proofs of the present full-box mechanisms are given -here. The final input-offset application explicitly imports the planar -processor and spatial clock of \cite{OpenAI379StationaryA2}, with the exact -parameters and conclusions stated in Section~\ref{bc:input-offset-section}. -The companion \cite{OpenAI379EulerianC} detects computation from an -integral of the velocity field on $\R^2\times\T$ via an -advection--diffusion estimate. The end of -Section~\ref{sec:bb-initialization} compares that event with the path of -one material particle studied here. - -\subsection{Equation, effectiveness and reading path} -Throughout $\nu>0$ is fixed, $\T=\R/\Z$, and the fluid equation on -$\R^3$ or the flat unit torus is -\begin{equation}\label{eq:box-NS} - u_t+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f, - \qquad \divg u=0,\qquad u(0)=0. -\end{equation} -A material label $a$ has path $\dot\Phi_t(a)=u(t,\Phi_t(a))$, -$\Phi_0(a)=a$. We also write $X(t;a)=\Phi_t(a)$. The smooth constructed -velocities have global material flows. On the torus pressure is normalized -to have mean zero. Whole-space pressure assumptions are specified in -Section~\ref{sec:model} and in the individual statements. Unless explicitly -projected, the force is a general body force and the constructed pressure -is zero. - -An effective field is a finite program that evaluates every requested -mixed derivative at computably supplied arguments to any prescribed -rational error and computes the bounds used in the proof. The finite -machine and its finite input determine the coefficients, cutoffs and -routing schedule. Evaluating the field cannot first compute the selected -machine run. All algorithmic statements concern computable $\nu$; -the same formulas are valid for arbitrary positive $\nu$, with evaluation -relative to that parameter. The encoded trajectories are exact real -trajectories. No uniform precision tolerance for an unbounded computation -is asserted. - -Section~\ref{sec:model} states the analytic comparison tools. -Section~\ref{sec:bb-boxes} proves the box motions, and -Section~\ref{sec:bb-balanced-three-stack} gives the balanced-stack application. -Sections~\ref{sec:bb-frontiers} and~\ref{ba:codes} establish reusable -recorders and full closed-rectangle coding; Section~\ref{ba:geometry} -supplies the common interpolation formulas. The subsequent constructions -can be read according to the requirement being changed: -\begin{center} -\begin{tabular}{@{}p{.24\textwidth}p{.71\textwidth}@{}} -\toprule -Choice & Where it is developed \\ -\midrule -Initialization and time periodicity & -Section~\ref{sec:bb-initialization} compares a separate loader, direct -placement of the initial code, and a loading branch repeated in every -period. An untargeted start square permits the last choice. \\[4pt] -History domains & -Sections~\ref{ba:marked-routes} and~\ref{sec:bb-guarded} distinguish -persistent marks, fresh history and neighboring-cell guards. Their -inverses on arbitrary allowed tapes determine the separated image boxes. -\\[4pt] -Observer geometry & -Section~\ref{ba:ordered-routes} treats bounded boxes, slabs and -half-spaces. Sections~\ref{sec:solid-routing} and~\ref{sec:torus-routing} -develop storage, detours and fixed torus charts with the corresponding -intermediate-time bounds. \\[4pt] -Decay and stationarity & -The logarithmic clock in Section~\ref{ba:euclidean} produces separate -decaying forces. Section~\ref{sec:clocks} treats further profile choices, -including the imported input-offset processor. \\ -\bottomrule -\end{tabular} -\end{center} -The five whole-space comparison classes are stated in -Lemma~\ref{lem:b-comparison}. Each alternative fixes its own label, -observer, support, pressure class and time dependence; these are separate -constructions rather than simultaneous guarantees for one force. +\section{Introduction}\label{sec:introduction} +A finite instruction can prescribe a volume-preserving affine map on a +box. A fluid realization must do more: it must move every point of that +box, preserve incompressibility outside it, and leave the other instructions +available. Source boxes and destination boxes can overlap. Moreover, a +particle used to report the answer must avoid the observation region +throughout every nonhalting transition. We study these requirements together. + +Our geometric result is Theorem~\ref{lem:bb-balanced-box-routing}. +It extends any finite collection of positive diagonal determinant-one maps +between separately disjoint rational solid boxes to the time-one map of +an effective smooth compactly supported incompressible velocity. The maps +hold on neighborhoods of the entire closed boxes. The proof separates +centers, changes shapes, evacuates the sources to storage, and delivers +boxes along paths avoiding everything still occupied. Positive margins make +the construction effective and permit additional observation guards. + +The first computational application, Theorem~\ref{thm:bb-balanced-three-stack}, +uses all three coordinates for symbolic data. Three stacks retain a tape +and its execution history. Every local instruction preserves the total +length of the prefixes it replaces, so its affine map has determinant one. +A fixed particle, initially in a fluid at rest, enters the fixed cube +$(-1,2)^3$ exactly when the encoded machine halts. The force is smooth, +compactly supported in space and periodic after a one-unit loading interval. +Its prescription uses the finite rule table, not the executed computation. +This application completes the main construction before we vary +initialization, history guards and observers. + +The passage from a symbolic instruction to a fluid observation begins +with retained history, which makes the instruction injective on its +entire domain of tapes, not just along the chosen computation. A rule +specifies finite prefixes and leaves their infinite tails free; gapped +radix expansions place the codes in separately disjoint closed source +and target boxes. Prefix replacement extends to an affine map of each +whole box, which smooth incompressible transport then realizes. +An additional geometric bound must exclude unintended visits to +the observer between successive rule times. Finally, the prescribed +velocity determines a body force, and an energy comparison identifies +the resulting solution in the declared class. Each later variant changes +one or more of these steps while retaining this separation of tasks. + +The remaining constructions answer more specific questions. Two tape +coordinates can be thickened into solid boxes, or their planar scale changes +can be compensated in the normal direction. A temporary mark, a persistent +mark, an explicit frontier and a boundary between occupied and unused +history give different partial instruction maps. We prove their inverses +on their full domains because this is what separates the target boxes. +A bounded observation box permits horizontal motion above it; a slab needs +a coordinate bound throughout that motion. Storage and obstacle detours +supply further alternatives. Loading once gives eventual periodicity, +whereas placing the initial code at the fixed label or repeating a loading +branch gives periodicity from time zero. Onto slow clocks give separate +decaying forces with derivative norms square-integrable in time. + +\subsection{Prior work and the role of forcing} +The computational starting point is Turing's finite machine model +\cite{Turing1936}. Landauer's discussion of retained intermediate information +\cite[Section~3]{Landauer1961} and Bennett's explicit history recording +\cite[Eqs.~(9)--(11)]{Bennett1973} explain how an irreversible computation +can be embedded in an invertible mechanism. We use that principle, but +prove the finite guarded tables and their complete inverses here. No +complexity bound or history-erasure procedure from those works is asserted +for these tables. + +Moore's generalized shifts turn symbolic updates into affine operations +on positional expansions \cite{Moore1990,Moore1991}. Our gapped expansions +serve the same purpose. Moore also realizes invertible generalized shifts +by smooth planar diffeomorphisms and three-dimensional flows +\cite[Section~6, Theorem~12 and its corollary]{Moore1991}. +Cardona, Miranda, Peralta-Salas and Presas extend the affine maps on +neighborhoods of separated planar blocks to an area-preserving disk +diffeomorphism, using contraction, transport, expansion and Moser's area +correction \cite[Proposition~5.1]{CardonaMirandaPeraltaSalasPresas2021}. +This is a geometric predecessor of the routing problem here. Our local +curl formulas implement positive diagonal determinant-one motions in +three dimensions directly; the proof supplies effective support and +neighborhood bounds and controls observation at intermediate times. + +Computational universality has been established for stationary Euler +flows with adapted geometry \cite{CardonaMirandaPeraltaSalasPresas2021}. +Cardona, Miranda and Peralta-Salas also construct universal Euclidean +Beltrami flows, with infinite energy, and robust simulations of +tape-bounded machines by unforced Navier--Stokes flows on a flat torus +\cite[arXiv version~3, Theorem~1 and Remark~29]{CardonaMirandaPeraltaSalas2023Beltrami}. +Dyhr, Gonz\'alez-Prieto, Miranda and Peralta-Salas obtain stationary +unforced Navier--Stokes universality using an adapted metric and the +Hodge Laplacian +\cite[Theorem~A]{DyhrGonzalezPrietoMirandaPeraltaSalas2026}. + +Here the ambient metric is fixed and flat, the viscosity is +prescribed, and the initial velocity is zero. The machine is placed in an +external body force. Given an explicit solenoidal velocity $U$, the identity +$f=U_t+(U\cdot\nabla)U-\nu\Delta U$ supplies that force. The mathematical +work is the effective construction of $U$ and its all-time observation, +followed by uniqueness in the stated comparison class. We do not assert a +regularity theorem for arbitrary fluid data. + +The companion \cite{OpenAI379Shear} gives an entry construction using +solenoidal shears, and \cite{OpenAI379Sheets} treats prefix instructions +and other planar or normal-compensated realizations. They explain adjacent +geometric choices; the proofs of the present full-box mechanisms are given +here. The final input-offset application explicitly imports the planar +processor and spatial clock of \cite{OpenAI379StationaryA2}, with the exact +parameters and conclusions stated in Section~\ref{bc:input-offset-section}. +The companion \cite{OpenAI379EulerianC} detects computation from an +integral of the velocity field on $\R^2\times\T$ via an +advection--diffusion estimate. The end of +Section~\ref{sec:bb-initialization} compares that event with the path of +one material particle studied here. + +\subsection{Equation, effectiveness and reading path} +Throughout $\nu>0$ is fixed, $\T=\R/\Z$, and the fluid equation on +$\R^3$ or the flat unit torus is +\begin{equation}\label{eq:box-NS} + u_t+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f, + \qquad \divg u=0,\qquad u(0)=0. +\end{equation} +A material label $a$ has path $\dot\Phi_t(a)=u(t,\Phi_t(a))$, +$\Phi_0(a)=a$. We also write $X(t;a)=\Phi_t(a)$. The smooth constructed +velocities have global material flows. On the torus pressure is normalized +to have mean zero. Whole-space pressure assumptions are specified in +Section~\ref{sec:model} and in the individual statements. Unless explicitly +projected, the force is a general body force and the constructed pressure +is zero. + +An effective field is a finite program that evaluates every requested +mixed derivative at computably supplied arguments to any prescribed +rational error and computes the bounds used in the proof. The finite +machine and its finite input determine the coefficients, cutoffs and +routing schedule. Evaluating the field cannot first compute the selected +machine run. All algorithmic statements concern computable $\nu$; +the same formulas are valid for arbitrary positive $\nu$, with evaluation +relative to that parameter. The encoded trajectories are exact real +trajectories. No uniform precision tolerance for an unbounded computation +is asserted. + +Section~\ref{sec:model} states the analytic comparison tools. +Section~\ref{sec:bb-boxes} proves the box motions, and +Section~\ref{sec:bb-balanced-three-stack} gives the balanced-stack application. +Sections~\ref{sec:bb-frontiers} and~\ref{ba:codes} establish reusable +recorders and full closed-rectangle coding; Section~\ref{ba:geometry} +supplies the common interpolation formulas. The subsequent constructions +can be read according to the requirement being changed: +\begin{center} +\begin{tabular}{@{}p{.24\textwidth}p{.71\textwidth}@{}} +\toprule +Choice & Where it is developed \\ +\midrule +Initialization and time periodicity & +Section~\ref{sec:bb-initialization} compares a separate loader, direct +placement of the initial code, and a loading branch repeated in every +period. An untargeted start square permits the last choice. \\[4pt] +History domains & +Sections~\ref{ba:marked-routes} and~\ref{sec:bb-guarded} distinguish +persistent marks, fresh history and neighboring-cell guards. Their +inverses on arbitrary allowed tapes determine the separated image boxes. +\\[4pt] +Observer geometry & +Section~\ref{ba:ordered-routes} treats bounded boxes, slabs and +half-spaces. Sections~\ref{sec:solid-routing} and~\ref{sec:torus-routing} +develop storage, detours and fixed torus charts with the corresponding +intermediate-time bounds. \\[4pt] +Decay and stationarity & +The logarithmic clock in Section~\ref{ba:euclidean} produces separate +decaying forces. Section~\ref{sec:clocks} treats further profile choices, +including the imported input-offset processor. \\ +\bottomrule +\end{tabular} +\end{center} +The five whole-space comparison classes are stated in +Lemma~\ref{lem:b-comparison}. Each alternative fixes its own label, +observer, support, pressure class and time dependence; these are separate +constructions rather than simultaneous guarantees for one force. diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/main.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/main.tex index 4e7906a98eb229d082ea574d5c2367b5645b34fc..900f7d70fcc200053cdf0a2a14884796d7295fd1 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/main.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/main.tex @@ -1,56 +1,56 @@ -\documentclass[11pt]{article} -\ifdefined\pdftrailerid\pdftrailerid{}\fi -\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi -\usepackage[T1]{fontenc} -\usepackage[utf8]{inputenc} -\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,enumitem,microtype,longtable,array,tikz} -\usepackage[margin=1in]{geometry} -\usetikzlibrary{arrows.meta,positioning} -\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref} -\newtheorem{theorem}{Theorem}[section] -\newtheorem{lemma}[theorem]{Lemma} -\newtheorem{proposition}[theorem]{Proposition} -\newtheorem{corollary}[theorem]{Corollary} -\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition} -\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark} -\newcommand{\R}{\mathbb R}\newcommand{\T}{\mathbb T}\newcommand{\Z}{\mathbb Z}\newcommand{\N}{\mathbb N}\newcommand{\Q}{\mathbb Q} -\DeclareMathOperator{\supp}{supp}\DeclareMathOperator{\divg}{div}\DeclareMathOperator{\diag}{diag}\DeclareMathOperator{\id}{id} -\hypersetup{pdftitle={Incompressible Box Transport and Finite Computation},pdfauthor={OpenAI}} -\title{Incompressible Box Transport and Finite Computation} -\author{OpenAI} -\date{October 6, 2026} -\begin{document}\maketitle -\begin{abstract} -We realize finite positive diagonal affine maps of determinant one by -effective smooth incompressible flows on neighborhoods of entire closed -rational solid boxes. The source and target families are each disjoint, but may -overlap each other. The construction uses localized curls, evacuation to -storage and obstacle detours. A balanced three-stack recorder then assigns -every machine and finite input a smooth Navier--Stokes force with one -compact spatial support, periodic after a loading interval, at any fixed -positive computable viscosity. The fluid starts at rest, and one fixed -particle enters one fixed open cube exactly when the machine halts. -Further constructions give fixed torus charts, periodicity from time zero, -alternative history guards and bounded or slab observers. Onto slow clocks -yield separate decaying forces. Each construction includes an all-time -observation proof, effective derivative bounds and a stated pressure -comparison class. -\end{abstract} -\input{introduction} -\input{analysis} -\input{geometry} -\input{balanced} -\input{compilers} -\input{coding-conventions} -\input{routing-conventions} -\input{initialization} -\input{compact-processors} -\input{persistent-history} -\input{guarded-history} -\input{observer-routing} -\input{solid-routing} -\input{torus-routing} -\input{clocks} -\bibliographystyle{plain} -\bibliography{references} -\end{document} +\documentclass[11pt]{article} +\ifdefined\pdftrailerid\pdftrailerid{}\fi +\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,enumitem,microtype,longtable,array,tikz} +\usepackage[margin=1in]{geometry} +\usetikzlibrary{arrows.meta,positioning} +\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R}\newcommand{\T}{\mathbb T}\newcommand{\Z}{\mathbb Z}\newcommand{\N}{\mathbb N}\newcommand{\Q}{\mathbb Q} +\DeclareMathOperator{\supp}{supp}\DeclareMathOperator{\divg}{div}\DeclareMathOperator{\diag}{diag}\DeclareMathOperator{\id}{id} +\hypersetup{pdftitle={Incompressible Box Transport and Finite Computation},pdfauthor={OpenAI}} +\title{Incompressible Box Transport and Finite Computation} +\author{OpenAI} +\date{October 6, 2026} +\begin{document}\maketitle +\begin{abstract} +We realize finite positive diagonal affine maps of determinant one by +effective smooth incompressible flows on neighborhoods of entire closed +rational solid boxes. The source and target families are each disjoint, but may +overlap each other. The construction uses localized curls, evacuation to +storage and obstacle detours. A balanced three-stack recorder then assigns +every machine and finite input a smooth Navier--Stokes force with one +compact spatial support, periodic after a loading interval, at any fixed +positive computable viscosity. The fluid starts at rest, and one fixed +particle enters one fixed open cube exactly when the machine halts. +Further constructions give fixed torus charts, periodicity from time zero, +alternative history guards and bounded or slab observers. Onto slow clocks +yield separate decaying forces. Each construction includes an all-time +observation proof, effective derivative bounds and a stated pressure +comparison class. +\end{abstract} +\input{introduction} +\input{analysis} +\input{geometry} +\input{balanced} +\input{compilers} +\input{coding-conventions} +\input{routing-conventions} +\input{initialization} +\input{compact-processors} +\input{persistent-history} +\input{guarded-history} +\input{observer-routing} +\input{solid-routing} +\input{torus-routing} +\input{clocks} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/observer-routing.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/observer-routing.tex index a5f6ba7fd12db7f57d5dce758dfda8556bac7df0..e89f86dd3079ed2263d98f8669f6787d73ace423 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/observer-routing.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/observer-routing.tex @@ -1,982 +1,982 @@ -\section{Routing choices dictated by the observer} -\label{ba:ordered-routes} - -The preceding processors move every branch by a prescribed affine map. -We now ask which parts of that motion must be controlled between the -integer times. A bounded observation box allows horizontal work above -the observer. A slab observer requires a bound on the horizontal -coordinate throughout the motion. Finally, a half-space observer can be -protected by moving all sources to storage before delivering any target. -These three requirements lead to the constructions below. - -\subsection{Four motions above a bounded observer} -\label{ba:four-stage-section} - -We first separate ascent, scaling, translation and descent. The two -planar motions occur at a fixed height, so their exclusion from the -bounded observer follows without a horizontal coordinate estimate. - -Let $D_i,E_i\subset\mathbb R^2$, $1\le i\le N$, be closed rational -axis-aligned rectangles with positive side lengths. Assume the $D_i$ -are pairwise disjoint and the $E_i$ are pairwise disjoint, and let -\[ - A_i(b)=c_i'+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(b-c_i), - \qquad \lambda_i\in\mathbb Q_{>0},\qquad A_i(D_i)=E_i, -\] -where $c_i,c_i'$ are their centers. No separation between a source and -a target is assumed. All distances used to choose coordinate padding -below are distances for the maximum norm. - -Let $\vartheta=\sigma$ be the effective flat switch in -\eqref{eq:p2-step}. For a closed interval, possibly reduced to one -point, define -\[ - \chi_{[a,b],\epsilon}(v)= - \vartheta\!\left(\frac{2(v-a+\epsilon)}\epsilon\right) - \vartheta\!\left(\frac{2(b+\epsilon-v)}\epsilon\right). -\] -For a box $K$, let $\chi_{K,\epsilon}$ be the product over its three -coordinates. It is one on the $\epsilon/2$ enlargement of $K$ and -zero outside its $\epsilon$ enlargement. The effective profile estimates -following \eqref{eq:p2-step} apply to these products and rescalings. - -\begin{lemma}[Four-stage realization]\label{ba:four-stage} -The rectangle data determine effectively a smooth divergence-free field -$W(s,x)$ supported in a compact subset of $(0,1)\times\mathbb R^3$. -Its time-one map is $(b,z)\mapsto(A_i(b),z)$ on a neighborhood of -each $D_i\times\{0\}$. For a point of that plate, every planar -motion occurs at height $4i$, while ascent and descent preserve its -source and target planar coordinates, respectively. The scaling can -use either linear interpolation of its positive first factor or linear -interpolation of the logarithm of that factor. -\end{lemma} - -\begin{proof}\label{ba:four-stage-proof} -The empty family is realized by zero. Otherwise put $h_i=4i$, -$w_i=c_i'-c_i$, and -\[ - \Theta_k(s)=\vartheta(10s-(2k-1)),\qquad 1\le k\le4. -\] -Only the $k$th motion is active on the interval -$[(2k-1)/10,2k/10]$; the four intervals are disjoint. If -$a_{ij},a'_{ij}$ are the source and target half-widths, respectively, -take the four swept boxes -\[ - \begin{aligned} - K_{i1}&=D_i\times[0,h_i],\\ - K_{i2}&=\{c_i+(u_1,u_2):|u_j|\le\max(a_{ij},a'_{ij})\} - \times\{h_i\},\\ - K_{i3}&=\operatorname{box}(E_i-w_i,E_i)\times\{h_i\},\\ - K_{i4}&=E_i\times[0,h_i]. - \end{aligned} -\] -Here $\operatorname{box}$ denotes the smallest axis-aligned rectangle -containing the two displayed rectangles. Within each of the four -families these boxes are disjoint: source separation handles the first, -target separation the fourth, and distinct heights the other two. -Choose a positive rational $\epsilon<1$ smaller than one quarter of -every pairwise distance within these families. A family with fewer than -two members contributes no restriction. Its $\epsilon$ enlargements -are then still disjoint. - -Choose either -\[ - \mu_i=1+(\lambda_i-1)\Theta_2 - \quad\hbox{or}\quad \mu_i=\lambda_i^{\Theta_2}, - \qquad \gamma_i=\dot\mu_i/\mu_i. -\] -The factors $\mu_i$ and $\mu_i^{-1}$ stay between their respective -endpoint values. Use the coordinate potentials -\[ - \begin{aligned} - B_{i1}&=(0,h_i x_1,0),& - B_{i2}&=(0,0,(x_1-c_{i1})(x_2-c_{i2})),\\ - B_{i3}&=(0,0,w_{i1}x_2-w_{i2}x_1),& - B_{i4}&=(0,-h_i x_1,0). - \end{aligned} -\] -For the logarithmic-scale construction we also retain the alternative -translation potentials, with $e_3=(0,0,1)$, -\[ - B_{i1}=\tfrac12(h_ie_3)\times x,\qquad - B_{i3}=\tfrac12(w_{i1},w_{i2},0)\times x,\qquad - B_{i4}=\tfrac12(-h_ie_3)\times x. -\] -Their curls equal the same constant translation vectors. Their localized -fields need not agree away from the plates. Either choice gives -\[ - W=\sum_i\left[ - \dot\Theta_1\nabla\times(\chi_{K_{i1},\epsilon}B_{i1}) - +\gamma_i\nabla\times(\chi_{K_{i2},\epsilon}B_{i2}) - +\dot\Theta_3\nabla\times(\chi_{K_{i3},\epsilon}B_{i3}) - +\dot\Theta_4\nabla\times(\chi_{K_{i4},\epsilon}B_{i4})\right]. -\] -This is a finite sum of spatial curls. Its spatial support is compact, -its time support is contained in $[1/10,8/10]$, and the lower bound -$\mu_i\ge\min(1,\lambda_i)>0$ proves smoothness and effectivity. - -For $b\in D_i$, the successive paths are -\[ - \begin{array}{c|c} - \text{motion}&\text{position}\\\hline - \text{ascent}&(b,h_i\Theta_1)\\ - \text{scaling}&(c_i+\operatorname{diag}(\mu_i,\mu_i^{-1})(b-c_i),h_i)\\ - \text{translation}&(A_i(b)-w_i+\Theta_3w_i,h_i)\\ - \text{descent}&(A_i(b),h_i(1-\Theta_4)). - \end{array} -\] -Each path stays in its swept box. On a neighborhood of that box the -uncut curls are, in order, -\[ - h_ie_3,\qquad (x_1-c_{i1},-(x_2-c_{i2}),0),\qquad - (w_{i1},w_{i2},0),\qquad -h_ie_3. -\] -After multiplication by the time coefficients, these are the prescribed -affine velocities. The motion matrices are positive diagonal of determinant -one, the plateau margin is $\epsilon/2$, and the support padding is -$\epsilon$. The disjoint swept boxes therefore permit the plateau -argument of Lemma~\ref{lem:bb-curl}, with $\eta=\epsilon/2$. -It identifies these particular localized fields with the displayed -paths on the plateaux. The normalized diagonal factors are bounded by -$L_i=\max(1,\lambda_i,\lambda_i^{-1})$, so that lemma gives the -full initial neighborhood of radius $\epsilon/(4L_i)$. -\end{proof} - -We use the following precise Euclidean comparison class in the applications: -on each $[0,T]$, -\begin{equation}\label{ba:euclidean-comparison-class} - \begin{gathered} - u\in C([0,T];H^2(\mathbb R^3))\cap C^1([0,T];L^2(\mathbb R^3)),\\ - u,\nabla u\in L^\infty([0,T]\times\mathbb R^3),\qquad - p\in C([0,T];H^1(\mathbb R^3)). - \end{gathered} -\end{equation} -The pressure representative is part of this condition; it is zero for -our constructed solution. These comparison velocities and pressures -satisfy case (i) of Lemma~\ref{lem:b-comparison}. We retain the stated -bounded-gradient requirement here, although that case does not need it. - -That lemma also gives a smooth material diffeomorphism on every finite -interval. Differentiating its trajectory equation gives -$\dot J=(D_xu)(t,\Phi_t(a))J$, $J(0)=I$, for $J=D_a\Phi_t$. -Consequently -\begin{equation}\label{ba:material-volume} - \frac d{dt}\det J=(\nabla\cdot u)(t,\Phi_t(a))\det J=0, - \qquad \det J=1. -\end{equation} -The same proof applies to periodic lifts on the torus. The residual -formulas below are affine in viscosity, so they also define the asserted -solutions for any real $\nu>0$, with effectivity relative to $\nu$. - -\begin{theorem}[Two bounded-box material tests]\label{ba:bounded-box-tests} -Fix a positive computable viscosity $\nu$. A deterministic one-tape -machine and finite input effectively determine each of the following -general body forces on $[0,\infty)\times\mathbb R^3$. Both have zero -initial velocity, a unique global smooth velocity in -\eqref{ba:euclidean-comparison-class}, and constructed pressure zero. -\begin{enumerate} -\item The force and velocity have one fixed compact spatial support, all -mixed derivatives bounded, and period one for $t\ge1$. The particle -from $0$ enters $(-5,-2)\times(-1,2)\times(-1,1)$ exactly when the -machine halts. -\item The force and velocity have one fixed compact spatial support and, -for all $k\ge0$ and spatial multi-indices $\alpha$, satisfy -\begin{equation}\label{ba:eighteen-decay} - \|\partial_t^k\partial_x^\alpha u(t)\|_\infty+ - \|\partial_t^k\partial_x^\alpha f(t)\|_\infty - \le C_{k,\alpha}(1+t)^{-1-k}. -\end{equation} -The particle from $(2,0,0)$ enters -$(-1/2,3/2)\times(-1/2,3/2)\times(-1/2,1/2)$ exactly when the -machine halts. In particular $f\in L^2([0,\infty);H^m(\mathbb R^3))$ -for every integer $m\ge0$. -\end{enumerate} -The supports and constants may depend on the machine and input. The -labels and observers do not. All force derivatives admit evaluation to -any prescribed rational error by a finite program. -\end{theorem} - -\begin{proof}\label{ba:bounded-box-tests-proof} -Normalize to a single halt state $q_H$ and a complete nonhalting table, -as in Section~\ref{ba:codes}. Apply the move-first recorder of -Lemma~\ref{lem:bb-movefirst}. Its history frontier starts at cell $2$; -all other history cells are empty and all mark bits are zero. At -checkpoint $n$, the frontier is $g_n=n+2$, the work head satisfies -$|h_n|\le n$, and the next work instruction ending at $h'$ is completed -in exactly $4+2(g_n-h')$ local steps. Thus every required checkpoint -is reached after finitely many steps. The incoming-direction and -written-symbol inverse properties hold on the whole partial table. -The records recover the successive work instructions and hence the -absolute head position, even though the geometric code is head-relative. - -Let $m$ be the full alphabet size. Use odd digits $\beta(a)$ in -$\{1,3,\ldots,2m-1\}$, with base $B=2m+1$ in the first construction -and $B=2m+2$ in the second. Set -\[ - C(a_0a_1\cdots)=\sum_{r\ge0}\beta(a_r)B^{-r-1},\quad - P_a(v)=\frac{\beta(a)+v}{B},\quad D_a(v)=Bv-\beta(a),\quad - I_a=P_a([0,1]). -\] -Let $\xi$ encode the left string, nearest cell first, and $\eta$ -the string starting at the head. The complete right, stay and left -branch maps are \eqref{eq:bb-one-cell-branches}, with its digit -$d(a)$ equal to $\beta(a)$ and its move $e$ equal to $d$ here. -The left move is split over every full symbol $c$. The shared lemma -proves the full-rectangle endpoint formulas, arbitrary-tail action and -linear part $\operatorname{diag}(B^{-d},B^d)$. - -Translate each state square horizontally. In the first construction -give the halt state offset $-4$ and the other states offsets $2+2r$, -$r\ge0$; in the second use halt offset $0$ and other offsets $3r$, -$r\ge1$. Thus the code is $(o_s+\xi,\eta,0)$. -Lemma~\ref{lem:bb-onecell} applies with exactly these odd digits and -either displayed base: its incoming conditions are those supplied by -the move-first recorder, and \eqref{eq:bb-one-cell-branches} is its minimal -left-split branch list. Thus both complete closed rectangle families -are separately disjoint, with within-state gaps at least $B^{-2}$. -This proves the hypotheses of Lemma~\ref{ba:four-stage}; no separation -between the source and target families is required. - -Use its linear scale and coordinate potentials for the first construction; -use its logarithmic scale and symmetric translation potentials for the -second. In the latter case the scaling coefficient is exactly -$(\log\lambda_i)\dot\Theta_2$ and the scale matrix is -$\operatorname{diag}(\lambda_i^{\Theta_2},\lambda_i^{-\Theta_2})$. -Denote the resulting step field by $W_{\rm step}$. - -The initial code $P_{\rm in}$ is rational. A constant full-symbol -tail of digit $\beta_0$ after $M$ positions contributes -$B^{-M}\beta_0/(B-1)$; all other contributions are finite. For the -appropriate fixed label $a_*$, put $v_0=P_{\rm in}-a_*$ and let -$J_0$ be the smallest box containing $[a_*,P_{\rm in}]$. The loader -\[ - W_{\rm in}(s,x)=\frac d{ds}\vartheta(2s-1/2)\, - \nabla\times\left(\chi_{J_0,1}(x)\frac{v_0\times x}{2}\right) -\] -has the exact marked path $a_*+\vartheta(2s-1/2)v_0$, since the -uncut curl is $v_0$ and the entire segment lies in the plateau. Form -\[ - W(s,x)=W_{\rm in}(s,x)+\sum_{n\ge1}W_{\rm step}(s-n,x). -\] -The zero time collars make this smooth, initially zero, and periodic for -$s\ge1$, with common compact spatial support and bounded mixed derivatives. - -At time $s=1+k$ the marked particle is exactly the code after $k$ local -instructions, through the first halt or indefinitely in a nonhalting run. -This follows by induction from the full-rectangle maps. Each halt code -lies in its stated observation box, including one reached by loading an -initially halted machine. In a nonhalting run, the first loader has -$x_1\ge0$ and later nonhalt codes have $x_1\ge2$; for the second these -bounds are $x_1\ge2$ and $x_1\ge3$. Ascent and descent leave those -planar coordinates unchanged. Every planar motion has height $4i\ge4$, -outside both observers. The intervening stationary intervals add no new -positions. This proves exclusion at every real time. - -For the first case take $u=W$ and its residual force. For the second set -\[ - s(t)=\log(1+t),\quad \theta(t)=(1+t)^{-1},\quad - u(t,x)=\theta(t)W(s(t),x), -\] -whose force is -\[ - f=\theta^2\{-W+\partial_sW+(W\cdot\nabla)W\}(s(t),x) - -\nu\theta\Delta W(s(t),x). -\] -Lemma~\ref{ba:log-clock} applies to this fixed-support, bounded-derivative -template. Its identity -$\partial_t(\theta^jY(s(t)))=\theta^{j+1}(\partial_s-j)Y(s(t))$ -proves \eqref{ba:eighteen-decay} for every $k,\alpha$. Fixed support -then gives the stated $L^2_tH^m_x$ conclusion. Its clock is onto, with -inverse $t=e^s-1$, so the material path is the old path composed with -$s(t)$ and the event is unchanged. - -In both cases smoothness, fixed support and bounded derivatives on -finite time intervals put the reference velocity in $CH^2\cap C^1L^2$ -with bounded velocity and gradient. It is solenoidal and initially zero. -Lemma~\ref{lem:b-comparison}(i) therefore gives the exact solution -$(u,0)$ and uniqueness in \eqref{ba:euclidean-comparison-class}. -The effective profiles following \eqref{eq:p2-step}, applied to the -finite table, rational initial code, finitely many boxes and repeated -template, give all derivative evaluations. A coarse time approximation -encloses finitely many possible active translates, avoiding equality -tests at seams. The prescription never computes the machine's run. -\end{proof} - -\subsection{Small plates and a slab observer on the unit torus} -\label{ba:torus-plates-section} - -A slab observer has no height restriction, so elevation alone cannot -protect it. We now keep each nonhalting branch inside a narrow range of -first coordinates throughout its motion. The delayed-head recorder also -allows arbitrary finitely supported work data on both sides of the head. - -\begin{theorem}[Torus plates and square-integrable forcing] -\label{ba:torus-plates} -Fix a positive computable $\nu$. Every deterministic one-tape machine -with finitely specified tape, blank outside a finite set, effectively -determines a smooth mean-zero general force on -$\mathbb T^3=\mathbb R^3/\mathbb Z^3$, with bounded mixed derivatives -and period one for $t\ge1$. Its zero-data Navier--Stokes solution is -globally smooth and unique among smooth solutions, with pressure normalized -to mean zero and constructed pressure zero. For -$P_*=(1/4,1/2,1/4)$ and $X(t)=\Phi_t(P_*)$, the event -\begin{equation}\label{ba:torus-plate-event} - \exists t\ge0:\quad X_1(t)\pmod1\in(1/2,7/8) -\end{equation} -is equivalent to halting. - -A separate effective force has the same zero data, event, uniqueness and -mean-zero property, is supported in one compact sub-box of $(0,1)^3$, -has all mixed derivatives bounded, and satisfies -\begin{equation}\label{ba:cube-rate} - \|\widehat f(t,\cdot)\|_\infty=O((1+t)^{-2/3}). -\end{equation} -In particular $\widehat f\in L^2([0,\infty)\times\mathbb T^3)$. -\end{theorem} - -\begin{proof}\label{ba:torus-plates-proof} -Use Lemma~\ref{ba:six-compiler} with the frontier initially at cell $1$. -Its main control $M_{q,e}$ records the last work displacement $e$, initially -zero. In one work step it writes and marks the old head, records -$(q,e,a)$ at the frontier, returns to the mark, and only then makes the -work displacement. All other initial history cells are blank and every -mark is zero. At checkpoint $n$ the frontier is $n+1$, $|h_n|\le n$, -and the next checkpoint takes exactly $2(n+1-h_n)+3$ local steps. -These scans test history and marks, so the proof is unaffected by work -symbols on either side of the head. Main controls with halting $q$ have -no outgoing rules. The full-domain incoming properties are those of -the cited lemma. - -Let $S$ be the local state set, $N=|S|$, let $S_h$ be its halting main -states, and enumerate $S$ by $n(s)=0,\ldots,N-1$. Put -\[ - \ell=\frac1{16(N+1)},\quad z_0=\frac14,\quad - x_s=\frac18+2\ell n(s)+\frac12\mathbf1_{s\in S_h},\quad - T_s(\xi,\eta)=(x_s+\ell\xi,1/4+\ell\eta). -\] -Nonhalting squares lie in $1/8\le x_1<1/4$ and halting squares in -$5/8\le x_1<3/4$; distinct squares have gap at least $\ell$. -Give the full alphabet, of size $m$, odd digits in base $\beta=2m+1$. -Conjugating \eqref{eq:bb-one-cell-branches} by the source and target charts -gives separately disjoint rectangle families $R_j^0,R_j^1$ with positive -diagonal factors $\alpha_j,\alpha_j^{-1}$, -$\alpha_j\in\{\beta^{-1},1,\beta\}$. Every chart has the same -scale, so this remains the physical linear part. -Lemma~\ref{lem:bb-onecell}, with this uniform chart scaling, gives -within-family coordinate gaps at least $\ell/\beta^2$, including the -two-letter target intervals for left moves. - -For $1\le j\le L$, write the centers as $p_j^0,p_j^1$, the source -half-widths as $a_j,b_j$, and the target half-widths as -$\alpha_ja_j,\alpha_j^{-1}b_j$. All endpoint half-widths are at -most $\ell/2$. Choose -\[ - h_j=\frac12+\frac{j}{4(L+1)},\qquad - \epsilon=\frac1{100}\min\left(1,\frac\ell{\beta^2}, - \frac1{4(L+1)}\right). -\] -With $\theta(s)=\vartheta(2s-1/2)$ and -$\Theta_k(\tau)=\theta(3\tau-k+1)$ for $k=1,2,3$, define -\[ - \begin{aligned} - C_j(\tau)&=\big((1-\Theta_2)p_j^0+\Theta_2p_j^1, - z_0+(\Theta_1-\Theta_3)(h_j-z_0)\big),\\ - \sigma_j&=1+(\alpha_j-1)\Theta_2,\qquad - \lambda_j=\dot\sigma_j/\sigma_j,\\ - (w_{j1},w_{j2},w_{j3})&=(a_j\sigma_j,b_j/\sigma_j,0). - \end{aligned} -\] -These centers and half-widths describe moving plates. Their -$\epsilon$ enlargements are disjoint: the first third uses source -projections, the middle third distinct heights, and the last third -target projections. The padding is at most one hundredth of every -relevant gap. All plates and padding lie in a compact sub-box of -$(0,1)^3$, since their centers interpolate between the stated chart -centers, widths are at most $\ell$, heights lie between $1/4$ and -$3/4$, and $\ell\le1/32$, $\epsilon\le1/100$. - -For $w\ge0$ put -\[ - \kappa_\epsilon(w,v)= - \vartheta\!\left(\frac{2(w+\epsilon-v)}\epsilon\right) - \vartheta\!\left(\frac{2(w+\epsilon+v)}\epsilon\right),\qquad - \chi_j=\prod_{a=1}^3\kappa_\epsilon(w_{ja},x_a-C_{ja}). -\] -Writing $y=x-C_j$, use -\[ - A_j=\tfrac12\dot C_j\times y+(0,0,\lambda_jy_1y_2), - \qquad V(\tau,x)=\sum_j\nabla\times(\chi_jA_j). -\] -Extend the potentials by zero outside the chart. The support remains -away from its faces, so this gives a smooth torus field, zero near the -time endpoints. On the whole plateau of plate $j$ its velocity is -$\dot C_j+(\lambda_jy_1,-\lambda_jy_2,0)$; other summands vanish -there. Hence the path from $(p_j^0+(r,s),z_0)$ is exactly -\begin{equation}\label{ba:plate-path} - C_j(\tau)+(\sigma_j(\tau)r,\sigma_j(\tau)^{-1}s,0), - \qquad |r|\le a_j,\quad |s|\le b_j. -\end{equation} -It ends at the prescribed affine image. The plateau argument of -Lemma~\ref{lem:bb-curl} applies with -$D_j=\operatorname{diag}(\sigma_j,\sigma_j^{-1},1)$, -$\eta=\epsilon/2$, and diagonal bound -$\max(1,\alpha_j,\alpha_j^{-1})$: the required enlarged-box separation -was just proved. It gives the same motion on an initial neighborhood -of radius $\epsilon/[4\max(1,\alpha_j,\alpha_j^{-1})]$, with constant -normal displacement, and therefore includes boundary codes. - -The initial code $P_{\rm in}$ is rational, since relative to constant -blank tails only finitely many work and frontier digits differ. Load it -from $P_*$ along $C_*=(1-\theta)P_*+\theta P_{\rm in}$ using -\[ - V_0(t,x)=\nabla\times\left[ - \prod_{a=1}^3\kappa_{1/64}(0,x_a-C_{*,a}(t)) - \frac{\dot C_*(t)\times(x-C_*(t))}{2}\right]. -\] -The segment and its collar stay within the chart: its first coordinate -stays between $1/8$ and $3/4$, its second between $1/4$ and -$1/2+\ell$, and its third is $1/4$. Its plateau curl is $\dot C_*$. -Take $U=V_0$ on $[0,1]$ and $U(t)=V(t-n)$ on $[n,n+1]$ for $n\ge1$. -Time collars make this smooth and initially zero. It has fixed compact -chart support, bounded mixed derivatives, and period one after loading. -Define $f=U_t+(U\cdot\nabla)U-\nu\Delta U$ using the stated physical -viscosity. Both $U$ and $f$ have mean zero: $U$ is a curl, the -Laplacian integrates to zero, and -$(U\cdot\nabla)U=\operatorname{div}(U\otimes U)$. -The torus clause of Lemma~\ref{lem:b-comparison} applies to this -smooth, initially zero, solenoidal reference field and gives the unique -smooth solution $(U,0)$ with mean-zero pressure. - -Equations~\eqref{eq:bb-one-cell-branches} and \eqref{ba:plate-path} identify -the particle at time $1+k$ with the $k$th local configuration. The -recorder invariant ensures that a nonhalting computation continues -indefinitely. A halt, including an initial halt reached by loading, -puts the particle in $5/8\le X_1<3/4$, proving the event. For a -nonhalting computation the loader has $X_1\le1/4$ and every later -branch has both endpoint centers below $1/4$. Its center is their -convex interpolation and its first half-width is at most $\ell/2$. -Thus $00$; the cutoff -estimates and zero collars allow derivative evaluation at approximate real -arguments without tests for exact boundary equality. Only positive -arguments enter the cube root. The program repeats the entire local -table and never uses a computed execution. -\end{proof} - -\subsection{Storage and periodic forcing without a loader} -\label{ba:parking-section} - -The next construction absorbs initialization into the positions of the -state rectangles. A zero digit for the full blank symbol places the -initial left stack at zero; an input-dependent second-coordinate translation then -places the entire initial code at one fixed particle. To protect a -half-space observer during every period, we first evacuate all sources -to separate storage rectangles and only afterwards fill the targets. - -\begin{lemma}[Zero-blank coding]\label{ba:zero-blank-code} -Let an alphabet of size $m>1$ have digits $0,\ldots,m-1$, with full -blank digit zero. Set $B=m+1$, $\kappa=(m-1)/m$, and -\[ - \operatorname{val}(a_0a_1\cdots)=\sum_{r\ge0}a_rB^{-r-1},\qquad - I_a=\frac{a+[0,\kappa]}B,\qquad I_{a,b}=\frac{a+I_b}B. -\] -The code is injective, takes values in $[0,\kappa]$, and its distinct -one-letter and two-letter cylinders have positive rational gaps. For a -deterministic partial table with target-state-determined incoming direction -and incoming rule determined by target state and written symbol, separated -state translates of $[0,\kappa]^2$ yield separately disjoint full source -and target rectangles by the following maps in state-relative coordinates: -\begin{equation}\label{ba:zero-digit-maps} - \begin{array}{c|c|c|c} - d&\text{source}&(\xi',\eta')&\text{target}\\\hline - 1&[0,\kappa]\times I_a&((b+\xi)/B,B\eta-a)&I_b\times[0,\kappa]\\ - 0&[0,\kappa]\times I_a&(\xi,\eta+(b-a)/B)&[0,\kappa]\times I_b\\ - -1&I_c\times I_a&(B\xi-c,c/B+(b+B\eta-a)/B^2)& - [0,\kappa]\times I_{c,b}. - \end{array} -\end{equation} -The last row is split over all $c$. Each map has positive diagonal -linear part of determinant one and acts exactly on the full rectangle. -\end{lemma} - -\begin{proof}\label{ba:zero-blank-code-proof} -The maximum tail is $(m-1)/(B-1)=\kappa$, and -$(m-1+\kappa)/B=\kappa$. Consecutive first-digit intervals have -gap $(1-\kappa)/B>0$. Within one first-digit interval the second-digit -gaps are this quantity divided by $B$; different first digits already -give a gap. The interval containing a code recovers its first digit, -and $v\mapsto Bv-a$ then exposes the tail. Repetition proves -injectivity, including endpoint codes. - -Writing $a$ to $b$ changes the first right digit. A right move pushes -$b$ onto the left string and removes $a$ from the right; a left move -removes $c$ from the left and prepends $c,b$ to the old right tail. -These identities give \eqref{ba:zero-digit-maps}, including its images -and reciprocal slopes. Determinism and digit gaps separate sources by -state, read digit, and $c$ when needed. At one target state the common -incoming direction determines where the written digit is visible: in -$I_b$ on the left or right, or in $I_{c,b}$. The incoming-rule -hypothesis and digit gaps then separate the closed target rectangles. -\end{proof} - -\begin{lemma}[Evacuation, storage and delivery]\label{ba:parking} -Suppose $A_i:D_i\to E_i$, $1\le i\le N$, are the rational reciprocal -rectangle maps of Lemma~\ref{ba:four-stage}, with $N\ge1$ and every -$D_i\subset\{x_1<0\}$. There is an effective smooth divergence-free -field on $\mathbb R\times\mathbb R^3$, one-periodic in time, zero near -each integer time, and uniformly compactly supported in space, whose -one-period map is $(b,z)\mapsto(A_i(b),z)$ on a neighborhood of every -$D_i\times\{0\}$. If $E_i\subset\{x_1<0\}$, the entire -one-period path of each point in $D_i\times\{0\}$ remains in that -half-space. -\end{lemma} - -\begin{proof}\label{ba:parking-proof} -Let $d_i,b_i$ be the source and target centers, and let $a_{ij},a'_{ij}$ -be their half-widths. Put $a^*_{ij}=\max(a_{ij},a'_{ij})$. -Choose $\epsilon$ to be one quarter of the minimum of $1$ and all -pairwise maximum-norm distances within the source and target families; -omit pair entries when $N=1$. Define -\[ - \begin{gathered} - x_{\min}=\min\{x_1:(x_1,x_2)\in D_i\cup E_i\text{ for some }i\},\\ - R=\max(1,\max_i a^*_{i1}),\qquad J=4(R+1),\qquad - s_i=(x_{\min}-iJ,0). - \end{gathered} -\] -The storage rectangle at $s_i$ with half-widths $a^*_{ij}$ lies in -$x_1<0$. Its first-coordinate projection is separated by more than one -from every source and target and from the other storage projections: -the relevant gaps are at least $J-R>1$ and $J-2R>1$. - -We need only a localized translation and a localized reciprocal scaling. -On a time segment $[t_0,t_0+\delta]$, use -$g(t)=\vartheta(3(t-t_0)/\delta-1)$, whose derivative is supported -in the middle third. Using the cutoff $\kappa_\epsilon$ defined in -the proof of Theorem~\ref{ba:torus-plates}, for half-widths $a,b$ write -\[ - \chi_{a,b}(y)=\kappa_\epsilon(a,y_1) - \kappa_\epsilon(b,y_2)\kappa_\epsilon(0,y_3). -\] -A plate translation from center $P$ to $Q$ is produced by -\[ - C(t)=(1-g)P+gQ,\qquad - \nabla\times\left[ - \tfrac12\chi_{a,b}(x-C(t))\dot C(t)\times(x-C(t))\right]. -\] -On its plateau the velocity is $\dot C$ and the exact path is -$C(t)+(r,s,0)$. At storage center $P=(s_i,0)$, take -$q(t)=1+(\lambda_i-1)g(t)$ and the field -\[ - \nabla\times\left(0,0, - \frac{\dot q}{q}(x_1-P_1)(x_2-P_2) - \chi_{a^*_{i1},a^*_{i2}}(x-P)\right). -\] -It has the exact path $P+(qr,q^{-1}s,0)$. The denominator has the -positive lower bound $\min(1,\lambda_i)$, and the whole path lies in -the storage rectangle, because each half-width stays between its source -and target values. - -Partition a period into $7N$ equal segments and take $Z=2$. For each -$i$ in order, use four segments for -\[ - (d_i,0)\longrightarrow(d_i,Z)\longrightarrow(s_i,Z) - \longrightarrow(s_i,0),\qquad\text{then scale at }s_i. -\] -Only after all these evacuations, use three segments for each $i$ to -deliver its target shape along -\[ - (s_i,0)\longrightarrow(s_i,Z)\longrightarrow(b_i,Z) - \longrightarrow(b_i,0). -\] -Sum these segment fields and extend periodically. The middle-third -time switches give zero neighborhoods of every segment endpoint and -every integer time. The result is a smooth finite sum of curls with -one compact spatial support. - -This is the $7N$ schedule of Lemma~\ref{lem:bb-lift-parking}(ii). -Its separation hypotheses have explicit margins here. Unpadded -source-source and target-target gaps are at least $4\epsilon$, so their -$\epsilon$ enlargements leave gap at least $2\epsilon$. Storage -gaps exceed one, leaving more than $1-2\epsilon>0$ after padding. -At horizontal travel height the enlarged moving plate and every -enlarged stationary plate have vertical gap $2-2\epsilon>0$. -Every scaling stays in its reserved storage rectangle. The lemma -therefore gives simultaneous noninterference for evacuation and delivery, -including when a source meets a target. - -The localized potentials above have exactly these affine generators on -their plateaux, so the plateau argument of Lemma~\ref{lem:bb-curl} -gives endpoint -$(d_i+(r,s),0)\mapsto(b_i+(\lambda_i r,\lambda_i^{-1}s),0)$, -with normal displacement unchanged. Put -$M=\max_i(1,\lambda_i,\lambda_i^{-1})$ and -$d_*=\min(2\epsilon,1-2\epsilon,2-2\epsilon)>0$. -A neighborhood radius smaller than both $\epsilon/(8M)$ and -$d_*/(4M)$ retains the plateaux and all exclusion margins throughout -the concatenation. This applies the shared proofs to the displayed -potentials and does not identify their fields away from the plateaux. - -If $E_i$ is negative, each translation of an individual point joins two -negative first coordinates, and its linear interpolation remains negative. -Scaling occurs inside the negative storage rectangle. The point is -stationary during all other segments. This proves the asserted -whole-period exclusion. -\end{proof} - -\begin{theorem}[A periodic force with no loading interval] -\label{ba:periodic-parking} -Fix a positive computable $\nu$. A deterministic one-tape machine and -finite input word in cells $0,\ldots,L-1$, with other cells blank and -initial head zero, effectively determine a general body force on -$\mathbb R^3$ extending smoothly and one-periodically to all real times. -It has uniformly compact spatial support and bounded mixed derivatives. -Its zero-data solution has a unique global smooth velocity in -\eqref{ba:euclidean-comparison-class}, with constructed pressure zero, and -\[ - \exists t\ge0:\quad (\Phi_t((-2,0,0)))_1>0 - \quad\Longleftrightarrow\quad\text{the machine halts}. -\] -The fixed label is already the initial code; periodicity begins at zero. -\end{theorem} - -\begin{proof}\label{ba:periodic-parking-proof} -Insert a fresh nonhalting start state that preserves the read letter, -stays put and enters the original start state. Replace missing -nonhalting instructions by letter-preserving stay instructions to a halt -state. This finite normalization preserves the halting question, -including an initially halted machine, and guarantees a nonhalting -initial control and at least one rule. - -Apply the six-row delayed-head recorder of Lemma~\ref{ba:six-compiler} -with initial frontier $1$. Initially every other history cell is blank -and every mark is zero. The main control records $(q,e)$, where $e$ -is the preceding displacement, and the history record is $(q,e,a)$. -The old head is marked before the right scan and cleared on return, -immediately before displacement into the next main state. At checkpoint -$k$ the frontier is $k+1$ and the head $i$ satisfies $|i|\le k$; -the next checkpoint uses $2(k+1-i)+3\le4k+5$ local steps. The -incoming-direction and written-symbol properties hold on the full -domain, as required by Lemma~\ref{ba:zero-blank-code}. - -Give the full blank symbol digit zero and the other full symbols digits -$1,\ldots,m-1$. Order nonhalting local states with the initial state -first and assign offsets $-2h$, $h=1,2,\ldots$; assign halting main -states offsets $2h$, $h=1,2,\ldots$. Let $\eta_{\rm in}$ be the -initial right-stack value and put $y_*=-\eta_{\rm in}$. Only finitely -many full symbols are initially nonblank, so this is a finite rational -sum. Every cell left of the initial head is fully blank. The code -\[ - (x_\sigma+\xi,y_*+\eta,0) -\] -therefore places the initial configuration exactly at $(-2,0,0)$. -The input changes the common second-coordinate translation, while the -observed label and zero initial velocity remain fixed. - -Every finitely reached tape still has finite full-symbol support, so its -code is rational and can be decoded by rational digit extraction until -both tails are zero. At a checkpoint the leftmost nonempty history site -has absolute index $1$; its relative position also recovers the head's -absolute index. Thus this coding retains the initialized computation. - -Translate the rectangles of \eqref{ba:zero-digit-maps} by the indicated -state offsets and $y_*$. Lemma~\ref{ba:zero-blank-code} gives their -full affine action and separate source and target gaps. Since -$\kappa<1$, every nonhalting state block is strictly negative in the -first coordinate and every halting block strictly positive. Halting -states have no outgoing rules, so all sources are negative. Apply -Lemma~\ref{ba:parking} to obtain its periodic field $v$. Zero tails -can place a code on a rectangle boundary, which is included in that -lemma's full-neighborhood conclusion. - -Set -\[ - f=f_0+\nu f_1,\qquad - f_0=\partial_tv+(v\cdot\nabla)v,\qquad f_1=-\Delta v. -\] -The field and force are smooth, periodic, uniformly compactly supported, -and have all mixed derivatives bounded. Since $v$ vanishes near integer -times, its initial value is zero. Fixed support and bounded derivatives -give $v\in CH^2\cap C^1L^2$ and bounded $v,\nabla v$ on every finite -interval. Lemma~\ref{lem:b-comparison}(i) therefore supplies the exact -solution $(v,0)$, uniqueness in -\eqref{ba:euclidean-comparison-class}, and its global material flow. - -At time zero the marked particle is the initial code. If it is a -nonhalting code at integer $n$, its digits select a source rectangle and -the next period applies exactly that local instruction. Induction -identifies the integer-time codes through the first halt or forever. -Finite completion of every recorder scan identifies the original -machine at the checkpoint integers. A halt therefore reaches a positive -code. If the machine never halts, each used branch has a negative target -rectangle. Lemma~\ref{ba:parking} keeps its marked path negative for -that whole period, ruling out an intermediate false visit. - -Finally the force program uses finite normalization, the finite table, -the rational input shift, a finite rectangle list, rational storage and -separation data, $7N$ time segments and explicit cutoffs. All scaling -denominators have known positive lower bounds. The zero-collar and flat -profile estimates following \eqref{eq:p2-step} permit evaluation of the -repeated formula and every derivative from approximate real arguments -without exact seam tests. -Every period executes all branch maps, independently of the marked -particle's current state; the program never computes the input run. -\end{proof} - -\subsection{Direction-split marking and a bounded observer} -\label{ba:bounded-box-native} - -A stay instruction need not clear and reinstall a persistent mark. -We separate that case in the next table. Its geometric realization -uses a bounded observation box: during horizontal transfers the routing -height itself prevents a false visit. - -\begin{lemma}[Direction-split persistent marking] -\label{ba:split-mark-compiler} -In the notation of Lemma~\ref{ba:persistent-compiler}, there is an -effective partial finite table with a persistent checkpoint mark, -pointer at $p_n=n+2$, and both full-domain incoming properties of -Lemma~\ref{ba:rectangles}. A work step ending at $w'$ takes -$2(p_n-w')+4+\mathbf1_{\{d_\tau\ne0\}}$ local steps. -\end{lemma} - -\begin{proof}\label{ba:split-mark-compiler-proof} -Use full symbols $[g,e,l]$, with work letter $g$, mark $e\in\{0,1\}$, -and $l\in\{E,P\}\sqcup\{J_\tau\}$. Controls are $A_q$, -$B_\tau$ for moving instructions, $C_\tau,S_\tau$, and $D_q,R_q$. -For $\tau=(q,a)$ writing $b_\tau$, the complete table is -\[ -\begin{array}{c|c|c|c|r} -\text{control}&\text{read and guard}&\text{target}&\text{write}&d\\\hline -A_q&[a,1,l],\ d_\tau=0&C_\tau&[b_\tau,1,l]&0\\ -A_q&[a,1,l],\ d_\tau\ne0&B_\tau&[b_\tau,0,l]&d_\tau\\ -B_\tau&[g,0,l]&C_\tau&[g,1,l]&0\\ -C_\tau&[g,1,l],\ l\ne P&S_\tau&[g,1,l]&1\\ -S_\tau&[g,0,l],\ l\ne P&S_\tau&[g,0,l]&1\\ -S_\tau&[g,0,P]&D_{q^+_\tau}&[g,0,J_\tau]&1\\ -D_q&[g,0,E]&R_q&[g,0,P]&-1\\ -R_q&[g,0,l],\ l\ne P&R_q&[g,0,l]&-1\\ -R_q&[g,1,l],\ l\ne P&A_q&[g,1,l]&0. -\end{array} -\] -The first two rows use nonhalting $q$. In those rows and the -$B_\tau$ row, $l$ is unrestricted, including $P$; no additional -history guard is imposed. Initialize in $A_{q_0}$ at zero, with the -input work tape, a unique mark at zero, pointer at two, and empty -history elsewhere. - -At checkpoint $n$, the work move ends at -$w'=w_n+d_\tau\le n+11$. These three exclusions cover every real -time and prove the event equivalence independently of horizontal -excursions in the middle stage. The repeated processor installs every -local branch from the finite table; it does not depend on the executed -run or its outcome. -\end{proof} - - -\subsection{Separating vertical and planar potentials} - -\begin{theorem}[A decaying processor with phasewise planar potentials] -\label{ba:phasewise-flow} -At fixed computable $\nu>0$ there is an effective force on $\mathbb R^3$ -with one compact spatial support and all mixed derivatives of time order -$k$ bounded by $C_{k,\alpha}(1+t)^{-1-k}$, whose unique smooth -zero-data solution satisfies -\[ - \exists t\ge0:\ (\Phi_t(0))_1<-1/2 - \quad\Longleftrightarrow\quad\text{halting}. -\] -It uses the six-rule recorder with initial frontier 1, full subdivision -of the rectangles, and separate vertical-translation and planar -Hamiltonian potentials. The constructed pressure is zero and uniqueness -is in Lemma~\ref{ba:realization}'s whole-space class. -\end{theorem} -\begin{proof}\label{ba:phasewise-flow-proof} -Let $\Gamma$ be the full alphabet of the six-rule table with $r_0=1$. -Use odd digits in base $B=2|\Gamma|+1$ and its fully subdivided maps. Put halt lanes at $-2,-4,\ldots$ -and the other lanes at $2,4,\ldots$, each of width one. The second -coordinate lies in $[0,1]$. For the resulting rectangles $P_i,Q_i$, -centers $p_i,q_i$, and first scale $\lambda_i$, choose rational -$0<\varepsilon\le1/4$ smaller than one quarter of all within-source -and within-target maximum-norm distances. Empty pair lists impose no -condition. Thicken by $[-1,1]$ and put $H_i=4i$. - -On each third of the cycle use a flat progress function $\vartheta$ -from zero to one. Lifting uses $z_i=H_i\vartheta$ and potential -$A_i=(0,z_i'X_1,0)$, whose curl is $(0,0,z_i')$. -At height $H_i$, set $c_i=(1-\vartheta)p_i+\vartheta q_i$ and -$\mu_i=1-\vartheta+\vartheta\lambda_i$. Use the potential -$A_i=(0,0,\psi_i)$, where -\[ - \psi_i=c_{i1}'(X_2-c_{i2})-c_{i2}'(X_1-c_{i1}) - +\frac{\mu_i'}{\mu_i}(X_1-c_{i1})(X_2-c_{i2}). -\] -Its curl is -$(c_{i1}'+(\mu_i'/\mu_i)(X_1-c_{i1}), - c_{i2}'-(\mu_i'/\mu_i)(X_2-c_{i2}),0)$. -Thus it realizes reciprocal scaling about the interpolated center. -Descending uses the first potential with $z_i=H_i(1-\vartheta)$. -Multiply each potential by its current-box $\varepsilon$ cutoff -before taking the curl. Source gaps, distinct middle heights, and -target gaps separate the three stages. On the whole boxes and a -neighborhood the uncut formulas are exact. The first coordinate is -constant during vertical motion and is the convex interpolation of its -endpoints during the planar stage. - -For loading put $\theta_0(s)=\sigma(2s-1/2)$, which is zero near zero -and one near one. Let $P_*$ be the rational initial code. Load it in the -first unit interval by the planar path -$m(s)=\theta_0(s)((P_*)_1,(P_*)_2)$, localized around -$(m(s),0)+[-1,1]^3$, using potential -$(0,0,m_1'(X_2-m_2)-m_2'(X_1-m_1))$. -Repeat the three-phase instruction field afterward to obtain $W$. -Code induction and the delayed-head invariant prove the correspondence -at times $1+n$. Nonhalting codes have first coordinate at least 2, -so loading and the convex-path bound exclude $X_1<-1/2$ for every real -time. A halt code has first coordinate at most $-1$, including one -reached by loading an initially halted machine. Apply -Lemma~\ref{ba:log-clock}; physical code times are $e^{1+n}-1$. -Its support, derivative, event, and uniqueness conclusions give precisely -the claimed separate force choice. -\end{proof} +\section{Routing choices dictated by the observer} +\label{ba:ordered-routes} + +The preceding processors move every branch by a prescribed affine map. +We now ask which parts of that motion must be controlled between the +integer times. A bounded observation box allows horizontal work above +the observer. A slab observer requires a bound on the horizontal +coordinate throughout the motion. Finally, a half-space observer can be +protected by moving all sources to storage before delivering any target. +These three requirements lead to the constructions below. + +\subsection{Four motions above a bounded observer} +\label{ba:four-stage-section} + +We first separate ascent, scaling, translation and descent. The two +planar motions occur at a fixed height, so their exclusion from the +bounded observer follows without a horizontal coordinate estimate. + +Let $D_i,E_i\subset\mathbb R^2$, $1\le i\le N$, be closed rational +axis-aligned rectangles with positive side lengths. Assume the $D_i$ +are pairwise disjoint and the $E_i$ are pairwise disjoint, and let +\[ + A_i(b)=c_i'+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(b-c_i), + \qquad \lambda_i\in\mathbb Q_{>0},\qquad A_i(D_i)=E_i, +\] +where $c_i,c_i'$ are their centers. No separation between a source and +a target is assumed. All distances used to choose coordinate padding +below are distances for the maximum norm. + +Let $\vartheta=\sigma$ be the effective flat switch in +\eqref{eq:p2-step}. For a closed interval, possibly reduced to one +point, define +\[ + \chi_{[a,b],\epsilon}(v)= + \vartheta\!\left(\frac{2(v-a+\epsilon)}\epsilon\right) + \vartheta\!\left(\frac{2(b+\epsilon-v)}\epsilon\right). +\] +For a box $K$, let $\chi_{K,\epsilon}$ be the product over its three +coordinates. It is one on the $\epsilon/2$ enlargement of $K$ and +zero outside its $\epsilon$ enlargement. The effective profile estimates +following \eqref{eq:p2-step} apply to these products and rescalings. + +\begin{lemma}[Four-stage realization]\label{ba:four-stage} +The rectangle data determine effectively a smooth divergence-free field +$W(s,x)$ supported in a compact subset of $(0,1)\times\mathbb R^3$. +Its time-one map is $(b,z)\mapsto(A_i(b),z)$ on a neighborhood of +each $D_i\times\{0\}$. For a point of that plate, every planar +motion occurs at height $4i$, while ascent and descent preserve its +source and target planar coordinates, respectively. The scaling can +use either linear interpolation of its positive first factor or linear +interpolation of the logarithm of that factor. +\end{lemma} + +\begin{proof}\label{ba:four-stage-proof} +The empty family is realized by zero. Otherwise put $h_i=4i$, +$w_i=c_i'-c_i$, and +\[ + \Theta_k(s)=\vartheta(10s-(2k-1)),\qquad 1\le k\le4. +\] +Only the $k$th motion is active on the interval +$[(2k-1)/10,2k/10]$; the four intervals are disjoint. If +$a_{ij},a'_{ij}$ are the source and target half-widths, respectively, +take the four swept boxes +\[ + \begin{aligned} + K_{i1}&=D_i\times[0,h_i],\\ + K_{i2}&=\{c_i+(u_1,u_2):|u_j|\le\max(a_{ij},a'_{ij})\} + \times\{h_i\},\\ + K_{i3}&=\operatorname{box}(E_i-w_i,E_i)\times\{h_i\},\\ + K_{i4}&=E_i\times[0,h_i]. + \end{aligned} +\] +Here $\operatorname{box}$ denotes the smallest axis-aligned rectangle +containing the two displayed rectangles. Within each of the four +families these boxes are disjoint: source separation handles the first, +target separation the fourth, and distinct heights the other two. +Choose a positive rational $\epsilon<1$ smaller than one quarter of +every pairwise distance within these families. A family with fewer than +two members contributes no restriction. Its $\epsilon$ enlargements +are then still disjoint. + +Choose either +\[ + \mu_i=1+(\lambda_i-1)\Theta_2 + \quad\hbox{or}\quad \mu_i=\lambda_i^{\Theta_2}, + \qquad \gamma_i=\dot\mu_i/\mu_i. +\] +The factors $\mu_i$ and $\mu_i^{-1}$ stay between their respective +endpoint values. Use the coordinate potentials +\[ + \begin{aligned} + B_{i1}&=(0,h_i x_1,0),& + B_{i2}&=(0,0,(x_1-c_{i1})(x_2-c_{i2})),\\ + B_{i3}&=(0,0,w_{i1}x_2-w_{i2}x_1),& + B_{i4}&=(0,-h_i x_1,0). + \end{aligned} +\] +For the logarithmic-scale construction we also retain the alternative +translation potentials, with $e_3=(0,0,1)$, +\[ + B_{i1}=\tfrac12(h_ie_3)\times x,\qquad + B_{i3}=\tfrac12(w_{i1},w_{i2},0)\times x,\qquad + B_{i4}=\tfrac12(-h_ie_3)\times x. +\] +Their curls equal the same constant translation vectors. Their localized +fields need not agree away from the plates. Either choice gives +\[ + W=\sum_i\left[ + \dot\Theta_1\nabla\times(\chi_{K_{i1},\epsilon}B_{i1}) + +\gamma_i\nabla\times(\chi_{K_{i2},\epsilon}B_{i2}) + +\dot\Theta_3\nabla\times(\chi_{K_{i3},\epsilon}B_{i3}) + +\dot\Theta_4\nabla\times(\chi_{K_{i4},\epsilon}B_{i4})\right]. +\] +This is a finite sum of spatial curls. Its spatial support is compact, +its time support is contained in $[1/10,8/10]$, and the lower bound +$\mu_i\ge\min(1,\lambda_i)>0$ proves smoothness and effectivity. + +For $b\in D_i$, the successive paths are +\[ + \begin{array}{c|c} + \text{motion}&\text{position}\\\hline + \text{ascent}&(b,h_i\Theta_1)\\ + \text{scaling}&(c_i+\operatorname{diag}(\mu_i,\mu_i^{-1})(b-c_i),h_i)\\ + \text{translation}&(A_i(b)-w_i+\Theta_3w_i,h_i)\\ + \text{descent}&(A_i(b),h_i(1-\Theta_4)). + \end{array} +\] +Each path stays in its swept box. On a neighborhood of that box the +uncut curls are, in order, +\[ + h_ie_3,\qquad (x_1-c_{i1},-(x_2-c_{i2}),0),\qquad + (w_{i1},w_{i2},0),\qquad -h_ie_3. +\] +After multiplication by the time coefficients, these are the prescribed +affine velocities. The motion matrices are positive diagonal of determinant +one, the plateau margin is $\epsilon/2$, and the support padding is +$\epsilon$. The disjoint swept boxes therefore permit the plateau +argument of Lemma~\ref{lem:bb-curl}, with $\eta=\epsilon/2$. +It identifies these particular localized fields with the displayed +paths on the plateaux. The normalized diagonal factors are bounded by +$L_i=\max(1,\lambda_i,\lambda_i^{-1})$, so that lemma gives the +full initial neighborhood of radius $\epsilon/(4L_i)$. +\end{proof} + +We use the following precise Euclidean comparison class in the applications: +on each $[0,T]$, +\begin{equation}\label{ba:euclidean-comparison-class} + \begin{gathered} + u\in C([0,T];H^2(\mathbb R^3))\cap C^1([0,T];L^2(\mathbb R^3)),\\ + u,\nabla u\in L^\infty([0,T]\times\mathbb R^3),\qquad + p\in C([0,T];H^1(\mathbb R^3)). + \end{gathered} +\end{equation} +The pressure representative is part of this condition; it is zero for +our constructed solution. These comparison velocities and pressures +satisfy case (i) of Lemma~\ref{lem:b-comparison}. We retain the stated +bounded-gradient requirement here, although that case does not need it. + +That lemma also gives a smooth material diffeomorphism on every finite +interval. Differentiating its trajectory equation gives +$\dot J=(D_xu)(t,\Phi_t(a))J$, $J(0)=I$, for $J=D_a\Phi_t$. +Consequently +\begin{equation}\label{ba:material-volume} + \frac d{dt}\det J=(\nabla\cdot u)(t,\Phi_t(a))\det J=0, + \qquad \det J=1. +\end{equation} +The same proof applies to periodic lifts on the torus. The residual +formulas below are affine in viscosity, so they also define the asserted +solutions for any real $\nu>0$, with effectivity relative to $\nu$. + +\begin{theorem}[Two bounded-box material tests]\label{ba:bounded-box-tests} +Fix a positive computable viscosity $\nu$. A deterministic one-tape +machine and finite input effectively determine each of the following +general body forces on $[0,\infty)\times\mathbb R^3$. Both have zero +initial velocity, a unique global smooth velocity in +\eqref{ba:euclidean-comparison-class}, and constructed pressure zero. +\begin{enumerate} +\item The force and velocity have one fixed compact spatial support, all +mixed derivatives bounded, and period one for $t\ge1$. The particle +from $0$ enters $(-5,-2)\times(-1,2)\times(-1,1)$ exactly when the +machine halts. +\item The force and velocity have one fixed compact spatial support and, +for all $k\ge0$ and spatial multi-indices $\alpha$, satisfy +\begin{equation}\label{ba:eighteen-decay} + \|\partial_t^k\partial_x^\alpha u(t)\|_\infty+ + \|\partial_t^k\partial_x^\alpha f(t)\|_\infty + \le C_{k,\alpha}(1+t)^{-1-k}. +\end{equation} +The particle from $(2,0,0)$ enters +$(-1/2,3/2)\times(-1/2,3/2)\times(-1/2,1/2)$ exactly when the +machine halts. In particular $f\in L^2([0,\infty);H^m(\mathbb R^3))$ +for every integer $m\ge0$. +\end{enumerate} +The supports and constants may depend on the machine and input. The +labels and observers do not. All force derivatives admit evaluation to +any prescribed rational error by a finite program. +\end{theorem} + +\begin{proof}\label{ba:bounded-box-tests-proof} +Normalize to a single halt state $q_H$ and a complete nonhalting table, +as in Section~\ref{ba:codes}. Apply the move-first recorder of +Lemma~\ref{lem:bb-movefirst}. Its history frontier starts at cell $2$; +all other history cells are empty and all mark bits are zero. At +checkpoint $n$, the frontier is $g_n=n+2$, the work head satisfies +$|h_n|\le n$, and the next work instruction ending at $h'$ is completed +in exactly $4+2(g_n-h')$ local steps. Thus every required checkpoint +is reached after finitely many steps. The incoming-direction and +written-symbol inverse properties hold on the whole partial table. +The records recover the successive work instructions and hence the +absolute head position, even though the geometric code is head-relative. + +Let $m$ be the full alphabet size. Use odd digits $\beta(a)$ in +$\{1,3,\ldots,2m-1\}$, with base $B=2m+1$ in the first construction +and $B=2m+2$ in the second. Set +\[ + C(a_0a_1\cdots)=\sum_{r\ge0}\beta(a_r)B^{-r-1},\quad + P_a(v)=\frac{\beta(a)+v}{B},\quad D_a(v)=Bv-\beta(a),\quad + I_a=P_a([0,1]). +\] +Let $\xi$ encode the left string, nearest cell first, and $\eta$ +the string starting at the head. The complete right, stay and left +branch maps are \eqref{eq:bb-one-cell-branches}, with its digit +$d(a)$ equal to $\beta(a)$ and its move $e$ equal to $d$ here. +The left move is split over every full symbol $c$. The shared lemma +proves the full-rectangle endpoint formulas, arbitrary-tail action and +linear part $\operatorname{diag}(B^{-d},B^d)$. + +Translate each state square horizontally. In the first construction +give the halt state offset $-4$ and the other states offsets $2+2r$, +$r\ge0$; in the second use halt offset $0$ and other offsets $3r$, +$r\ge1$. Thus the code is $(o_s+\xi,\eta,0)$. +Lemma~\ref{lem:bb-onecell} applies with exactly these odd digits and +either displayed base: its incoming conditions are those supplied by +the move-first recorder, and \eqref{eq:bb-one-cell-branches} is its minimal +left-split branch list. Thus both complete closed rectangle families +are separately disjoint, with within-state gaps at least $B^{-2}$. +This proves the hypotheses of Lemma~\ref{ba:four-stage}; no separation +between the source and target families is required. + +Use its linear scale and coordinate potentials for the first construction; +use its logarithmic scale and symmetric translation potentials for the +second. In the latter case the scaling coefficient is exactly +$(\log\lambda_i)\dot\Theta_2$ and the scale matrix is +$\operatorname{diag}(\lambda_i^{\Theta_2},\lambda_i^{-\Theta_2})$. +Denote the resulting step field by $W_{\rm step}$. + +The initial code $P_{\rm in}$ is rational. A constant full-symbol +tail of digit $\beta_0$ after $M$ positions contributes +$B^{-M}\beta_0/(B-1)$; all other contributions are finite. For the +appropriate fixed label $a_*$, put $v_0=P_{\rm in}-a_*$ and let +$J_0$ be the smallest box containing $[a_*,P_{\rm in}]$. The loader +\[ + W_{\rm in}(s,x)=\frac d{ds}\vartheta(2s-1/2)\, + \nabla\times\left(\chi_{J_0,1}(x)\frac{v_0\times x}{2}\right) +\] +has the exact marked path $a_*+\vartheta(2s-1/2)v_0$, since the +uncut curl is $v_0$ and the entire segment lies in the plateau. Form +\[ + W(s,x)=W_{\rm in}(s,x)+\sum_{n\ge1}W_{\rm step}(s-n,x). +\] +The zero time collars make this smooth, initially zero, and periodic for +$s\ge1$, with common compact spatial support and bounded mixed derivatives. + +At time $s=1+k$ the marked particle is exactly the code after $k$ local +instructions, through the first halt or indefinitely in a nonhalting run. +This follows by induction from the full-rectangle maps. Each halt code +lies in its stated observation box, including one reached by loading an +initially halted machine. In a nonhalting run, the first loader has +$x_1\ge0$ and later nonhalt codes have $x_1\ge2$; for the second these +bounds are $x_1\ge2$ and $x_1\ge3$. Ascent and descent leave those +planar coordinates unchanged. Every planar motion has height $4i\ge4$, +outside both observers. The intervening stationary intervals add no new +positions. This proves exclusion at every real time. + +For the first case take $u=W$ and its residual force. For the second set +\[ + s(t)=\log(1+t),\quad \theta(t)=(1+t)^{-1},\quad + u(t,x)=\theta(t)W(s(t),x), +\] +whose force is +\[ + f=\theta^2\{-W+\partial_sW+(W\cdot\nabla)W\}(s(t),x) + -\nu\theta\Delta W(s(t),x). +\] +Lemma~\ref{ba:log-clock} applies to this fixed-support, bounded-derivative +template. Its identity +$\partial_t(\theta^jY(s(t)))=\theta^{j+1}(\partial_s-j)Y(s(t))$ +proves \eqref{ba:eighteen-decay} for every $k,\alpha$. Fixed support +then gives the stated $L^2_tH^m_x$ conclusion. Its clock is onto, with +inverse $t=e^s-1$, so the material path is the old path composed with +$s(t)$ and the event is unchanged. + +In both cases smoothness, fixed support and bounded derivatives on +finite time intervals put the reference velocity in $CH^2\cap C^1L^2$ +with bounded velocity and gradient. It is solenoidal and initially zero. +Lemma~\ref{lem:b-comparison}(i) therefore gives the exact solution +$(u,0)$ and uniqueness in \eqref{ba:euclidean-comparison-class}. +The effective profiles following \eqref{eq:p2-step}, applied to the +finite table, rational initial code, finitely many boxes and repeated +template, give all derivative evaluations. A coarse time approximation +encloses finitely many possible active translates, avoiding equality +tests at seams. The prescription never computes the machine's run. +\end{proof} + +\subsection{Small plates and a slab observer on the unit torus} +\label{ba:torus-plates-section} + +A slab observer has no height restriction, so elevation alone cannot +protect it. We now keep each nonhalting branch inside a narrow range of +first coordinates throughout its motion. The delayed-head recorder also +allows arbitrary finitely supported work data on both sides of the head. + +\begin{theorem}[Torus plates and square-integrable forcing] +\label{ba:torus-plates} +Fix a positive computable $\nu$. Every deterministic one-tape machine +with finitely specified tape, blank outside a finite set, effectively +determines a smooth mean-zero general force on +$\mathbb T^3=\mathbb R^3/\mathbb Z^3$, with bounded mixed derivatives +and period one for $t\ge1$. Its zero-data Navier--Stokes solution is +globally smooth and unique among smooth solutions, with pressure normalized +to mean zero and constructed pressure zero. For +$P_*=(1/4,1/2,1/4)$ and $X(t)=\Phi_t(P_*)$, the event +\begin{equation}\label{ba:torus-plate-event} + \exists t\ge0:\quad X_1(t)\pmod1\in(1/2,7/8) +\end{equation} +is equivalent to halting. + +A separate effective force has the same zero data, event, uniqueness and +mean-zero property, is supported in one compact sub-box of $(0,1)^3$, +has all mixed derivatives bounded, and satisfies +\begin{equation}\label{ba:cube-rate} + \|\widehat f(t,\cdot)\|_\infty=O((1+t)^{-2/3}). +\end{equation} +In particular $\widehat f\in L^2([0,\infty)\times\mathbb T^3)$. +\end{theorem} + +\begin{proof}\label{ba:torus-plates-proof} +Use Lemma~\ref{ba:six-compiler} with the frontier initially at cell $1$. +Its main control $M_{q,e}$ records the last work displacement $e$, initially +zero. In one work step it writes and marks the old head, records +$(q,e,a)$ at the frontier, returns to the mark, and only then makes the +work displacement. All other initial history cells are blank and every +mark is zero. At checkpoint $n$ the frontier is $n+1$, $|h_n|\le n$, +and the next checkpoint takes exactly $2(n+1-h_n)+3$ local steps. +These scans test history and marks, so the proof is unaffected by work +symbols on either side of the head. Main controls with halting $q$ have +no outgoing rules. The full-domain incoming properties are those of +the cited lemma. + +Let $S$ be the local state set, $N=|S|$, let $S_h$ be its halting main +states, and enumerate $S$ by $n(s)=0,\ldots,N-1$. Put +\[ + \ell=\frac1{16(N+1)},\quad z_0=\frac14,\quad + x_s=\frac18+2\ell n(s)+\frac12\mathbf1_{s\in S_h},\quad + T_s(\xi,\eta)=(x_s+\ell\xi,1/4+\ell\eta). +\] +Nonhalting squares lie in $1/8\le x_1<1/4$ and halting squares in +$5/8\le x_1<3/4$; distinct squares have gap at least $\ell$. +Give the full alphabet, of size $m$, odd digits in base $\beta=2m+1$. +Conjugating \eqref{eq:bb-one-cell-branches} by the source and target charts +gives separately disjoint rectangle families $R_j^0,R_j^1$ with positive +diagonal factors $\alpha_j,\alpha_j^{-1}$, +$\alpha_j\in\{\beta^{-1},1,\beta\}$. Every chart has the same +scale, so this remains the physical linear part. +Lemma~\ref{lem:bb-onecell}, with this uniform chart scaling, gives +within-family coordinate gaps at least $\ell/\beta^2$, including the +two-letter target intervals for left moves. + +For $1\le j\le L$, write the centers as $p_j^0,p_j^1$, the source +half-widths as $a_j,b_j$, and the target half-widths as +$\alpha_ja_j,\alpha_j^{-1}b_j$. All endpoint half-widths are at +most $\ell/2$. Choose +\[ + h_j=\frac12+\frac{j}{4(L+1)},\qquad + \epsilon=\frac1{100}\min\left(1,\frac\ell{\beta^2}, + \frac1{4(L+1)}\right). +\] +With $\theta(s)=\vartheta(2s-1/2)$ and +$\Theta_k(\tau)=\theta(3\tau-k+1)$ for $k=1,2,3$, define +\[ + \begin{aligned} + C_j(\tau)&=\big((1-\Theta_2)p_j^0+\Theta_2p_j^1, + z_0+(\Theta_1-\Theta_3)(h_j-z_0)\big),\\ + \sigma_j&=1+(\alpha_j-1)\Theta_2,\qquad + \lambda_j=\dot\sigma_j/\sigma_j,\\ + (w_{j1},w_{j2},w_{j3})&=(a_j\sigma_j,b_j/\sigma_j,0). + \end{aligned} +\] +These centers and half-widths describe moving plates. Their +$\epsilon$ enlargements are disjoint: the first third uses source +projections, the middle third distinct heights, and the last third +target projections. The padding is at most one hundredth of every +relevant gap. All plates and padding lie in a compact sub-box of +$(0,1)^3$, since their centers interpolate between the stated chart +centers, widths are at most $\ell$, heights lie between $1/4$ and +$3/4$, and $\ell\le1/32$, $\epsilon\le1/100$. + +For $w\ge0$ put +\[ + \kappa_\epsilon(w,v)= + \vartheta\!\left(\frac{2(w+\epsilon-v)}\epsilon\right) + \vartheta\!\left(\frac{2(w+\epsilon+v)}\epsilon\right),\qquad + \chi_j=\prod_{a=1}^3\kappa_\epsilon(w_{ja},x_a-C_{ja}). +\] +Writing $y=x-C_j$, use +\[ + A_j=\tfrac12\dot C_j\times y+(0,0,\lambda_jy_1y_2), + \qquad V(\tau,x)=\sum_j\nabla\times(\chi_jA_j). +\] +Extend the potentials by zero outside the chart. The support remains +away from its faces, so this gives a smooth torus field, zero near the +time endpoints. On the whole plateau of plate $j$ its velocity is +$\dot C_j+(\lambda_jy_1,-\lambda_jy_2,0)$; other summands vanish +there. Hence the path from $(p_j^0+(r,s),z_0)$ is exactly +\begin{equation}\label{ba:plate-path} + C_j(\tau)+(\sigma_j(\tau)r,\sigma_j(\tau)^{-1}s,0), + \qquad |r|\le a_j,\quad |s|\le b_j. +\end{equation} +It ends at the prescribed affine image. The plateau argument of +Lemma~\ref{lem:bb-curl} applies with +$D_j=\operatorname{diag}(\sigma_j,\sigma_j^{-1},1)$, +$\eta=\epsilon/2$, and diagonal bound +$\max(1,\alpha_j,\alpha_j^{-1})$: the required enlarged-box separation +was just proved. It gives the same motion on an initial neighborhood +of radius $\epsilon/[4\max(1,\alpha_j,\alpha_j^{-1})]$, with constant +normal displacement, and therefore includes boundary codes. + +The initial code $P_{\rm in}$ is rational, since relative to constant +blank tails only finitely many work and frontier digits differ. Load it +from $P_*$ along $C_*=(1-\theta)P_*+\theta P_{\rm in}$ using +\[ + V_0(t,x)=\nabla\times\left[ + \prod_{a=1}^3\kappa_{1/64}(0,x_a-C_{*,a}(t)) + \frac{\dot C_*(t)\times(x-C_*(t))}{2}\right]. +\] +The segment and its collar stay within the chart: its first coordinate +stays between $1/8$ and $3/4$, its second between $1/4$ and +$1/2+\ell$, and its third is $1/4$. Its plateau curl is $\dot C_*$. +Take $U=V_0$ on $[0,1]$ and $U(t)=V(t-n)$ on $[n,n+1]$ for $n\ge1$. +Time collars make this smooth and initially zero. It has fixed compact +chart support, bounded mixed derivatives, and period one after loading. +Define $f=U_t+(U\cdot\nabla)U-\nu\Delta U$ using the stated physical +viscosity. Both $U$ and $f$ have mean zero: $U$ is a curl, the +Laplacian integrates to zero, and +$(U\cdot\nabla)U=\operatorname{div}(U\otimes U)$. +The torus clause of Lemma~\ref{lem:b-comparison} applies to this +smooth, initially zero, solenoidal reference field and gives the unique +smooth solution $(U,0)$ with mean-zero pressure. + +Equations~\eqref{eq:bb-one-cell-branches} and \eqref{ba:plate-path} identify +the particle at time $1+k$ with the $k$th local configuration. The +recorder invariant ensures that a nonhalting computation continues +indefinitely. A halt, including an initial halt reached by loading, +puts the particle in $5/8\le X_1<3/4$, proving the event. For a +nonhalting computation the loader has $X_1\le1/4$ and every later +branch has both endpoint centers below $1/4$. Its center is their +convex interpolation and its first half-width is at most $\ell/2$. +Thus $00$; the cutoff +estimates and zero collars allow derivative evaluation at approximate real +arguments without tests for exact boundary equality. Only positive +arguments enter the cube root. The program repeats the entire local +table and never uses a computed execution. +\end{proof} + +\subsection{Storage and periodic forcing without a loader} +\label{ba:parking-section} + +The next construction absorbs initialization into the positions of the +state rectangles. A zero digit for the full blank symbol places the +initial left stack at zero; an input-dependent second-coordinate translation then +places the entire initial code at one fixed particle. To protect a +half-space observer during every period, we first evacuate all sources +to separate storage rectangles and only afterwards fill the targets. + +\begin{lemma}[Zero-blank coding]\label{ba:zero-blank-code} +Let an alphabet of size $m>1$ have digits $0,\ldots,m-1$, with full +blank digit zero. Set $B=m+1$, $\kappa=(m-1)/m$, and +\[ + \operatorname{val}(a_0a_1\cdots)=\sum_{r\ge0}a_rB^{-r-1},\qquad + I_a=\frac{a+[0,\kappa]}B,\qquad I_{a,b}=\frac{a+I_b}B. +\] +The code is injective, takes values in $[0,\kappa]$, and its distinct +one-letter and two-letter cylinders have positive rational gaps. For a +deterministic partial table with target-state-determined incoming direction +and incoming rule determined by target state and written symbol, separated +state translates of $[0,\kappa]^2$ yield separately disjoint full source +and target rectangles by the following maps in state-relative coordinates: +\begin{equation}\label{ba:zero-digit-maps} + \begin{array}{c|c|c|c} + d&\text{source}&(\xi',\eta')&\text{target}\\\hline + 1&[0,\kappa]\times I_a&((b+\xi)/B,B\eta-a)&I_b\times[0,\kappa]\\ + 0&[0,\kappa]\times I_a&(\xi,\eta+(b-a)/B)&[0,\kappa]\times I_b\\ + -1&I_c\times I_a&(B\xi-c,c/B+(b+B\eta-a)/B^2)& + [0,\kappa]\times I_{c,b}. + \end{array} +\end{equation} +The last row is split over all $c$. Each map has positive diagonal +linear part of determinant one and acts exactly on the full rectangle. +\end{lemma} + +\begin{proof}\label{ba:zero-blank-code-proof} +The maximum tail is $(m-1)/(B-1)=\kappa$, and +$(m-1+\kappa)/B=\kappa$. Consecutive first-digit intervals have +gap $(1-\kappa)/B>0$. Within one first-digit interval the second-digit +gaps are this quantity divided by $B$; different first digits already +give a gap. The interval containing a code recovers its first digit, +and $v\mapsto Bv-a$ then exposes the tail. Repetition proves +injectivity, including endpoint codes. + +Writing $a$ to $b$ changes the first right digit. A right move pushes +$b$ onto the left string and removes $a$ from the right; a left move +removes $c$ from the left and prepends $c,b$ to the old right tail. +These identities give \eqref{ba:zero-digit-maps}, including its images +and reciprocal slopes. Determinism and digit gaps separate sources by +state, read digit, and $c$ when needed. At one target state the common +incoming direction determines where the written digit is visible: in +$I_b$ on the left or right, or in $I_{c,b}$. The incoming-rule +hypothesis and digit gaps then separate the closed target rectangles. +\end{proof} + +\begin{lemma}[Evacuation, storage and delivery]\label{ba:parking} +Suppose $A_i:D_i\to E_i$, $1\le i\le N$, are the rational reciprocal +rectangle maps of Lemma~\ref{ba:four-stage}, with $N\ge1$ and every +$D_i\subset\{x_1<0\}$. There is an effective smooth divergence-free +field on $\mathbb R\times\mathbb R^3$, one-periodic in time, zero near +each integer time, and uniformly compactly supported in space, whose +one-period map is $(b,z)\mapsto(A_i(b),z)$ on a neighborhood of every +$D_i\times\{0\}$. If $E_i\subset\{x_1<0\}$, the entire +one-period path of each point in $D_i\times\{0\}$ remains in that +half-space. +\end{lemma} + +\begin{proof}\label{ba:parking-proof} +Let $d_i,b_i$ be the source and target centers, and let $a_{ij},a'_{ij}$ +be their half-widths. Put $a^*_{ij}=\max(a_{ij},a'_{ij})$. +Choose $\epsilon$ to be one quarter of the minimum of $1$ and all +pairwise maximum-norm distances within the source and target families; +omit pair entries when $N=1$. Define +\[ + \begin{gathered} + x_{\min}=\min\{x_1:(x_1,x_2)\in D_i\cup E_i\text{ for some }i\},\\ + R=\max(1,\max_i a^*_{i1}),\qquad J=4(R+1),\qquad + s_i=(x_{\min}-iJ,0). + \end{gathered} +\] +The storage rectangle at $s_i$ with half-widths $a^*_{ij}$ lies in +$x_1<0$. Its first-coordinate projection is separated by more than one +from every source and target and from the other storage projections: +the relevant gaps are at least $J-R>1$ and $J-2R>1$. + +We need only a localized translation and a localized reciprocal scaling. +On a time segment $[t_0,t_0+\delta]$, use +$g(t)=\vartheta(3(t-t_0)/\delta-1)$, whose derivative is supported +in the middle third. Using the cutoff $\kappa_\epsilon$ defined in +the proof of Theorem~\ref{ba:torus-plates}, for half-widths $a,b$ write +\[ + \chi_{a,b}(y)=\kappa_\epsilon(a,y_1) + \kappa_\epsilon(b,y_2)\kappa_\epsilon(0,y_3). +\] +A plate translation from center $P$ to $Q$ is produced by +\[ + C(t)=(1-g)P+gQ,\qquad + \nabla\times\left[ + \tfrac12\chi_{a,b}(x-C(t))\dot C(t)\times(x-C(t))\right]. +\] +On its plateau the velocity is $\dot C$ and the exact path is +$C(t)+(r,s,0)$. At storage center $P=(s_i,0)$, take +$q(t)=1+(\lambda_i-1)g(t)$ and the field +\[ + \nabla\times\left(0,0, + \frac{\dot q}{q}(x_1-P_1)(x_2-P_2) + \chi_{a^*_{i1},a^*_{i2}}(x-P)\right). +\] +It has the exact path $P+(qr,q^{-1}s,0)$. The denominator has the +positive lower bound $\min(1,\lambda_i)$, and the whole path lies in +the storage rectangle, because each half-width stays between its source +and target values. + +Partition a period into $7N$ equal segments and take $Z=2$. For each +$i$ in order, use four segments for +\[ + (d_i,0)\longrightarrow(d_i,Z)\longrightarrow(s_i,Z) + \longrightarrow(s_i,0),\qquad\text{then scale at }s_i. +\] +Only after all these evacuations, use three segments for each $i$ to +deliver its target shape along +\[ + (s_i,0)\longrightarrow(s_i,Z)\longrightarrow(b_i,Z) + \longrightarrow(b_i,0). +\] +Sum these segment fields and extend periodically. The middle-third +time switches give zero neighborhoods of every segment endpoint and +every integer time. The result is a smooth finite sum of curls with +one compact spatial support. + +This is the $7N$ schedule of Lemma~\ref{lem:bb-lift-parking}(ii). +Its separation hypotheses have explicit margins here. Unpadded +source-source and target-target gaps are at least $4\epsilon$, so their +$\epsilon$ enlargements leave gap at least $2\epsilon$. Storage +gaps exceed one, leaving more than $1-2\epsilon>0$ after padding. +At horizontal travel height the enlarged moving plate and every +enlarged stationary plate have vertical gap $2-2\epsilon>0$. +Every scaling stays in its reserved storage rectangle. The lemma +therefore gives simultaneous noninterference for evacuation and delivery, +including when a source meets a target. + +The localized potentials above have exactly these affine generators on +their plateaux, so the plateau argument of Lemma~\ref{lem:bb-curl} +gives endpoint +$(d_i+(r,s),0)\mapsto(b_i+(\lambda_i r,\lambda_i^{-1}s),0)$, +with normal displacement unchanged. Put +$M=\max_i(1,\lambda_i,\lambda_i^{-1})$ and +$d_*=\min(2\epsilon,1-2\epsilon,2-2\epsilon)>0$. +A neighborhood radius smaller than both $\epsilon/(8M)$ and +$d_*/(4M)$ retains the plateaux and all exclusion margins throughout +the concatenation. This applies the shared proofs to the displayed +potentials and does not identify their fields away from the plateaux. + +If $E_i$ is negative, each translation of an individual point joins two +negative first coordinates, and its linear interpolation remains negative. +Scaling occurs inside the negative storage rectangle. The point is +stationary during all other segments. This proves the asserted +whole-period exclusion. +\end{proof} + +\begin{theorem}[A periodic force with no loading interval] +\label{ba:periodic-parking} +Fix a positive computable $\nu$. A deterministic one-tape machine and +finite input word in cells $0,\ldots,L-1$, with other cells blank and +initial head zero, effectively determine a general body force on +$\mathbb R^3$ extending smoothly and one-periodically to all real times. +It has uniformly compact spatial support and bounded mixed derivatives. +Its zero-data solution has a unique global smooth velocity in +\eqref{ba:euclidean-comparison-class}, with constructed pressure zero, and +\[ + \exists t\ge0:\quad (\Phi_t((-2,0,0)))_1>0 + \quad\Longleftrightarrow\quad\text{the machine halts}. +\] +The fixed label is already the initial code; periodicity begins at zero. +\end{theorem} + +\begin{proof}\label{ba:periodic-parking-proof} +Insert a fresh nonhalting start state that preserves the read letter, +stays put and enters the original start state. Replace missing +nonhalting instructions by letter-preserving stay instructions to a halt +state. This finite normalization preserves the halting question, +including an initially halted machine, and guarantees a nonhalting +initial control and at least one rule. + +Apply the six-row delayed-head recorder of Lemma~\ref{ba:six-compiler} +with initial frontier $1$. Initially every other history cell is blank +and every mark is zero. The main control records $(q,e)$, where $e$ +is the preceding displacement, and the history record is $(q,e,a)$. +The old head is marked before the right scan and cleared on return, +immediately before displacement into the next main state. At checkpoint +$k$ the frontier is $k+1$ and the head $i$ satisfies $|i|\le k$; +the next checkpoint uses $2(k+1-i)+3\le4k+5$ local steps. The +incoming-direction and written-symbol properties hold on the full +domain, as required by Lemma~\ref{ba:zero-blank-code}. + +Give the full blank symbol digit zero and the other full symbols digits +$1,\ldots,m-1$. Order nonhalting local states with the initial state +first and assign offsets $-2h$, $h=1,2,\ldots$; assign halting main +states offsets $2h$, $h=1,2,\ldots$. Let $\eta_{\rm in}$ be the +initial right-stack value and put $y_*=-\eta_{\rm in}$. Only finitely +many full symbols are initially nonblank, so this is a finite rational +sum. Every cell left of the initial head is fully blank. The code +\[ + (x_\sigma+\xi,y_*+\eta,0) +\] +therefore places the initial configuration exactly at $(-2,0,0)$. +The input changes the common second-coordinate translation, while the +observed label and zero initial velocity remain fixed. + +Every finitely reached tape still has finite full-symbol support, so its +code is rational and can be decoded by rational digit extraction until +both tails are zero. At a checkpoint the leftmost nonempty history site +has absolute index $1$; its relative position also recovers the head's +absolute index. Thus this coding retains the initialized computation. + +Translate the rectangles of \eqref{ba:zero-digit-maps} by the indicated +state offsets and $y_*$. Lemma~\ref{ba:zero-blank-code} gives their +full affine action and separate source and target gaps. Since +$\kappa<1$, every nonhalting state block is strictly negative in the +first coordinate and every halting block strictly positive. Halting +states have no outgoing rules, so all sources are negative. Apply +Lemma~\ref{ba:parking} to obtain its periodic field $v$. Zero tails +can place a code on a rectangle boundary, which is included in that +lemma's full-neighborhood conclusion. + +Set +\[ + f=f_0+\nu f_1,\qquad + f_0=\partial_tv+(v\cdot\nabla)v,\qquad f_1=-\Delta v. +\] +The field and force are smooth, periodic, uniformly compactly supported, +and have all mixed derivatives bounded. Since $v$ vanishes near integer +times, its initial value is zero. Fixed support and bounded derivatives +give $v\in CH^2\cap C^1L^2$ and bounded $v,\nabla v$ on every finite +interval. Lemma~\ref{lem:b-comparison}(i) therefore supplies the exact +solution $(v,0)$, uniqueness in +\eqref{ba:euclidean-comparison-class}, and its global material flow. + +At time zero the marked particle is the initial code. If it is a +nonhalting code at integer $n$, its digits select a source rectangle and +the next period applies exactly that local instruction. Induction +identifies the integer-time codes through the first halt or forever. +Finite completion of every recorder scan identifies the original +machine at the checkpoint integers. A halt therefore reaches a positive +code. If the machine never halts, each used branch has a negative target +rectangle. Lemma~\ref{ba:parking} keeps its marked path negative for +that whole period, ruling out an intermediate false visit. + +Finally the force program uses finite normalization, the finite table, +the rational input shift, a finite rectangle list, rational storage and +separation data, $7N$ time segments and explicit cutoffs. All scaling +denominators have known positive lower bounds. The zero-collar and flat +profile estimates following \eqref{eq:p2-step} permit evaluation of the +repeated formula and every derivative from approximate real arguments +without exact seam tests. +Every period executes all branch maps, independently of the marked +particle's current state; the program never computes the input run. +\end{proof} + +\subsection{Direction-split marking and a bounded observer} +\label{ba:bounded-box-native} + +A stay instruction need not clear and reinstall a persistent mark. +We separate that case in the next table. Its geometric realization +uses a bounded observation box: during horizontal transfers the routing +height itself prevents a false visit. + +\begin{lemma}[Direction-split persistent marking] +\label{ba:split-mark-compiler} +In the notation of Lemma~\ref{ba:persistent-compiler}, there is an +effective partial finite table with a persistent checkpoint mark, +pointer at $p_n=n+2$, and both full-domain incoming properties of +Lemma~\ref{ba:rectangles}. A work step ending at $w'$ takes +$2(p_n-w')+4+\mathbf1_{\{d_\tau\ne0\}}$ local steps. +\end{lemma} + +\begin{proof}\label{ba:split-mark-compiler-proof} +Use full symbols $[g,e,l]$, with work letter $g$, mark $e\in\{0,1\}$, +and $l\in\{E,P\}\sqcup\{J_\tau\}$. Controls are $A_q$, +$B_\tau$ for moving instructions, $C_\tau,S_\tau$, and $D_q,R_q$. +For $\tau=(q,a)$ writing $b_\tau$, the complete table is +\[ +\begin{array}{c|c|c|c|r} +\text{control}&\text{read and guard}&\text{target}&\text{write}&d\\\hline +A_q&[a,1,l],\ d_\tau=0&C_\tau&[b_\tau,1,l]&0\\ +A_q&[a,1,l],\ d_\tau\ne0&B_\tau&[b_\tau,0,l]&d_\tau\\ +B_\tau&[g,0,l]&C_\tau&[g,1,l]&0\\ +C_\tau&[g,1,l],\ l\ne P&S_\tau&[g,1,l]&1\\ +S_\tau&[g,0,l],\ l\ne P&S_\tau&[g,0,l]&1\\ +S_\tau&[g,0,P]&D_{q^+_\tau}&[g,0,J_\tau]&1\\ +D_q&[g,0,E]&R_q&[g,0,P]&-1\\ +R_q&[g,0,l],\ l\ne P&R_q&[g,0,l]&-1\\ +R_q&[g,1,l],\ l\ne P&A_q&[g,1,l]&0. +\end{array} +\] +The first two rows use nonhalting $q$. In those rows and the +$B_\tau$ row, $l$ is unrestricted, including $P$; no additional +history guard is imposed. Initialize in $A_{q_0}$ at zero, with the +input work tape, a unique mark at zero, pointer at two, and empty +history elsewhere. + +At checkpoint $n$, the work move ends at +$w'=w_n+d_\tau\le n+11$. These three exclusions cover every real +time and prove the event equivalence independently of horizontal +excursions in the middle stage. The repeated processor installs every +local branch from the finite table; it does not depend on the executed +run or its outcome. +\end{proof} + + +\subsection{Separating vertical and planar potentials} + +\begin{theorem}[A decaying processor with phasewise planar potentials] +\label{ba:phasewise-flow} +At fixed computable $\nu>0$ there is an effective force on $\mathbb R^3$ +with one compact spatial support and all mixed derivatives of time order +$k$ bounded by $C_{k,\alpha}(1+t)^{-1-k}$, whose unique smooth +zero-data solution satisfies +\[ + \exists t\ge0:\ (\Phi_t(0))_1<-1/2 + \quad\Longleftrightarrow\quad\text{halting}. +\] +It uses the six-rule recorder with initial frontier 1, full subdivision +of the rectangles, and separate vertical-translation and planar +Hamiltonian potentials. The constructed pressure is zero and uniqueness +is in Lemma~\ref{ba:realization}'s whole-space class. +\end{theorem} +\begin{proof}\label{ba:phasewise-flow-proof} +Let $\Gamma$ be the full alphabet of the six-rule table with $r_0=1$. +Use odd digits in base $B=2|\Gamma|+1$ and its fully subdivided maps. Put halt lanes at $-2,-4,\ldots$ +and the other lanes at $2,4,\ldots$, each of width one. The second +coordinate lies in $[0,1]$. For the resulting rectangles $P_i,Q_i$, +centers $p_i,q_i$, and first scale $\lambda_i$, choose rational +$0<\varepsilon\le1/4$ smaller than one quarter of all within-source +and within-target maximum-norm distances. Empty pair lists impose no +condition. Thicken by $[-1,1]$ and put $H_i=4i$. + +On each third of the cycle use a flat progress function $\vartheta$ +from zero to one. Lifting uses $z_i=H_i\vartheta$ and potential +$A_i=(0,z_i'X_1,0)$, whose curl is $(0,0,z_i')$. +At height $H_i$, set $c_i=(1-\vartheta)p_i+\vartheta q_i$ and +$\mu_i=1-\vartheta+\vartheta\lambda_i$. Use the potential +$A_i=(0,0,\psi_i)$, where +\[ + \psi_i=c_{i1}'(X_2-c_{i2})-c_{i2}'(X_1-c_{i1}) + +\frac{\mu_i'}{\mu_i}(X_1-c_{i1})(X_2-c_{i2}). +\] +Its curl is +$(c_{i1}'+(\mu_i'/\mu_i)(X_1-c_{i1}), + c_{i2}'-(\mu_i'/\mu_i)(X_2-c_{i2}),0)$. +Thus it realizes reciprocal scaling about the interpolated center. +Descending uses the first potential with $z_i=H_i(1-\vartheta)$. +Multiply each potential by its current-box $\varepsilon$ cutoff +before taking the curl. Source gaps, distinct middle heights, and +target gaps separate the three stages. On the whole boxes and a +neighborhood the uncut formulas are exact. The first coordinate is +constant during vertical motion and is the convex interpolation of its +endpoints during the planar stage. + +For loading put $\theta_0(s)=\sigma(2s-1/2)$, which is zero near zero +and one near one. Let $P_*$ be the rational initial code. Load it in the +first unit interval by the planar path +$m(s)=\theta_0(s)((P_*)_1,(P_*)_2)$, localized around +$(m(s),0)+[-1,1]^3$, using potential +$(0,0,m_1'(X_2-m_2)-m_2'(X_1-m_1))$. +Repeat the three-phase instruction field afterward to obtain $W$. +Code induction and the delayed-head invariant prove the correspondence +at times $1+n$. Nonhalting codes have first coordinate at least 2, +so loading and the convex-path bound exclude $X_1<-1/2$ for every real +time. A halt code has first coordinate at most $-1$, including one +reached by loading an initially halted machine. Apply +Lemma~\ref{ba:log-clock}; physical code times are $e^{1+n}-1$. +Its support, derivative, event, and uniqueness conclusions give precisely +the claimed separate force choice. +\end{proof} diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/persistent-history.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/persistent-history.tex index aabafc523af314f1b57807feeab325e694c5fff8..d616a4c812e5c688e30b062c2a60c69ac8004303 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/persistent-history.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/persistent-history.tex @@ -1,512 +1,512 @@ -\section{Persistent marks and guarded history routes} -\label{ba:marked-routes} - -The preceding recorders clear the work-head mark at each checkpoint. -We now retain that mark, or replace the moving history pointer by a -guarded boundary between occupied and unused cells. These changes give -different partial maps on arbitrary tapes, even though their initialized -executions simulate the same machine. We prove each inverse on its -declared domain before using it to separate the geometric branch images. -The use of retained history follows the reversible-computation idea of -Bennett~\cite{Bennett1973}; the local guards and geometric interfaces -are established here. - -In this section an input word occupies cells $0,1,\ldots$, the other -work cells are blank, and the initial head is at zero. All constructions -include initial halting. When one halt state $h$ is used, identify the -original halting states with $h$ and redirect incoming instructions; -there are no outgoing halting instructions to preserve. If necessary -adjoin an unreachable halt state. -We write $X(t;a)=\Phi_t(a)$ for the trajectory from $a$. - -\subsection{A persistent mark and input-dependent placement} -\label{ba:persistent-periodic} - -A force can be periodic from time zero if no separate loading interval -is needed. We achieve this by translating the initial control's coding -square so that the prescribed particle already represents the input. -The translation depends effectively on the finite input. First we give -the recorder that this periodic processor implements. - -\begin{lemma}[A recorder with a persistent work-head mark] -\label{ba:persistent-compiler} -Let a machine have work alphabet $\Gamma$, halt state $h$, and -instructions -$\gamma=(q,a)\mapsto(q^+_\gamma,a^+_\gamma,d_\gamma)$, -where $q\ne h$ and $d_\gamma\in\{-1,0,1\}$. -There is an effective finite partial one-head table with these properties. -At the checkpoint after $n$ work steps, it has the correct work tape, -state and head, a unique mark at that head, and a history pointer at -$p_n=n+2$. Each target control determines the incoming displacement; -the target control and written full cell symbol determine the source -control and read full symbol on the entire partial domain. If the next -work-head position is $w'=w_n+d_\gamma$, the next checkpoint is reached -after exactly $2(p_n-w')+4$ local steps. -\end{lemma} - -\begin{proof}\label{ba:persistent-compiler-proof} -A full symbol is $(a,e,s)\in\Gamma\times\{0,1\}\times\mathcal H$, -where $\mathcal H=\{E,P\}\sqcup\{r_\gamma\}$ records empty history, -a pointer, or an instruction. All displayed control families are -disjoint. The following is the complete table; free cell indices range -over their finite alphabets: -\[ -\begin{array}{c|c|c|c|r} -\text{control}&\text{read and guard}&\text{target}&\text{write}&d\\\hline -A_q&(a,1,s),\ s\ne P&B_\gamma&(a^+_\gamma,0,s)&d_\gamma\\ -B_\gamma&(c,0,s),\ s\ne P&R_\gamma&(c,1,s)&1\\ -R_\gamma&(c,0,s),\ s\ne P&R_\gamma&(c,0,s)&1\\ -R_\gamma&(c,0,P)&C_{q^+_\gamma}&(c,0,r_\gamma)&1\\ -C_q&(c,0,E)&L_q&(c,0,P)&-1\\ -L_q&(c,0,s),\ s\ne P&L_q&(c,0,s)&-1\\ -L_q&(c,1,s),\ s\ne P&A_q&(c,1,s)&0 -\end{array} -\] -The first row has $q\ne h$ and $\gamma=(q,a)$; there is no rule from -$A_h$. Initially the actual head and the unique mark are at zero, -the work tape is the input, and history is $E$ except for $P$ at cell 2. - -At checkpoint $n$, $|w_n|\le n$, so after the first row the head is at -$w'\le n+10}, -\] -maps $S_i$ onto $T_i$, with $c_i^S,c_i^T$ their centers. There is a -smooth nondecreasing switch $\eta:[0,1]\to[0,1]$, zero near 0 and one -near 1, for which either scale -path $r_i=\lambda_i^\eta$ or $r_i=1-\eta+\eta\lambda_i$ yields an -effective smooth compactly supported solenoidal unit-time field, zero -on time collars, realizing $(y,z)\mapsto(F_i(y),z)$ on the whole box -$S_i\times[-1,1]$ and a neighborhood of it. The paths are affine; -each horizontal half-width remains between its endpoint values, and the -horizontal sweep of each original box lies in the coordinate bounding -rectangle of $S_i$ and $T_i$. With the linear choice, a particle's first -coordinate is the convex -interpolation of its endpoint first coordinates. -\end{lemma} - -\begin{proof}\label{ba:two-interpolations-proof} -For $N=0$ take the zero field. For $N\ge1$ apply -Lemma~\ref{lem:bb-lift-parking}(i) with vertical half-width one. -Use the shared phase switches with $\eta=w_2$ and set -\[ - C_i(s)=\big((1-\eta)c_i^S+\eta c_i^T,\,6i(w_1-w_3)\big), - \qquad D_i(s)=\operatorname{diag}(r_i,r_i^{-1},1). -\] -The affine path is -$G_i(s,y)=C_i(s)+D_i(s)(y-(c_i^S,0))$. -Choose $\epsilon$ to be one quarter of the minimum of 1 and all -pairwise max-norm distances in the two planar lists, omitting empty -pair lists. The $\epsilon$-padded moving boxes are disjoint: source -and target gaps are at least $4\epsilon$ during the vertical phases, -and the middle-phase height spacing is $6>2(1+\epsilon)$. - -For later loading formulas, record the centered cutoff explicitly: -\[ - \chi(C,L,\epsilon;x)= - \prod_{j=1}^3\sigma\!\left(\frac{L_j+\epsilon+x_j-C_j}{\epsilon/2}\right) - \sigma\!\left(\frac{L_j+\epsilon-x_j+C_j}{\epsilon/2}\right). -\] -Here $L$ is the vector of half-widths; the plateau has padding -$\epsilon/2$ and the support has padding $\epsilon$. -If $L_i(s)$ are the moving half-widths, $Y=x-C_i(s)$ and -$A_i=D_i'D_i^{-1}$, the field supplied by -\eqref{eq:bb-potential} is -\begin{equation}\label{ba:affine-box-field} - V(s,x)=\sum_i\nabla\times\left\{ - \chi(C_i,L_i,\epsilon;x) - \left[\tfrac12 C_i'\times Y+\tfrac13(A_iY)\times Y\right] - \right\}. -\end{equation} -Indeed $\det D_i=1$, and the preceding separation verifies -Lemma~\ref{lem:bb-curl} with localization parameter $\epsilon/2$. -Its neighborhood conclusion applies with -$L_* =\max_i\max(1,\lambda_i,\lambda_i^{-1})$ because -$\|D_i(s)\|_\infty\le L_*$ and $D_i(0)=I$. -It therefore gives the stated affine motion on the full boxes and -on their initial max-norm neighborhoods of radius $\epsilon/(4L_*)$. -The paths are stationary for $s\le1/12$ and $s\ge11/12$. -Their finite center ranges and endpoint width bounds give common -compact support. The effective positive lower bound -$r_i\ge\min(1,\lambda_i)$ verifies the scale requirement. -For the linear scale, the first-coordinate identity is -\[ - (G_i(s,y))_1=(1-\eta)y_1+\eta(F_i(y_1,y_2))_1. -\] -The second coordinate need not interpolate its own endpoint values. -Instead, if $b_{i2}$ is the source's second half-width, convexity of the -reciprocal function gives -\begin{equation}\label{eq:ba-reciprocal-envelope} - \frac{b_{i2}}{(1-\eta)+\eta\lambda_i} - \le (1-\eta)b_{i2}+\eta\frac{b_{i2}}{\lambda_i}. -\end{equation} -The first half-width is exactly its endpoint interpolation. For the -exponential scale, convexity of the exponential gives the corresponding -upper bounds in both coordinates, since -$\lambda_i^{\pm\eta}\le(1-\eta)+\eta\lambda_i^{\pm1}$. -In either case the center is interpolated linearly and each half-width -is at most the interpolation of its endpoint half-widths. Therefore each -lower edge is at least the interpolation of the endpoint lower edges, -and each upper edge is at most the corresponding interpolation of upper -edges. This proves the claimed bounding rectangle for the entire sweep. -The individual endpoint-width bounds also follow directly from monotonicity -of the positive scales and their reciprocals. -\end{proof} - -The same construction applies to $P_i\times[-h,h]$ for any common -$h\ge0$, including plates and rectangles with zero side lengths, under -the hypotheses of Lemma~\ref{lem:bb-lift-parking}(i). Replace $6i$ by -heights $H_i$ whose pairwise gaps exceed $2h+2\varepsilon$, and choose -$\varepsilon$ below one third of every within-family planar gap. The -formula for $C_i$ uses the source and target centers in place of -$c_i^S,c_i^T$; its support padding is $\varepsilon$ and plateau padding -is $\varepsilon/2$. Source projections, distinct heights and target -projections separate the three phases in that order. Empty pair lists -impose no gap restriction, and the empty family gives the zero field. -Arbitrary real data give smooth existence; the effective-data assumptions -of the shared lemma give computable scales, derivative bounds and margins. -The same perturbation argument gives an open neighborhood even when a -side length is zero. - -\subsection{Transporting an initial cutoff} -\label{sec:ba-transported-localization} - -The transported-initial-cutoff convention is also useful. Write the -affine path as $G_i(t)x=J_i(t)x+g_i(t)$ and localize by -$\psi_i(G_i(t)^{-1}x)$, where $\psi_i$ has support padding -$\varepsilon_0$ around the initial box and equals one on a smaller -positive padding. -If $\lambda_i\in[B^{-1},B]$, its initial padding -$\varepsilon_0$ expands horizontally by at most $B$. Choose -$2B\varepsilon_0$ below every target gap, -$2\varepsilon_0$ below every source gap, and -$\varepsilon_0\le\varepsilon$. These transported supports stay -separated in the same three phases. The affine generator is -$v_i+A_ix$, with $A_i=\dot J_iJ_i^{-1}$ and -$v_i=\dot g_i-A_ig_i$. Its potential is -\[ - \tfrac12v_i\times x+\tfrac13(A_ix)\times x. -\] -The identity proved in Lemma~\ref{lem:bb-curl} gives the exact -generator on the transported plateau. Its neighborhood argument then -proves the same full-box endpoint map. The factor $B$ in the margin -choice is essential: transported padding does not stay fixed under -stretching. - -\subsection{Independent planar interpolation} - -For an observer that needs both planar coordinates to follow their own -endpoint interpolations, the reciprocal path is insufficient. The next -construction interpolates the two factors independently and compensates -the temporary planar area change in the third direction. Here the named -targets may be disjoint containers of the actual affine images; only the -image centers enter the motion. - -\begin{lemma}[Independent planar scales and normal compensation] -\label{ba:plate-routing} -Let $P_i,Q_i\subset\R^2$ be finite families of closed rectangles, -separately pairwise disjoint, and let assigned positive diagonal affine -maps $F_i$ satisfy $F_i(P_i)\subseteq Q_i$. Write their diagonal factors as -$\lambda_{i1}^*,\lambda_{i2}^*>0$ with product one. -There is a smooth compactly supported solenoidal unit-time field, zero on -time collars, whose endpoint map is $(y,z)\mapsto(F_i(y),z)$ on a -neighborhood of each entire plate $P_i\times\{0\}$. During the horizontal -phase both planar factors interpolate linearly, while the normal factor -compensates their product. Each planar coordinate of a point on the plate -is its convex endpoint interpolation. -\end{lemma} -\begin{proof}\label{ba:plate-routing-proof} -Use the height phases in Section~\ref{sec:ba-localization}. For source -center $p_i$, end its center path at its actual image $F_i(p_i)$; in the -onto case this is the named target center. Choose separated heights $H_i$ -as above and set -\[ - c_i=\bigl((1-w_2)p_i+w_2F_i(p_i),\,H_i(w_1-w_3)\bigr), -\] -\[ - \lambda_{ij}=1-w_2+w_2\lambda_{ij}^*,\qquad - D_i=\operatorname{diag}\bigl( - \lambda_{i1},\lambda_{i2},(\lambda_{i1}\lambda_{i2})^{-1}\bigr). -\] -The volume-preserving path has logarithmic derivative -\[ - T_i=\dot D_iD_i^{-1}=\operatorname{diag}\left( - \frac{\dot\lambda_{i1}}{\lambda_{i1}}, - \frac{\dot\lambda_{i2}}{\lambda_{i2}}, - -\frac{\dot\lambda_{i1}}{\lambda_{i1}} - -\frac{\dot\lambda_{i2}}{\lambda_{i2}}\right). -\] -This matrix has trace zero, so \eqref{eq:bb-potential} localized -around the moving plate gives the desired field. Its planar motion is -$c_i+(\lambda_{i1}(y_1-p_{i1}),\lambda_{i2}(y_2-p_{i2}),0)$. -Normal displacement changes by the factor -$(\lambda_{i1}\lambda_{i2})^{-1}$. The factors have positive -lower bounds; choose the initial thickness less than their product's -minimum times the normal plateau width. A small enlargement of the -planar rectangle is allowed by the positive collars as well. Thus a -whole thin neighborhood follows the volume-preserving affine motion. -The normal factor returns to one at the endpoint. The final plate image -is contained in $Q_i\times\{0\}$. Source and target-container -gaps protect the vertical phases and distinct heights protect the -middle phase. Linear interpolation of each planar factor gives both -convex-coordinate identities on the plate. -This proves smooth existence for arbitrary real data. Under the effective -input assumptions of Lemma~\ref{lem:bb-curl}, the positive scale bounds and -finite gap margins can be computed and the construction is effective. -\end{proof} - -\subsection{Forcing and effective evaluation} - -\begin{lemma}[Residual forcing for the finite processors]\label{ba:realization} -Fix computable $\nu>0$. Let $W$ be smooth, solenoidal, initially zero, -with support in one compact subset of $\mathbb R^3$ and every mixed -derivative bounded. Then -\begin{equation}\label{ba:residual} - F_W=W_t+(W\cdot\nabla)W-\nu\Delta W -\end{equation} -has the same support and derivative properties. The zero-data solution -$(W,0)$ is unique among smooth solutions with -$u\in C_tH^2\cap C_t^1L^2$, bounded $u$ on finite intervals, and -$p\in C_tH^1$. The corresponding torus assertion holds in the -classical comparison class of Section~\ref{sec:model}. A sum of curls -in a torus chart gives both $W$ and $F_W$ mean zero. Finite formulas -from the effective-data instances of the preceding constructions, finite -loading, and repetition of one cycle give effective evaluation of every -force derivative. -\end{lemma} -\begin{proof}\label{ba:realization-proof} -Fixed compact support and bounded mixed derivatives imply all reference -velocity hypotheses of Lemma~\ref{lem:b-comparison}. Case (i) gives -exactly the stated whole-space comparison class, without a competing -gradient bound; its torus clause gives the stated classical comparison -class. The same lemma provides the pressure normalization and global -material flow. The product rule applied to \eqref{ba:residual} -proves the force's support and derivative bounds. A chart curl has -zero mean, and integration of the residual uses -$\int\Delta W=0$ and -$\int(W\cdot\nabla)W=\int\operatorname{div}(W\otimes W)=0$. -For these effective-data instances, gate derivatives and seam evaluation are -those of \eqref{eq:p2-step} and Proposition~\ref{prop:bb-realization}. -The explicit affine scales and every reciprocal denominator have positive -computable bounds. Thus differentiating the finite formulas, using a -finite superset of possible period indices, evaluates every derivative -without a machine execution or a real equality test at a seam. -\end{proof} +\section{Affine motion on neighborhoods of boxes}\label{ba:geometry} + +Rectangle separation controls the beginning and end of a transfer; distinct +heights separate its middle. We now specify the paths and cutoff margins +used by the applications. Interpolating one planar scale and its reciprocal +preserves thickness and, for a linear first scale, gives a convex formula +for each first coordinate. Interpolating both planar scales independently +gives that formula in both coordinates, at the cost of a compensating +normal strain during the motion. + +\subsection{Reciprocal paths and their localization} +\label{sec:ba-localization} + +Use the flat step $\sigma$ of \eqref{eq:p2-step} and put +\[ + \theta(s)=\sigma(2s-1/2),\qquad + w_j(s)=\theta(3s-j+1)\quad(j=1,2,3). +\] +The switches $w_1,w_2,w_3$ perform the lift, horizontal transfer and +descent in separate thirds of the unit interval. All are constant near +their phase endpoints. The following version of +Lemma~\ref{lem:bb-lift-parking}(i) records a common vertical half-width +of one and explicit neighborhoods; the same formulas allow other heights +and thicknesses as described immediately afterward. + +\begin{lemma}[Two explicit interpolations of separated boxes] +\label{ba:two-interpolations} +Let $S_i,T_i\subset\mathbb R^2$, $1\le i\le N$, be closed rational +rectangles of positive side lengths. Suppose each list is pairwise +disjoint and +\[ + F_i(y)=c_i^T+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(y-c_i^S), + \qquad\lambda_i\in\mathbb Q_{>0}, +\] +maps $S_i$ onto $T_i$, with $c_i^S,c_i^T$ their centers. There is a +smooth nondecreasing switch $\eta:[0,1]\to[0,1]$, zero near 0 and one +near 1, for which either scale +path $r_i=\lambda_i^\eta$ or $r_i=1-\eta+\eta\lambda_i$ yields an +effective smooth compactly supported solenoidal unit-time field, zero +on time collars, realizing $(y,z)\mapsto(F_i(y),z)$ on the whole box +$S_i\times[-1,1]$ and a neighborhood of it. The paths are affine; +each horizontal half-width remains between its endpoint values, and the +horizontal sweep of each original box lies in the coordinate bounding +rectangle of $S_i$ and $T_i$. With the linear choice, a particle's first +coordinate is the convex +interpolation of its endpoint first coordinates. +\end{lemma} + +\begin{proof}\label{ba:two-interpolations-proof} +For $N=0$ take the zero field. For $N\ge1$ apply +Lemma~\ref{lem:bb-lift-parking}(i) with vertical half-width one. +Use the shared phase switches with $\eta=w_2$ and set +\[ + C_i(s)=\big((1-\eta)c_i^S+\eta c_i^T,\,6i(w_1-w_3)\big), + \qquad D_i(s)=\operatorname{diag}(r_i,r_i^{-1},1). +\] +The affine path is +$G_i(s,y)=C_i(s)+D_i(s)(y-(c_i^S,0))$. +Choose $\epsilon$ to be one quarter of the minimum of 1 and all +pairwise max-norm distances in the two planar lists, omitting empty +pair lists. The $\epsilon$-padded moving boxes are disjoint: source +and target gaps are at least $4\epsilon$ during the vertical phases, +and the middle-phase height spacing is $6>2(1+\epsilon)$. + +For later loading formulas, record the centered cutoff explicitly: +\[ + \chi(C,L,\epsilon;x)= + \prod_{j=1}^3\sigma\!\left(\frac{L_j+\epsilon+x_j-C_j}{\epsilon/2}\right) + \sigma\!\left(\frac{L_j+\epsilon-x_j+C_j}{\epsilon/2}\right). +\] +Here $L$ is the vector of half-widths; the plateau has padding +$\epsilon/2$ and the support has padding $\epsilon$. +If $L_i(s)$ are the moving half-widths, $Y=x-C_i(s)$ and +$A_i=D_i'D_i^{-1}$, the field supplied by +\eqref{eq:bb-potential} is +\begin{equation}\label{ba:affine-box-field} + V(s,x)=\sum_i\nabla\times\left\{ + \chi(C_i,L_i,\epsilon;x) + \left[\tfrac12 C_i'\times Y+\tfrac13(A_iY)\times Y\right] + \right\}. +\end{equation} +Indeed $\det D_i=1$, and the preceding separation verifies +Lemma~\ref{lem:bb-curl} with localization parameter $\epsilon/2$. +Its neighborhood conclusion applies with +$L_* =\max_i\max(1,\lambda_i,\lambda_i^{-1})$ because +$\|D_i(s)\|_\infty\le L_*$ and $D_i(0)=I$. +It therefore gives the stated affine motion on the full boxes and +on their initial max-norm neighborhoods of radius $\epsilon/(4L_*)$. +The paths are stationary for $s\le1/12$ and $s\ge11/12$. +Their finite center ranges and endpoint width bounds give common +compact support. The effective positive lower bound +$r_i\ge\min(1,\lambda_i)$ verifies the scale requirement. +For the linear scale, the first-coordinate identity is +\[ + (G_i(s,y))_1=(1-\eta)y_1+\eta(F_i(y_1,y_2))_1. +\] +The second coordinate need not interpolate its own endpoint values. +Instead, if $b_{i2}$ is the source's second half-width, convexity of the +reciprocal function gives +\begin{equation}\label{eq:ba-reciprocal-envelope} + \frac{b_{i2}}{(1-\eta)+\eta\lambda_i} + \le (1-\eta)b_{i2}+\eta\frac{b_{i2}}{\lambda_i}. +\end{equation} +The first half-width is exactly its endpoint interpolation. For the +exponential scale, convexity of the exponential gives the corresponding +upper bounds in both coordinates, since +$\lambda_i^{\pm\eta}\le(1-\eta)+\eta\lambda_i^{\pm1}$. +In either case the center is interpolated linearly and each half-width +is at most the interpolation of its endpoint half-widths. Therefore each +lower edge is at least the interpolation of the endpoint lower edges, +and each upper edge is at most the corresponding interpolation of upper +edges. This proves the claimed bounding rectangle for the entire sweep. +The individual endpoint-width bounds also follow directly from monotonicity +of the positive scales and their reciprocals. +\end{proof} + +The same construction applies to $P_i\times[-h,h]$ for any common +$h\ge0$, including plates and rectangles with zero side lengths, under +the hypotheses of Lemma~\ref{lem:bb-lift-parking}(i). Replace $6i$ by +heights $H_i$ whose pairwise gaps exceed $2h+2\varepsilon$, and choose +$\varepsilon$ below one third of every within-family planar gap. The +formula for $C_i$ uses the source and target centers in place of +$c_i^S,c_i^T$; its support padding is $\varepsilon$ and plateau padding +is $\varepsilon/2$. Source projections, distinct heights and target +projections separate the three phases in that order. Empty pair lists +impose no gap restriction, and the empty family gives the zero field. +Arbitrary real data give smooth existence; the effective-data assumptions +of the shared lemma give computable scales, derivative bounds and margins. +The same perturbation argument gives an open neighborhood even when a +side length is zero. + +\subsection{Transporting an initial cutoff} +\label{sec:ba-transported-localization} + +The transported-initial-cutoff convention is also useful. Write the +affine path as $G_i(t)x=J_i(t)x+g_i(t)$ and localize by +$\psi_i(G_i(t)^{-1}x)$, where $\psi_i$ has support padding +$\varepsilon_0$ around the initial box and equals one on a smaller +positive padding. +If $\lambda_i\in[B^{-1},B]$, its initial padding +$\varepsilon_0$ expands horizontally by at most $B$. Choose +$2B\varepsilon_0$ below every target gap, +$2\varepsilon_0$ below every source gap, and +$\varepsilon_0\le\varepsilon$. These transported supports stay +separated in the same three phases. The affine generator is +$v_i+A_ix$, with $A_i=\dot J_iJ_i^{-1}$ and +$v_i=\dot g_i-A_ig_i$. Its potential is +\[ + \tfrac12v_i\times x+\tfrac13(A_ix)\times x. +\] +The identity proved in Lemma~\ref{lem:bb-curl} gives the exact +generator on the transported plateau. Its neighborhood argument then +proves the same full-box endpoint map. The factor $B$ in the margin +choice is essential: transported padding does not stay fixed under +stretching. + +\subsection{Independent planar interpolation} + +For an observer that needs both planar coordinates to follow their own +endpoint interpolations, the reciprocal path is insufficient. The next +construction interpolates the two factors independently and compensates +the temporary planar area change in the third direction. Here the named +targets may be disjoint containers of the actual affine images; only the +image centers enter the motion. + +\begin{lemma}[Independent planar scales and normal compensation] +\label{ba:plate-routing} +Let $P_i,Q_i\subset\R^2$ be finite families of closed rectangles, +separately pairwise disjoint, and let assigned positive diagonal affine +maps $F_i$ satisfy $F_i(P_i)\subseteq Q_i$. Write their diagonal factors as +$\lambda_{i1}^*,\lambda_{i2}^*>0$ with product one. +There is a smooth compactly supported solenoidal unit-time field, zero on +time collars, whose endpoint map is $(y,z)\mapsto(F_i(y),z)$ on a +neighborhood of each entire plate $P_i\times\{0\}$. During the horizontal +phase both planar factors interpolate linearly, while the normal factor +compensates their product. Each planar coordinate of a point on the plate +is its convex endpoint interpolation. +\end{lemma} +\begin{proof}\label{ba:plate-routing-proof} +Use the height phases in Section~\ref{sec:ba-localization}. For source +center $p_i$, end its center path at its actual image $F_i(p_i)$; in the +onto case this is the named target center. Choose separated heights $H_i$ +as above and set +\[ + c_i=\bigl((1-w_2)p_i+w_2F_i(p_i),\,H_i(w_1-w_3)\bigr), +\] +\[ + \lambda_{ij}=1-w_2+w_2\lambda_{ij}^*,\qquad + D_i=\operatorname{diag}\bigl( + \lambda_{i1},\lambda_{i2},(\lambda_{i1}\lambda_{i2})^{-1}\bigr). +\] +The volume-preserving path has logarithmic derivative +\[ + T_i=\dot D_iD_i^{-1}=\operatorname{diag}\left( + \frac{\dot\lambda_{i1}}{\lambda_{i1}}, + \frac{\dot\lambda_{i2}}{\lambda_{i2}}, + -\frac{\dot\lambda_{i1}}{\lambda_{i1}} + -\frac{\dot\lambda_{i2}}{\lambda_{i2}}\right). +\] +This matrix has trace zero, so \eqref{eq:bb-potential} localized +around the moving plate gives the desired field. Its planar motion is +$c_i+(\lambda_{i1}(y_1-p_{i1}),\lambda_{i2}(y_2-p_{i2}),0)$. +Normal displacement changes by the factor +$(\lambda_{i1}\lambda_{i2})^{-1}$. The factors have positive +lower bounds; choose the initial thickness less than their product's +minimum times the normal plateau width. A small enlargement of the +planar rectangle is allowed by the positive collars as well. Thus a +whole thin neighborhood follows the volume-preserving affine motion. +The normal factor returns to one at the endpoint. The final plate image +is contained in $Q_i\times\{0\}$. Source and target-container +gaps protect the vertical phases and distinct heights protect the +middle phase. Linear interpolation of each planar factor gives both +convex-coordinate identities on the plate. +This proves smooth existence for arbitrary real data. Under the effective +input assumptions of Lemma~\ref{lem:bb-curl}, the positive scale bounds and +finite gap margins can be computed and the construction is effective. +\end{proof} + +\subsection{Forcing and effective evaluation} + +\begin{lemma}[Residual forcing for the finite processors]\label{ba:realization} +Fix computable $\nu>0$. Let $W$ be smooth, solenoidal, initially zero, +with support in one compact subset of $\mathbb R^3$ and every mixed +derivative bounded. Then +\begin{equation}\label{ba:residual} + F_W=W_t+(W\cdot\nabla)W-\nu\Delta W +\end{equation} +has the same support and derivative properties. The zero-data solution +$(W,0)$ is unique among smooth solutions with +$u\in C_tH^2\cap C_t^1L^2$, bounded $u$ on finite intervals, and +$p\in C_tH^1$. The corresponding torus assertion holds in the +classical comparison class of Section~\ref{sec:model}. A sum of curls +in a torus chart gives both $W$ and $F_W$ mean zero. Finite formulas +from the effective-data instances of the preceding constructions, finite +loading, and repetition of one cycle give effective evaluation of every +force derivative. +\end{lemma} +\begin{proof}\label{ba:realization-proof} +Fixed compact support and bounded mixed derivatives imply all reference +velocity hypotheses of Lemma~\ref{lem:b-comparison}. Case (i) gives +exactly the stated whole-space comparison class, without a competing +gradient bound; its torus clause gives the stated classical comparison +class. The same lemma provides the pressure normalization and global +material flow. The product rule applied to \eqref{ba:residual} +proves the force's support and derivative bounds. A chart curl has +zero mean, and integration of the residual uses +$\int\Delta W=0$ and +$\int(W\cdot\nabla)W=\int\operatorname{div}(W\otimes W)=0$. +For these effective-data instances, gate derivatives and seam evaluation are +those of \eqref{eq:p2-step} and Proposition~\ref{prop:bb-realization}. +The explicit affine scales and every reciprocal denominator have positive +computable bounds. Thus differentiating the finite formulas, using a +finite superset of possible period indices, evaluates every derivative +without a machine execution or a real equality test at a seam. +\end{proof} diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/solid-routing.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/solid-routing.tex index a59ad0bc94d783a0b841ca7471473b1ef390f571..921546d966504bd9479811576be27fa929d51d87 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/solid-routing.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/solid-routing.tex @@ -1,668 +1,668 @@ -\section{Evacuation, columns and obstacle detours}\label{sec:solid-routing} -The following refinements choose the occupied regions and the storage order -explicitly. Ordered low extraction controls a height observer; simultaneous -lifting controls a first-coordinate observer; expanded centers permit -three-dimensional detours. Fixed columns and recorded pointers give further -ways to keep a whole transition outside a detector. Each proof specifies -which parts of the geometry act on solid neighborhoods and which bounds -are needed only for the coded central plane. - -Unless an individual theorem specifies another class, whole-space -comparison means smooth $u\in CH^2\cap C^1L^2$, bounded $u,\nabla u$ -on finite intervals, and a pressure representative in $CH^1$, as in -Section~\ref{bc:analytic-section}. The torus comparison class is the -classical class of Lemma~\ref{lem:b-comparison}. The distinct pressure -conditions stated below remain part of their respective results. - -\subsection{Ordered extraction below the computation}\label{bc:ordered-section} -\begin{proposition}[Ordered full-box transport]\label{bc:ordered} -Let $D_i,S_i$ be finite families of closed rational axis-aligned boxes in $\R^3$ -with positive side lengths, pairwise disjoint within each family. -Suppose the assigned center-to-center affine map $F_i$ satisfies -$F_i(D_i)=S_i$ and has positive diagonal linear part $A_i$ of -determinant one. There is an effective compactly supported solenoidal -unit-time field, zero near the time endpoints, whose time-one map has -this action on every full $D_i$. If both centers of a pair have negative -third coordinate, every point of $D_i$ whose third coordinate equals that -of its center follows a path with negative third coordinate. -\end{proposition} -\begin{proof}\label{bc:ordered-proof} -Choose an integer $L>1$ larger than the absolute values of all endpoint -coordinates. The storage columns have first coordinates $U_i=4(i+1)L$ -and common height $Z=-4L$. Write $[\ell_i,r_i]$ for the first-coordinate -interval of $D_i$. Extract sources in decreasing order of $r_i$. -Translate each horizontally right to its column and then vertically to -$Z$, retaining its second coordinate. If an unextracted box meets the -active box in both the second and third projections, disjointness forces -their first intervals to be separated. That box cannot lie to the right: -its right endpoint would then exceed $r_i$, contrary to the extraction -order. It lies strictly to the left of $\ell_i$ and misses the entire -rightward sweep. Stored boxes lie below the original boxes; vertical -moves occur in distinct columns outside them. - -At storage height, adjust the second center coordinate to its target -value and apply $A_i^{\theta}$. Each intermediate half-width lies between -its two endpoint values, hence is less than $L$. The columns remain -disjoint throughout these operations. Deliver boxes in increasing order -of the right endpoint of their target first interval: raise a box to its -target height and translate it left into its target. The raising column -is empty. As above, a previously delivered box whose last two intervals -meet the active ones must be separated in the first coordinate. Its -smaller right endpoint puts it wholly to the left of the active target -interval, so the leftward sweep stops before reaching it. -Undelivered boxes remain at storage height. These observations give -positive rational clearances for all swept bounding boxes. Choose cutoff -padding below each clearance and use Lemma~\ref{lem:bb-curl} in successive -flat slots. Induction over the slots proves the full-box assertion. -A point initially in the central third-coordinate plane retains zero -third-coordinate offset during each diagonal deformation. Its height is -interpolated only between the two endpoint heights and $Z$. If those -endpoint heights are negative, every intermediate height is negative. -\end{proof} - -The onto hypothesis in the preceding proposition is a useful exact-box -specialization. Its target-container version, including the effective -input conditions, is given in Section~\ref{bc:contained-targets}. - -\begin{theorem}[A positive-height observation]\label{bc:positive-height} -On $\R^3$, the fixed label $0$ and observation $\{x_3>0\}$ realize -halting effectively by a compactly supported general force with zero -initial velocity and pressure, bounded mixed derivatives, and period one -after time one. The force has a separate alternative in -$L^2(0,\infty;H^j)$ for every $j$, tending to zero in every spatial -supremum norm at rate $O((1+t)^{-1})$. -\end{theorem} -\begin{proof}\label{bc:positive-height-proof} -Use Lemma~\ref{lem:bb-movefirst} and odd digits in base $2m+2$. Give every -simulator state a distinct positive integer index $i_s$ and put its -third-coordinate center at $z_s=i_s$ for ready halting states and -$z_s=-i_s$ otherwise. Thicken the full rectangles of -Lemma~\ref{lem:bb-onecell} by $[z_s-1/4,z_s+1/4]$. The assigned third -scale is one, so the maps satisfy Proposition~\ref{bc:ordered}. -The rational initial code $Y_0$ is loaded from the origin along the -straight segment, by a localized translation during $[0,1]$. Repeat the -unit-time mechanism after time one. At time $1+k$ the particle is exactly -the code after $k$ defined simulator transitions. If the initial machine -halts, the loading endpoint already has positive height. If it never -halts, loading has nonpositive height and all later transitions have -negative endpoint centers; the proposition excludes a positive height -at every intervening time. Conversely, a halting checkpoint has positive -height. Lemma~\ref{bc:analytic} gives the force and uniqueness, and -Lemma~\ref{ba:log-clock} gives the separate temporal alternative. -\end{proof} - -\subsection{Simultaneous lifting with a convex-coordinate guard} -\label{bc:simultaneous-section} -\begin{proposition}[Simultaneous full-box transport]\label{bc:simultaneous} -Suppose finite planar source and target rectangle families $S_i,T_i$ -have positive side lengths and endpoints on the grid $B^{-2}\Z^2$, -with integer $B\ge2$, and are separately -disjoint. Require their assigned center-to-center maps to satisfy -$F_i(S_i)=T_i$ and to have linear part $\diag(\lambda_i,\lambda_i^{-1})$ with -$\lambda_i\in\{B^{-1},1,B\}$. There is an effective solenoidal motion -of the full boxes obtained by taking the product with $[-1,1]$. -For every transported point its first coordinate is a convex combination -of its initial and final first coordinates at every time. -\end{proposition} -\begin{proof}\label{bc:simultaneous-proof} -Let $p_i^-,p_i^+$ be the centers in the plane $z=0$, set $h_i=5i$, and -choose a smooth $\Theta$ equal to zero on $(-\infty,1/4]$ and one on -$[3/4,\infty)$. For $0\le t\le1$, put -\[ - \theta=\Theta(3t-1),\quad l_i=1-\theta+\theta\lambda_i, - \quad c_i=(1-\theta)p_i^-+\theta p_i^+ - +h_i\{\Theta(3t)-\Theta(3t-2)\}e_3. -\] -The desired path is $g_i(t,y)=c_i(t)+\diag(l_i,l_i^{-1},1)(y-p_i^-)$. -Use a source padding $\varepsilon=1/(10B^3)$, transported by this same -affine map, for the cutoff neighborhood. During lifting the planar -source gap is at least $B^{-2}$. During lowering the enlarged target -padding is at most $B\varepsilon20R$ so large that the -expanded source centers $A_i^*=Ha_i$ and target centers $B_i^*=Hb_i$ -are separately more than $20R$ apart in the sup norm. Set -\[ - Z_0=1+100R+\max_i\{\|A_i^*\|_\infty,\|B_i^*\|_\infty\}, - \qquad P_i=(0,0,-Z_0-100Ri). -\] -Each of the unions $\{A_i^*\}\cup\{P_i\}$ and -$\{B_i^*\}\cup\{P_i\}$ has separation greater than $20R$. -The construction has five phases: expand source centers by a factor -from one to $H$ while keeping shapes fixed; move them successively to -$P_i$; change each parked shape by the prescribed positive diagonal -power; move them successively to $B_i^*$; contract target centers to -$b_i$ while keeping the final shapes fixed. - -The simultaneous expansion is collision free. For each pair of source -boxes some coordinate has positive gap -$|a_{i,j}-a_{k,j}|-r_{i,j}-r_{k,j}$; multiplication of the center -difference by a factor at least one preserves that gap. The same -argument applies to contraction toward the disjoint targets. Choose -padding less than one quarter of the minimum such gap, and less than -one. During a parked deformation the $j$th half-width is -$r_{i,j}^{1-\theta}w_{i,j}^{\theta}\le\max(r_{i,j},w_{i,j})0$. -Every boundary used around a negative-height obstacle lies below -$-H+4R<0$. Storage heights are negative as well. Expansion, contraction, -and positive diagonal deformation preserve the central-plane condition. -Thus all these paths are negative. Finitely many rational vertices, -endpoint widths, and cutoff paddings give a rational support bound. -\end{proof} - -\begin{theorem}[A height test inside one torus chart]\label{bc:expanded-height-event} -On the unit torus, the fixed label $o=(1/2,1/2,1/2)$ and event -\begin{equation}\label{bc:expanded-height-fixed-event} - \{x:x_3\pmod1\in(1/2,1)\} -\end{equation} -realize halting by a smooth general mean-zero force, periodic after time -one, with zero initial velocity and pressure and bounded mixed derivatives. -The repeated mechanism depends on the machine alone; only initialization -depends on its finite input. The solution is unique in the torus class. -\end{theorem} -\begin{proof}\label{bc:expanded-height-event-proof} -Use the frontier recorder, odd base $B=2m+1$, distinct negative integer -heights for nonterminal states and height one for the terminal ready state. -Thicken every branch by a third half-width $1/4$. The reciprocal tape -map is onto its target rectangle; adjoining the unchanged third interval -gives an onto box map with linear part $\diag(B^{-e},B^e,1)$ and -determinant one. All source centers are -negative because terminal controls have no outgoing rules. Apply the -preceding proposition. The logical initial code $Y_0$ is rational and has -a uniform machine-dependent bound as the input varies: its two stack -coordinates lie in $[0,1]$ and its height is fixed by the initial control. -A moving translation loads the logical origin to $Y_0$, with a fixed -source half-width one and padding one. This gives a machine-dependent -rational bound $K$ for the supports of all such loaders and the repeated -mechanism; for example take $K=1+2R+M$ with $M$ an upper bound for all -routing vertices and loading coordinates and $R$ enlarged to cover their -half-widths. Choose $\lambda=1/(4K)$ and use the chart $x=o+\lambda Y$. -Everything remains strictly inside $(1/4,3/4)^3$. - -Periodize the resulting compactly supported curl fields, loading during -the first unit interval and repeating thereafter. Compute the residual -at physical viscosity $\nu$, after the chart change. The fields and force -have zero mean and all the bounds in Lemma~\ref{bc:analytic}. -For a nonhalting run, loading has nonpositive logical height and each -later segment remains negative by the proposition. A terminal code has -physical height $1/2+\lambda$, which lies in the event. There is no -chart-boundary crossing that could create a spurious observation. -An initially terminal input is detected at the loading endpoint. -The initialized correspondence is exact; at checkpoints the number of -history records gives $n$, and the frontier's relative position together -with its absolute index $2+n$ also recovers the absolute work-head position. -\end{proof} - -\subsection{Fixed columns and separate layers}\label{bc:columns-section} -\begin{proposition}[Column and layer transport]\label{bc:columns} -Let $S_i,T_i$ be finite rational planar rectangle families of positive side -lengths, each separately disjoint, with endpoints on the $B^{-2}$ grid, -where $B\ge2$ is an integer. -Let their assigned center-to-center positive reciprocal diagonal affine -maps satisfy $F_i(S_i)=T_i$. A finite effective -solenoidal field in $\R^3$ realizes all these maps on the rectangles in -$z=0$. Along each tracked path the first coordinate stays between its -initial and final values. The field has compact support and flat time -collars. -\end{proposition} -\begin{proof}\label{bc:columns-proof} -Give rectangle $i$ height $h_i=4i$. A planar cutoff equal to one near -its source and supported within padding $1/(4B^2)$ has support disjoint -from the other source cutoffs. Multiply it by a third-coordinate cutoff -which is one near the entire interval $[0,h_i]$. The curl of the potential -$(0,h_i x,0)$ times these cutoffs equals $(0,0,h_i)$ along that source -column. A common flat schedule lifts all rectangles. The corresponding -target cutoffs give a later descent by $-h_i$. - -In the middle phase, let $p_i,q_i$ be the centers and $a_i>0$ the -first-coordinate scale. With phase parameter $\mu\in[0,1]$, set -$c_i=p_i+\mu(q_i-p_i)$, $l_i=1+\mu(a_i-1)$ and -\[ - H_i(\mu,x,y)=d_{i,1}y-d_{i,2}x+ - \frac{a_i-1}{l_i}(x-c_{i,1})(y-c_{i,2}), - \qquad d_i=q_i-p_i. -\] -On a neighborhood of the desired path, the planar Hamiltonian field -$(\partial_yH_i,-\partial_xH_i)$ generates -$c_i+\diag(l_i,l_i^{-1})(\zeta-p_i)$. -The first coordinate is the convex interpolation of its endpoint values. -The second half-width obeys -\[ - \frac{r_{i,2}}{(1-\mu)+\mu a_i} - \le (1-\mu)r_{i,2}+\mu r_{i,2}/a_i, -\] -by convexity of the reciprocal function. Consequently the swept rectangle -lies in the coordinate bounding rectangle of the two endpoints. Localize -$H_i$ around that rectangle in the layer $z=h_i$, with third padding less -than one, and take the curl of $(0,0,H_i)$ times that cutoff. For a -flat middle ramp $\mu=\eta_2(t)$, multiply the resulting phase field by -$\eta_2'(t)$. Layers are disjoint, so the sum has the required action. Lift and descent preserve -planar coordinates. The exact paths and ODE uniqueness establish the -full-rectangle and intermediate-coordinate assertions. -\end{proof} - -\subsection{Target containers and prescribed affine images} -\label{bc:contained-targets} -The named targets in Propositions~\ref{bc:ordered}, \ref{bc:simultaneous} -and~\ref{bc:columns} can also be rational \emph{containers} for the actual -images. More precisely, replace the onto condition by -\[ - F_i(D_i)\subseteq S_i\quad\text{in Proposition~\ref{bc:ordered}}, - \qquad - F_i(S_i)\subseteq T_i\quad\text{in the other two propositions}. -\] -Keep the other geometric hypotheses. For ordered transport retain the -center-to-center condition. For simultaneous and column transport the -actual endpoint center is $F_i(p_i)$, where $p_i$ is the source center; -it need not be the center of the container. The maps have the same positive -diagonal determinant-one linear parts as before, including the specified -three scale choices in Proposition~\ref{bc:simultaneous}. -All transport and path conclusions remain valid. For arbitrary real map -coefficients this is a smooth existence statement. The construction is -effective when those coefficients are computable, and otherwise its -evaluation is relative to them. An empty list again uses the zero field. - -Here are the necessary modifications of the proofs. In ordered transport, -write $r_{i,j}$ for a source half-width, $w_{i,j}$ for the corresponding -container half-width, and $a_{i,j}>0$ for the diagonal factor. Containment -gives $a_{i,j}r_{i,j}\le w_{i,j}$, hence -\[ - r_{i,j}a_{i,j}^{\theta} - \le\max(r_{i,j},a_{i,j}r_{i,j}) - \le\max(r_{i,j},w_{i,j})0$ be the minimum sup-norm separation within the -two center families. Put $K=1+32R/m_0$. Expand all source centers by -factors from one to $K$, leaving their shapes unchanged. Let -$M=\max_i\{|(Kc_i)_1|,|(Kq_i)_1|\}$ and choose parking centers -$P_i=(M+32Ri,0)$. Each union of expanded endpoint centers and parking -centers has separation at least $32R$. - -Move expanded sources successively to their parking centers, deform them -there using the linear reciprocal scale, move them successively to the -expanded target centers, then contract those centers to their targets. -For each successive transfer put a square of sup-radius $4R$ around -every stationary center. Replace each nontrivial segment intersection -with an obstacle square by a counterclockwise path along its boundary. -The intersection points and corners are rational; the squares are disjoint, -so the resulting finite polygon stays at distance at least $4R$ from -all stationary centers. Flat schedules on its segments give smooth center -motion. The simultaneous expansion and contraction preserve a positive -coordinate gap for every pair, exactly as in the solid-box argument. -If $g_0,g_1$ are their minimum positive gaps, choose padding -$\varepsilon=\min(g_0,g_1,2R)/4$ for these phases. During individual -transfers use padding below $R$; the stationary half-widths are below $R$. -Thus all required cutoff supports avoid stationary rectangles. - -For a moving center $c$ and horizontal scale $l$, the potential -\[ - H=\chi\left[\dot c_x(y-c_y)-\dot c_y(x-c_x) - +\frac{\dot l}{l}(x-c_x)(y-c_y)\right] -\] -gives the required motion on the cutoff plateau. The other current -rectangles are fixed. This proves every full planar branch, including -its boundary, by induction through the operations. A one-square loading -translation with half-width $1/4$ takes $(-4,0)$ to the initial rational -code. All these supports lie in a square $[-L,L]^2$, where one can choose -$L=10+S_0+4m_Q$ with $S_0$ a finite bound for routing vertices and cutoff -widths. The same $L$ covers every input of a fixed machine, since initial -codes are uniformly bounded. - -The planar detours need not avoid the horizontal observation box. -We instead make them at height three. Choose a switch $\eta$ with -collars and set -$h(s)=3\{\eta(3s)-\eta(3s-2)\}$ on a unit interval. It lifts in the -first third, stays at three in the middle, and lowers in the last third. -Choose $\Gamma$ equal to one near $[-L,L]^2\times[-1,4]$, with compact -support, and $\beta(z)$ compactly supported and equal to one near three. -For any one of the planar fields $w$, zero outside its phase interval, -put -\[ - Z(s,x)=h'(s)\nabla\times(0,x_1\Gamma(x),0),\qquad - V_w(s,x)=Z(s,x)+3\beta(x_3)(w(3s-1,x_1,x_2),0). -\] -Along each tracked path the first term is exact vertical translation. -The second term acts only in the middle third, at height three, and runs -the full planar phase. It is divergence free because $w$ is planar -solenoidal and $\beta$ is independent of the planar coordinates. -Thus the trajectory lifts at its exact source, performs every detour -outside the vertical detector interval, and lowers at its exact target. -Use this construction for the loader and for each repeated phase. While -its height can lie in $(-1,1)$, a nonterminal computational particle has -first coordinate at least four; the initial loading source has first -coordinate $-4$. A terminal code lies in $(0,1)^2\times\{0\}$. -These facts prove the all-real-time observation equivalence, including -an initially terminal input. - -For the comparison assertion, the constructed field has compact support -and bounded mixed derivatives, so it satisfies the reference hypotheses -of Lemma~\ref{lem:b-comparison}. The displayed competitor class is exactly -case (v) of that lemma. Its proof obtains $\nabla p\in L^2$ from the -equation and cancels the pressure term by Lemma~\ref{lem:b-gradient}; it -does not require an integrable pressure representative. Hence the velocity -is unique, and pressure is determined up to a function of time. - -Finally let $U(s)$ denote the eventually periodic field just constructed -and set $\rho(t)=(1+t)^{1/3}-1$, $\widehat U=\rho' U(\rho(t))$. -The new residual is -\[ - \widehat f=\rho''U+(\rho')^2\{U_s+(U\cdot\nabla)U\} - -\nu\rho'\Delta U, - \quad \rho'=\tfrac13(1+t)^{-2/3},\quad - \rho''=-\tfrac29(1+t)^{-5/3}. -\] -Uniform derivative bounds and the fixed compact support give the stated -decay and space-time $L^2$ estimate. Chain differentiation also gives all -mixed bounds. The clock is effective on $[0,\infty)$, starts at zero, -is finite at finite time and is onto; hence it preserves precisely the -same material event. The initial zero collar is preserved as well. -This slower force is an alternative to, not a periodic version of, the -first force. -\end{proof} +\section{Evacuation, columns and obstacle detours}\label{sec:solid-routing} +The following refinements choose the occupied regions and the storage order +explicitly. Ordered low extraction controls a height observer; simultaneous +lifting controls a first-coordinate observer; expanded centers permit +three-dimensional detours. Fixed columns and recorded pointers give further +ways to keep a whole transition outside a detector. Each proof specifies +which parts of the geometry act on solid neighborhoods and which bounds +are needed only for the coded central plane. + +Unless an individual theorem specifies another class, whole-space +comparison means smooth $u\in CH^2\cap C^1L^2$, bounded $u,\nabla u$ +on finite intervals, and a pressure representative in $CH^1$, as in +Section~\ref{bc:analytic-section}. The torus comparison class is the +classical class of Lemma~\ref{lem:b-comparison}. The distinct pressure +conditions stated below remain part of their respective results. + +\subsection{Ordered extraction below the computation}\label{bc:ordered-section} +\begin{proposition}[Ordered full-box transport]\label{bc:ordered} +Let $D_i,S_i$ be finite families of closed rational axis-aligned boxes in $\R^3$ +with positive side lengths, pairwise disjoint within each family. +Suppose the assigned center-to-center affine map $F_i$ satisfies +$F_i(D_i)=S_i$ and has positive diagonal linear part $A_i$ of +determinant one. There is an effective compactly supported solenoidal +unit-time field, zero near the time endpoints, whose time-one map has +this action on every full $D_i$. If both centers of a pair have negative +third coordinate, every point of $D_i$ whose third coordinate equals that +of its center follows a path with negative third coordinate. +\end{proposition} +\begin{proof}\label{bc:ordered-proof} +Choose an integer $L>1$ larger than the absolute values of all endpoint +coordinates. The storage columns have first coordinates $U_i=4(i+1)L$ +and common height $Z=-4L$. Write $[\ell_i,r_i]$ for the first-coordinate +interval of $D_i$. Extract sources in decreasing order of $r_i$. +Translate each horizontally right to its column and then vertically to +$Z$, retaining its second coordinate. If an unextracted box meets the +active box in both the second and third projections, disjointness forces +their first intervals to be separated. That box cannot lie to the right: +its right endpoint would then exceed $r_i$, contrary to the extraction +order. It lies strictly to the left of $\ell_i$ and misses the entire +rightward sweep. Stored boxes lie below the original boxes; vertical +moves occur in distinct columns outside them. + +At storage height, adjust the second center coordinate to its target +value and apply $A_i^{\theta}$. Each intermediate half-width lies between +its two endpoint values, hence is less than $L$. The columns remain +disjoint throughout these operations. Deliver boxes in increasing order +of the right endpoint of their target first interval: raise a box to its +target height and translate it left into its target. The raising column +is empty. As above, a previously delivered box whose last two intervals +meet the active ones must be separated in the first coordinate. Its +smaller right endpoint puts it wholly to the left of the active target +interval, so the leftward sweep stops before reaching it. +Undelivered boxes remain at storage height. These observations give +positive rational clearances for all swept bounding boxes. Choose cutoff +padding below each clearance and use Lemma~\ref{lem:bb-curl} in successive +flat slots. Induction over the slots proves the full-box assertion. +A point initially in the central third-coordinate plane retains zero +third-coordinate offset during each diagonal deformation. Its height is +interpolated only between the two endpoint heights and $Z$. If those +endpoint heights are negative, every intermediate height is negative. +\end{proof} + +The onto hypothesis in the preceding proposition is a useful exact-box +specialization. Its target-container version, including the effective +input conditions, is given in Section~\ref{bc:contained-targets}. + +\begin{theorem}[A positive-height observation]\label{bc:positive-height} +On $\R^3$, the fixed label $0$ and observation $\{x_3>0\}$ realize +halting effectively by a compactly supported general force with zero +initial velocity and pressure, bounded mixed derivatives, and period one +after time one. The force has a separate alternative in +$L^2(0,\infty;H^j)$ for every $j$, tending to zero in every spatial +supremum norm at rate $O((1+t)^{-1})$. +\end{theorem} +\begin{proof}\label{bc:positive-height-proof} +Use Lemma~\ref{lem:bb-movefirst} and odd digits in base $2m+2$. Give every +simulator state a distinct positive integer index $i_s$ and put its +third-coordinate center at $z_s=i_s$ for ready halting states and +$z_s=-i_s$ otherwise. Thicken the full rectangles of +Lemma~\ref{lem:bb-onecell} by $[z_s-1/4,z_s+1/4]$. The assigned third +scale is one, so the maps satisfy Proposition~\ref{bc:ordered}. +The rational initial code $Y_0$ is loaded from the origin along the +straight segment, by a localized translation during $[0,1]$. Repeat the +unit-time mechanism after time one. At time $1+k$ the particle is exactly +the code after $k$ defined simulator transitions. If the initial machine +halts, the loading endpoint already has positive height. If it never +halts, loading has nonpositive height and all later transitions have +negative endpoint centers; the proposition excludes a positive height +at every intervening time. Conversely, a halting checkpoint has positive +height. Lemma~\ref{bc:analytic} gives the force and uniqueness, and +Lemma~\ref{ba:log-clock} gives the separate temporal alternative. +\end{proof} + +\subsection{Simultaneous lifting with a convex-coordinate guard} +\label{bc:simultaneous-section} +\begin{proposition}[Simultaneous full-box transport]\label{bc:simultaneous} +Suppose finite planar source and target rectangle families $S_i,T_i$ +have positive side lengths and endpoints on the grid $B^{-2}\Z^2$, +with integer $B\ge2$, and are separately +disjoint. Require their assigned center-to-center maps to satisfy +$F_i(S_i)=T_i$ and to have linear part $\diag(\lambda_i,\lambda_i^{-1})$ with +$\lambda_i\in\{B^{-1},1,B\}$. There is an effective solenoidal motion +of the full boxes obtained by taking the product with $[-1,1]$. +For every transported point its first coordinate is a convex combination +of its initial and final first coordinates at every time. +\end{proposition} +\begin{proof}\label{bc:simultaneous-proof} +Let $p_i^-,p_i^+$ be the centers in the plane $z=0$, set $h_i=5i$, and +choose a smooth $\Theta$ equal to zero on $(-\infty,1/4]$ and one on +$[3/4,\infty)$. For $0\le t\le1$, put +\[ + \theta=\Theta(3t-1),\quad l_i=1-\theta+\theta\lambda_i, + \quad c_i=(1-\theta)p_i^-+\theta p_i^+ + +h_i\{\Theta(3t)-\Theta(3t-2)\}e_3. +\] +The desired path is $g_i(t,y)=c_i(t)+\diag(l_i,l_i^{-1},1)(y-p_i^-)$. +Use a source padding $\varepsilon=1/(10B^3)$, transported by this same +affine map, for the cutoff neighborhood. During lifting the planar +source gap is at least $B^{-2}$. During lowering the enlarged target +padding is at most $B\varepsilon20R$ so large that the +expanded source centers $A_i^*=Ha_i$ and target centers $B_i^*=Hb_i$ +are separately more than $20R$ apart in the sup norm. Set +\[ + Z_0=1+100R+\max_i\{\|A_i^*\|_\infty,\|B_i^*\|_\infty\}, + \qquad P_i=(0,0,-Z_0-100Ri). +\] +Each of the unions $\{A_i^*\}\cup\{P_i\}$ and +$\{B_i^*\}\cup\{P_i\}$ has separation greater than $20R$. +The construction has five phases: expand source centers by a factor +from one to $H$ while keeping shapes fixed; move them successively to +$P_i$; change each parked shape by the prescribed positive diagonal +power; move them successively to $B_i^*$; contract target centers to +$b_i$ while keeping the final shapes fixed. + +The simultaneous expansion is collision free. For each pair of source +boxes some coordinate has positive gap +$|a_{i,j}-a_{k,j}|-r_{i,j}-r_{k,j}$; multiplication of the center +difference by a factor at least one preserves that gap. The same +argument applies to contraction toward the disjoint targets. Choose +padding less than one quarter of the minimum such gap, and less than +one. During a parked deformation the $j$th half-width is +$r_{i,j}^{1-\theta}w_{i,j}^{\theta}\le\max(r_{i,j},w_{i,j})0$. +Every boundary used around a negative-height obstacle lies below +$-H+4R<0$. Storage heights are negative as well. Expansion, contraction, +and positive diagonal deformation preserve the central-plane condition. +Thus all these paths are negative. Finitely many rational vertices, +endpoint widths, and cutoff paddings give a rational support bound. +\end{proof} + +\begin{theorem}[A height test inside one torus chart]\label{bc:expanded-height-event} +On the unit torus, the fixed label $o=(1/2,1/2,1/2)$ and event +\begin{equation}\label{bc:expanded-height-fixed-event} + \{x:x_3\pmod1\in(1/2,1)\} +\end{equation} +realize halting by a smooth general mean-zero force, periodic after time +one, with zero initial velocity and pressure and bounded mixed derivatives. +The repeated mechanism depends on the machine alone; only initialization +depends on its finite input. The solution is unique in the torus class. +\end{theorem} +\begin{proof}\label{bc:expanded-height-event-proof} +Use the frontier recorder, odd base $B=2m+1$, distinct negative integer +heights for nonterminal states and height one for the terminal ready state. +Thicken every branch by a third half-width $1/4$. The reciprocal tape +map is onto its target rectangle; adjoining the unchanged third interval +gives an onto box map with linear part $\diag(B^{-e},B^e,1)$ and +determinant one. All source centers are +negative because terminal controls have no outgoing rules. Apply the +preceding proposition. The logical initial code $Y_0$ is rational and has +a uniform machine-dependent bound as the input varies: its two stack +coordinates lie in $[0,1]$ and its height is fixed by the initial control. +A moving translation loads the logical origin to $Y_0$, with a fixed +source half-width one and padding one. This gives a machine-dependent +rational bound $K$ for the supports of all such loaders and the repeated +mechanism; for example take $K=1+2R+M$ with $M$ an upper bound for all +routing vertices and loading coordinates and $R$ enlarged to cover their +half-widths. Choose $\lambda=1/(4K)$ and use the chart $x=o+\lambda Y$. +Everything remains strictly inside $(1/4,3/4)^3$. + +Periodize the resulting compactly supported curl fields, loading during +the first unit interval and repeating thereafter. Compute the residual +at physical viscosity $\nu$, after the chart change. The fields and force +have zero mean and all the bounds in Lemma~\ref{bc:analytic}. +For a nonhalting run, loading has nonpositive logical height and each +later segment remains negative by the proposition. A terminal code has +physical height $1/2+\lambda$, which lies in the event. There is no +chart-boundary crossing that could create a spurious observation. +An initially terminal input is detected at the loading endpoint. +The initialized correspondence is exact; at checkpoints the number of +history records gives $n$, and the frontier's relative position together +with its absolute index $2+n$ also recovers the absolute work-head position. +\end{proof} + +\subsection{Fixed columns and separate layers}\label{bc:columns-section} +\begin{proposition}[Column and layer transport]\label{bc:columns} +Let $S_i,T_i$ be finite rational planar rectangle families of positive side +lengths, each separately disjoint, with endpoints on the $B^{-2}$ grid, +where $B\ge2$ is an integer. +Let their assigned center-to-center positive reciprocal diagonal affine +maps satisfy $F_i(S_i)=T_i$. A finite effective +solenoidal field in $\R^3$ realizes all these maps on the rectangles in +$z=0$. Along each tracked path the first coordinate stays between its +initial and final values. The field has compact support and flat time +collars. +\end{proposition} +\begin{proof}\label{bc:columns-proof} +Give rectangle $i$ height $h_i=4i$. A planar cutoff equal to one near +its source and supported within padding $1/(4B^2)$ has support disjoint +from the other source cutoffs. Multiply it by a third-coordinate cutoff +which is one near the entire interval $[0,h_i]$. The curl of the potential +$(0,h_i x,0)$ times these cutoffs equals $(0,0,h_i)$ along that source +column. A common flat schedule lifts all rectangles. The corresponding +target cutoffs give a later descent by $-h_i$. + +In the middle phase, let $p_i,q_i$ be the centers and $a_i>0$ the +first-coordinate scale. With phase parameter $\mu\in[0,1]$, set +$c_i=p_i+\mu(q_i-p_i)$, $l_i=1+\mu(a_i-1)$ and +\[ + H_i(\mu,x,y)=d_{i,1}y-d_{i,2}x+ + \frac{a_i-1}{l_i}(x-c_{i,1})(y-c_{i,2}), + \qquad d_i=q_i-p_i. +\] +On a neighborhood of the desired path, the planar Hamiltonian field +$(\partial_yH_i,-\partial_xH_i)$ generates +$c_i+\diag(l_i,l_i^{-1})(\zeta-p_i)$. +The first coordinate is the convex interpolation of its endpoint values. +The second half-width obeys +\[ + \frac{r_{i,2}}{(1-\mu)+\mu a_i} + \le (1-\mu)r_{i,2}+\mu r_{i,2}/a_i, +\] +by convexity of the reciprocal function. Consequently the swept rectangle +lies in the coordinate bounding rectangle of the two endpoints. Localize +$H_i$ around that rectangle in the layer $z=h_i$, with third padding less +than one, and take the curl of $(0,0,H_i)$ times that cutoff. For a +flat middle ramp $\mu=\eta_2(t)$, multiply the resulting phase field by +$\eta_2'(t)$. Layers are disjoint, so the sum has the required action. Lift and descent preserve +planar coordinates. The exact paths and ODE uniqueness establish the +full-rectangle and intermediate-coordinate assertions. +\end{proof} + +\subsection{Target containers and prescribed affine images} +\label{bc:contained-targets} +The named targets in Propositions~\ref{bc:ordered}, \ref{bc:simultaneous} +and~\ref{bc:columns} can also be rational \emph{containers} for the actual +images. More precisely, replace the onto condition by +\[ + F_i(D_i)\subseteq S_i\quad\text{in Proposition~\ref{bc:ordered}}, + \qquad + F_i(S_i)\subseteq T_i\quad\text{in the other two propositions}. +\] +Keep the other geometric hypotheses. For ordered transport retain the +center-to-center condition. For simultaneous and column transport the +actual endpoint center is $F_i(p_i)$, where $p_i$ is the source center; +it need not be the center of the container. The maps have the same positive +diagonal determinant-one linear parts as before, including the specified +three scale choices in Proposition~\ref{bc:simultaneous}. +All transport and path conclusions remain valid. For arbitrary real map +coefficients this is a smooth existence statement. The construction is +effective when those coefficients are computable, and otherwise its +evaluation is relative to them. An empty list again uses the zero field. + +Here are the necessary modifications of the proofs. In ordered transport, +write $r_{i,j}$ for a source half-width, $w_{i,j}$ for the corresponding +container half-width, and $a_{i,j}>0$ for the diagonal factor. Containment +gives $a_{i,j}r_{i,j}\le w_{i,j}$, hence +\[ + r_{i,j}a_{i,j}^{\theta} + \le\max(r_{i,j},a_{i,j}r_{i,j}) + \le\max(r_{i,j},w_{i,j})0$ be the minimum sup-norm separation within the +two center families. Put $K=1+32R/m_0$. Expand all source centers by +factors from one to $K$, leaving their shapes unchanged. Let +$M=\max_i\{|(Kc_i)_1|,|(Kq_i)_1|\}$ and choose parking centers +$P_i=(M+32Ri,0)$. Each union of expanded endpoint centers and parking +centers has separation at least $32R$. + +Move expanded sources successively to their parking centers, deform them +there using the linear reciprocal scale, move them successively to the +expanded target centers, then contract those centers to their targets. +For each successive transfer put a square of sup-radius $4R$ around +every stationary center. Replace each nontrivial segment intersection +with an obstacle square by a counterclockwise path along its boundary. +The intersection points and corners are rational; the squares are disjoint, +so the resulting finite polygon stays at distance at least $4R$ from +all stationary centers. Flat schedules on its segments give smooth center +motion. The simultaneous expansion and contraction preserve a positive +coordinate gap for every pair, exactly as in the solid-box argument. +If $g_0,g_1$ are their minimum positive gaps, choose padding +$\varepsilon=\min(g_0,g_1,2R)/4$ for these phases. During individual +transfers use padding below $R$; the stationary half-widths are below $R$. +Thus all required cutoff supports avoid stationary rectangles. + +For a moving center $c$ and horizontal scale $l$, the potential +\[ + H=\chi\left[\dot c_x(y-c_y)-\dot c_y(x-c_x) + +\frac{\dot l}{l}(x-c_x)(y-c_y)\right] +\] +gives the required motion on the cutoff plateau. The other current +rectangles are fixed. This proves every full planar branch, including +its boundary, by induction through the operations. A one-square loading +translation with half-width $1/4$ takes $(-4,0)$ to the initial rational +code. All these supports lie in a square $[-L,L]^2$, where one can choose +$L=10+S_0+4m_Q$ with $S_0$ a finite bound for routing vertices and cutoff +widths. The same $L$ covers every input of a fixed machine, since initial +codes are uniformly bounded. + +The planar detours need not avoid the horizontal observation box. +We instead make them at height three. Choose a switch $\eta$ with +collars and set +$h(s)=3\{\eta(3s)-\eta(3s-2)\}$ on a unit interval. It lifts in the +first third, stays at three in the middle, and lowers in the last third. +Choose $\Gamma$ equal to one near $[-L,L]^2\times[-1,4]$, with compact +support, and $\beta(z)$ compactly supported and equal to one near three. +For any one of the planar fields $w$, zero outside its phase interval, +put +\[ + Z(s,x)=h'(s)\nabla\times(0,x_1\Gamma(x),0),\qquad + V_w(s,x)=Z(s,x)+3\beta(x_3)(w(3s-1,x_1,x_2),0). +\] +Along each tracked path the first term is exact vertical translation. +The second term acts only in the middle third, at height three, and runs +the full planar phase. It is divergence free because $w$ is planar +solenoidal and $\beta$ is independent of the planar coordinates. +Thus the trajectory lifts at its exact source, performs every detour +outside the vertical detector interval, and lowers at its exact target. +Use this construction for the loader and for each repeated phase. While +its height can lie in $(-1,1)$, a nonterminal computational particle has +first coordinate at least four; the initial loading source has first +coordinate $-4$. A terminal code lies in $(0,1)^2\times\{0\}$. +These facts prove the all-real-time observation equivalence, including +an initially terminal input. + +For the comparison assertion, the constructed field has compact support +and bounded mixed derivatives, so it satisfies the reference hypotheses +of Lemma~\ref{lem:b-comparison}. The displayed competitor class is exactly +case (v) of that lemma. Its proof obtains $\nabla p\in L^2$ from the +equation and cancels the pressure term by Lemma~\ref{lem:b-gradient}; it +does not require an integrable pressure representative. Hence the velocity +is unique, and pressure is determined up to a function of time. + +Finally let $U(s)$ denote the eventually periodic field just constructed +and set $\rho(t)=(1+t)^{1/3}-1$, $\widehat U=\rho' U(\rho(t))$. +The new residual is +\[ + \widehat f=\rho''U+(\rho')^2\{U_s+(U\cdot\nabla)U\} + -\nu\rho'\Delta U, + \quad \rho'=\tfrac13(1+t)^{-2/3},\quad + \rho''=-\tfrac29(1+t)^{-5/3}. +\] +Uniform derivative bounds and the fixed compact support give the stated +decay and space-time $L^2$ estimate. Chain differentiation also gives all +mixed bounds. The clock is effective on $[0,\infty)$, starts at zero, +is finite at finite time and is onto; hence it preserves precisely the +same material event. The initial zero collar is preserved as well. +This slower force is an alternative to, not a periodic version of, the +first force. +\end{proof} diff --git a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/torus-routing.tex b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/torus-routing.tex index c18dc50cf82ad5f425a868824b40d419afffd9c1..42d13725693fbe7a5b84fbbcbc335824b73162e1 100644 --- a/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/torus-routing.tex +++ b/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/torus-routing.tex @@ -1,365 +1,365 @@ -\section{Torus charts and phasewise localization}\label{sec:torus-routing} - -The next constructions keep the whole motion inside a specified torus -chart. A scaled linear interpolation preserves a convex first coordinate; -an exponential interpolation in a fixed chart instead uses endpoint-width -bounds. A phasewise construction then exhibits the vertical and planar -potentials separately, and the final variant retains a permanent origin -track. These choices give different path estimates for their observers. -In every case the force is recomputed at the prescribed physical viscosity -after any spatial rescaling. - -\subsection{Physical chart scaling and thickened torus boxes} - -\begin{proposition}[Linear scaling into the torus] -\label{prop:bb-mid-scaled-torus} -The direction recorder without its passive track has a unit-torus -realization periodic after $t=1$, with fixed label $(1/8,0,0)$ and -halting event $x_1\in(-1/4,0)\pmod1$. Each period acts on full -solid boxes; the pressure is zero and the force has zero mean. -\end{proposition} -\begin{proof}\label{proof:bb-mid-scaled-torus} -Use the controls of Lemma~\ref{lem:bb-direction}. Let -$\Sigma=\Gamma\times(\{\bot,F\}\sqcup\mathcal R)\times\{0,1\}$ -be its full cell alphabet after deleting the singleton passive track. -Let $S$ be the finite set of recorder controls and choose a bijection -$j:S\to\{1,\ldots,|S|\}$. -Use odd digits with $B=2|\Sigma|+1$, and offsets $o_s=-2j(s)$ for halting main -controls and $o_s=2j(s)$ for the others. Number the resulting reciprocal branches -$i=1,\ldots,N$, and write $k_i$ for branch $i$'s first-coordinate scale. -Thicken each reciprocal branch by -$[-1,1]$. Its horizontal family gaps are at least $B^{-2}$. -Lift branch $i$ to $4i$, use the linear first scale -$\kappa_i=1+\theta(k_i-1)$, and descend. Padding smaller than -$(4B^2)^{-1}$ separates all phases: in the middle the half-height -is one and height gaps are four; in the other phases horizontal -gaps suffice. The two horizontal half-widths are at most $1/2$. -Use the true-plateau cutoff of Lemma~\ref{lem:bb-curl}; it supplies -at least the full-box guarantee, together with a neighborhood extension. -The exact first coordinate obeys -\begin{equation}\label{eq:bb-mid-scaled-convex} - X_1(\tau)=(1-\theta)X_1(0)+\theta X_1(1). -\end{equation} - -Let $V$ be the period-one extension of the unit-time field just constructed. -All codes, paths and padded supports lie in $(-L,L)^3$ for -$L=10+2|S|+4N$. Put $a=1/(16L)$ and define in the torus patch -$(-1/2,1/2)^3$ the repeated field $U(t,x)=aV(t,x/a)$. -Its support lies in $(-1/16,1/16)^3$ and its curl potential is the -scaled potential $a^2A(t,x/a)$. In particular its mean is zero. -The residual must be recomputed at the physical viscosity: -\begin{equation}\label{eq:bb-mid-viscosity-scale} - \mathcal F_\nu[U](t,x)=a\{V_t+(V\cdot\nabla)V-(\nu/a^2)\Delta V\}(t,x/a). -\end{equation} -This keeps the specified viscosity rather than silently rescaling it. -During the first unit interval load from $(1/8,0,0)$ to the scaled -initial code using a curl translation with a box of half-width $1/32$ -and padding $1/32$. Its support lies inside $(-1/4,1/4)^3$. - -For a nonhalting run both loading endpoints have positive first -coordinate, and every subsequent pair of coded endpoints is positive. -Equation~\eqref{eq:bb-mid-scaled-convex} preserves that sign throughout. -A halting checkpoint has negative first coordinate in $(-1/16,0)$. -All paths stay in the stated patch, so passage modulo one introduces -no extra visit. Checkpoint times are -$1+\sum_{l0,\\0,&r\le0,\end{cases} - \qquad \Theta(r)=\frac{E(r)}{E(r)+E(1-r)}. -\] -Use the single-halt normalization and the seven-row recorder in -Lemma~\ref{lem:bb-movefirst}, with frontier initially at two and all marks -initially zero. In the present notation its controls are -$S_q,U_r,W_r,C_q,D_q$ (ready, work-move completed, right scan, -frontier extension, left scan), and its full symbols are $(a,g,m)$ -with history $g\in\{E,F\}\sqcup\{\widehat r\}$. -The first row writes and moves, the second marks and starts the scan, -and the last clears the mark and stays. These are exactly the seven -rows already proved, after renaming. Thus the new work head $j'$ is -less than $n+2$, a work step takes $2(n+2-j')+4$ local steps, and -both full-domain incoming properties hold. - -Let $M$ count the partial controls, let $H=S_h$, and put -\[ - \ell=\frac1{100(M+1)},\qquad o_H=(3/4,1/2),\qquad - o_{s_i}=(1/4+2i\ell,1/2)\quad(0\le i0$. Curls have zero mean, and incompressibility -gives zero mean for the residual. Lemma~\ref{ba:realization} supplies -the unique zero-data PDE solution and its global trajectories. - -At times $1+k$, induction through the full branch maps gives the exact -local code through the first halt, or for all $k$ in the nonhalting -case. A halt code has first coordinate in -$[3/4,3/4+\ell]\subset(2/3,5/6)$, including a code reached solely by -loading an initially halted machine. For a nonhalting input the loader -stays below $1/2$ in first coordinate. Lift and descent retain the -nonhalting planar endpoint codes. In the middle phase the centers are -convex combinations of nonhalting centers and the first half-width is -at most $\ell/2$, so -\[ - X_1\le1/4+2M\ell+\ell/2<1/2. -\] -This excludes the height-independent observer at every intermediate -time. Only the loader depends on the input; the repeated field depends -on the finite machine table. -\end{proof} - - -\subsection{A permanent origin marker and moving cutoff neighborhoods} -\label{bc:height-separated-section} -\begin{theorem}[A height-separated mechanism with an origin track] -\label{bc:height-separated-event} -On the unit torus, the fixed label $(1/8,3/8,1/4)$ and strip -$2/30,\\0,&r\le0,\end{cases} + \qquad \Theta(r)=\frac{E(r)}{E(r)+E(1-r)}. +\] +Use the single-halt normalization and the seven-row recorder in +Lemma~\ref{lem:bb-movefirst}, with frontier initially at two and all marks +initially zero. In the present notation its controls are +$S_q,U_r,W_r,C_q,D_q$ (ready, work-move completed, right scan, +frontier extension, left scan), and its full symbols are $(a,g,m)$ +with history $g\in\{E,F\}\sqcup\{\widehat r\}$. +The first row writes and moves, the second marks and starts the scan, +and the last clears the mark and stays. These are exactly the seven +rows already proved, after renaming. Thus the new work head $j'$ is +less than $n+2$, a work step takes $2(n+2-j')+4$ local steps, and +both full-domain incoming properties hold. + +Let $M$ count the partial controls, let $H=S_h$, and put +\[ + \ell=\frac1{100(M+1)},\qquad o_H=(3/4,1/2),\qquad + o_{s_i}=(1/4+2i\ell,1/2)\quad(0\le i0$. Curls have zero mean, and incompressibility +gives zero mean for the residual. Lemma~\ref{ba:realization} supplies +the unique zero-data PDE solution and its global trajectories. + +At times $1+k$, induction through the full branch maps gives the exact +local code through the first halt, or for all $k$ in the nonhalting +case. A halt code has first coordinate in +$[3/4,3/4+\ell]\subset(2/3,5/6)$, including a code reached solely by +loading an initially halted machine. For a nonhalting input the loader +stays below $1/2$ in first coordinate. Lift and descent retain the +nonhalting planar endpoint codes. In the middle phase the centers are +convex combinations of nonhalting centers and the first half-width is +at most $\ell/2$, so +\[ + X_1\le1/4+2M\ell+\ell/2<1/2. +\] +This excludes the height-independent observer at every intermediate +time. Only the loader depends on the input; the repeated field depends +on the finite machine table. +\end{proof} + + +\subsection{A permanent origin marker and moving cutoff neighborhoods} +\label{bc:height-separated-section} +\begin{theorem}[A height-separated mechanism with an origin track] +\label{bc:height-separated-event} +On the unit torus, the fixed label $(1/8,3/8,1/4)$ and strip +$2/30\) for which \(rL\) is a line -bundle. For every \(\varepsilon>0\), that line has a smooth Hermitian metric -\(h_\varepsilon\) satisfying -\[ - \frac{\sqrt{-1}}{2\pi r}\Theta_{h_\varepsilon} - \geq-\varepsilon\omega. -\] -Smooth metrics and forms on a singular space are understood through local -embeddings. We call \(L\) semiample if an actual positive integral multiple -is a holomorphic line bundle generated by its global sections. - -\begin{assumption}[Logarithmic Iitaka subadditivity] -\label{asm:log-iitaka} -Let \(f:X\to Y\) be a surjective morphism with connected fibers between -smooth connected projective complex varieties. Let \(D_X\) and \(D_Y\) be -reduced effective simple normal crossing divisors, allowing zero divisors, -such that -\[ - \Supp(f^*D_Y)\subseteq\Supp(D_X). -\] -For a very general smooth fiber \(F\), put \(D_F=D_X|_F\). Then -\[ - \kappa(X,K_X+D_X)\geq - \kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y). -\] -Here \(D_F\) is reduced SNC and -\(K_F+D_F\sim (K_X+D_X)|_F\). The Iitaka dimension is \(-\infty\) when -all positive integral systems are empty, with -\((-\infty)+b=-\infty\) also for \(b=-\infty\); the zero divisor on a -point has Iitaka dimension zero. -\end{assumption} - -\begin{theorem}[Log abundance for compact Kähler spaces]\label{thm:main} -Assume Assumption~\ref{asm:log-iitaka}. Let \(X\) be a normal irreducible -compact Kähler complex analytic space, and let \(\Delta\geq0\) be a rational -divisor such that \((X,\Delta)\) is log canonical and \(K_X+\Delta\) is -rational Cartier. If \(K_X+\Delta\) is analytically nef, then it is -semiample. Explicitly, there is an integer \(m>0\) such that -\(m(K_X+\Delta)\) is Cartier and -\[ - H^0\bigl(X,\OO_X(m(K_X+\Delta))\bigr)\otimes_{\C}\OO_X - \longrightarrow \OO_X(m(K_X+\Delta)) -\] -is surjective at every point of \(X\). -\end{theorem} - -The theorem includes every finite dimension and the zero-boundary case. -Assumption~\ref{asm:log-iitaka} is used through the projective good-model -theorem stated in Proposition~\ref{proj:good-models}. The cited proof of -that projective theorem uses only its zero-boundary case, on a resolved -projective Albanese fibration. - -\subsection{Context and relation to earlier work} - -In the minimal model program, abundance complements the construction of -minimal models: the nef adjoint on a minimal model should determine a -canonical fibration. For minimal projective threefolds, Miyaoka proved the -case of numerical dimension one and Kawamata completed canonical abundance -\cite{Miyaoka1988,Kawamata1992}. Keel, Matsuki, and McKernan established -log abundance for threefolds -\cite{KMM1994,KMM2004}. In arbitrary dimension, Birkar, Cascini, Hacon, and -McKernan established log terminal models for projective klt pairs with big -boundary and pseudo-effective adjoint, and finite generation for big -adjoints \cite{BCHM2010}. - -The compact Kähler history already points to the importance of simple -spaces. Peternell's threefold theorem isolated the possible case of a simple -space not bimeromorphic to a finite quotient of a torus -\cite{Peternell2001}; Demailly and Peternell subsequently proved canonical -nonvanishing for nef terminal Kähler threefolds, including that case -\cite{DemaillyPeternell2003}. Höring and Peternell constructed minimal models -for normal \(\Q\)-factorial terminal compact Kähler threefolds with pseudo-effective canonical class -\cite{HoeringPeternell2016}. Campana, Höring, and Peternell treated canonical abundance -\cite{CHP2016,CHP2023}, and Das and Ou -established log abundance for compact Kähler log canonical threefolds -\cite[Corollary~1.3]{DasOu2026}. - -With \(D_X=D_Y=0\), Assumption~\ref{asm:log-iitaka} is Iitaka's classical -\(C_{n,m}\) inequality -\(\kappa(X)\geq\kappa(F)+\kappa(Y)\) for algebraic fiber spaces -\cite[Equation~(1.0.1)]{CaoPaun2017}. Cao and Păun proved the analogous -logarithmic inequality over an abelian base when the pair on the total space -has an effective rational boundary and is klt -\cite[Theorem~1.1]{CaoPaun2017}. Fujino derives -logarithmic subadditivity from the conjectured equality -\(\kappa_\sigma(X,K_X+D)=\kappa(X,K_X+D)\) for smooth projective -\(X\) with reduced SNC \(D\) -\cite{Fujino2017Subadditivity,Fujino2020Corrigendum}. Hashizume proves the -reduced-SNC subadditivity statement when the log canonical divisor of the -very general fiber is abundant \cite[Theorem~1.2]{Hashizume2020}. Here the -implication is used in the other direction: the projective companion -\cite{LI} derives good models for pseudo-effective projective lc adjoints -from Assumption~\ref{asm:log-iitaka}, and those good models anchor the -Kähler induction. - -Hacon and Xie's cone theorem, adjunction, relative positivity, and -modified-big preparation \cite{HX} provide analytic inputs for our -program constructions. Section~\ref{sec:programs} constructs effective -generalized base data on a higher carrier, with an exact adjoint -pullback identity and a multiplier-ideal inclusion. Fibrewise -$L^2$ positivity and regularization give this presentation; its -Stein-factor consequence supplies the non-klt induction. -The section then proves the restricted semiampleness, proper relative -good-model, and finite-model statements needed here, following the -dimension-induction strategy of \cite{HX}. -The nef data remain globally nef on a fixed carrier, and the total -generalized boundary is globally modified big. Actual rational lines -make the selected ray contractions projective; ordinary projective -analytic results \cite{FujinoAnalyticBCHM} then construct their flips -and the relative models. This produces a Kähler pullback description of -the relevant Bott--Chern class. Generation of the prescribed holomorphic -line requires the additional divisorial decomposition and boundary -arguments. The fibration comparison and period positivity come from -\cite{CI}; the boundary and lifting constructions adapt inputs in -\cite{CB,BL}. The precise imported results are stated at their points -of use. - -\subsection{The proof and its main constructions} - -For a pseudo-effective \((1,1)\)-class \(\alpha\), \(N(\alpha)\) records -the least generic vanishing forced along prime divisors. Given a smooth -compact Kähler manifold \(X\) and a rational SNC boundary \(B\) with -\(J=K_X+B\) pseudo-effective, the induction constructs a modification -\(\mu:Y\to X\) with \(Y\) smooth and -\[ - \mu^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad - R=N(c_1(\mu^*J)), -\] -as an identity of rational holomorphic line bundles, with \(R\) an effective -rational divisor. Section~\ref{sec:inductive-reduction} uses transfer rules, -algebraic reduction, and generically finite covers to reduce the induction -to projective manifolds, nontrivial fibrations, and simple spaces of -algebraic dimension zero. Here simple means that no positive-dimensional -proper compact subvariety passes through a very general point. - -Section~\ref{sec:programs} supplies two program results. For the fibration -case it contracts a known negative part while preserving the prescribed nef -line. For the simple case it proves special termination for a chosen scaling -and reaches a nef model when the pseudo-effective adjoint is rationally -equivalent to a divisor supported on the reduced boundary, allowing negative -coefficients. The termination argument uses lower-dimensional induction to -construct one model on which an interval of perturbed adjoints is nef. On a -common resolution their negative multiplicities are affine, whereas every -nontrivial wall of the restricted program changes a slope. Hence only -finitely many such walls occur. - -Section~\ref{sec:boundary} turns generation on separate strata into generation -on the entire reduced dlt boundary, adapting Fujino's method of admissible -sections \cite[Section~4]{Fujino2000}. When a comparison is needed, a Mori -contraction realizes Kollár's residue comparison at the two coefficient-one -points of a general \(\PP^1\) fiber -\cite[Definition~13 and Proposition~14]{Kollar2013Sources}. On a common -resolution, the pulled-back restrictions of the ambient Kähler class differ -by a real linear combination of Chern classes of line bundles. On a stratum -whose boundary dominates the image of the map -defined by its semiample adjoint, restriction determines sections. For the -remaining strata, we prove finite image for self-comparisons on sufficiently -divisible pluricanonical systems using an invariant integral, period and -lattice arguments, and a uniform cohomological bound for cyclic covers. -Products over these finite images give compatible generating sections, used -in both closing arguments. - -Section~\ref{sec:rank-one} first reduces to fibrations whose very general -fiber has log Kodaira dimension zero. On a prepared fibration \(g:X\to W\), -a relative generating form gives -\[ - K_X+B\sim_{\Q}g^*H+A_*,\qquad H=K_W+T+M, -\] -where \(T\) is a rational SNC boundary and \(M\) is pulled back from a nef -rational line on a projective quotient of \(W\). Fiber induction allows us -to subtract the part of \(A_*\) dominating \(W\) from a positive current for -\(K_X+B\). The resulting metric descends along connected smooth fibers; a -zero among the nonnegative coefficients at primes dominating each base prime -allows extension, proving \(H\) pseudo-effective. If \(a(W)=0\), the -projective quotient is a point, so \(M\sim_{\Q}0\) and lower-dimensional -induction makes \(H\) rationally equivalent to its negative divisor. For -projective \(W\), a reduction via a chosen generalized program derived from -Assumption~\ref{asm:log-iitaka} either lowers the positive base dimension, -handled by secondary induction, or reaches a nef line that is big or torsion. -At a stopping case, an intersection argument identifies the full negative -divisor upstairs. A torsion positive part completes the decomposition -directly. In the big nef case, Section~\ref{sec:programs} contracts that -divisor while preserving the nef line; boundary generation, extension, and -a new log canonical place at a hypothetical base locus prove semiampleness. - -Section~\ref{sec:nonvanishing} proves, independently of -Assumption~\ref{asm:log-iitaka}, that a positive canonical power has a nonzero -meromorphic section on every smooth connected simple compact Kähler manifold -of algebraic dimension zero. Ou's uniruledness and foliation results supply -the cotangent slope control \cite{Ou2025}. A point-threshold bound for big -classes of volume one is contradicted using an auxiliary projective bundle -over \(X\times X\). The diagonal in its square produces a subsheaf occupying -a fixed positive fraction of a symmetric cotangent power of that bundle. -A second construction over the diagonal of \(X\times X\) forces determinant -vanishing which, after restriction to a blowup of \(X\), violates the point -bound. - -Finally, Section~\ref{sec:signed} uses the meromorphic section to obtain, -after resolving and enlarging to a reduced SNC boundary, a pseudo-effective -adjoint with a signed representative supported there. -Section~\ref{sec:programs} gives a nef dlt model, and -Section~\ref{sec:boundary} gives generation on its reduced boundary. -Pseudo-effectivity of the canonical pullback and an intersection argument -isolate positive-dimensional boundary fibers away from components whose -signed coefficients are nonpositive. A local root construction adapted from -\cite[Proposition~3.3]{LI} separates the positive and negative divisor -supports. An SNC Hodge-module calculation extends the lifting method of -\cite[Sections~10--12]{BL} to residual poles and kills every finite-order -obstruction to lifting such a fiber. Douady space and Artin approximation -give compact deformations leaving the boundary; compactness of the cycle-space -components and relative compactness of bounded-volume cycles allow a Baire -argument to produce a covering family. Simplicity -forces the actual nef line to be torsion. The program comparison and -Lemma~\ref{bir:unique-current} then permit subtraction of the added boundary, -yielding the inductive decomposition. +\section{Introduction}\label{sec:introduction} + +The abundance problem asks when the numerical positivity of a log canonical +bundle produces a holomorphic map. More precisely, a nef adjoint should have +a positive multiple generated by global sections. On a compact Kähler space, +nefness is an analytic condition on a cohomology class, whereas generation +is a statement about an actual holomorphic line bundle. The distinction is +essential in the nonprojective setting. + +We prove log abundance for compact Kähler spaces under logarithmic Iitaka +subadditivity for smooth projective varieties with reduced boundary. The +argument passes through a stronger birational statement: every +pseudo-effective smooth log adjoint has a semiample positive part and an +effective rational fixed divisor equal to its analytic divisorial negative +part. This form both supports the induction and retains the line-bundle +information needed for generation on the original space. + +\subsection{The assumption and the theorem} + +A compact Kähler space is a compact complex analytic space with a Kähler +form in the sense of local strictly plurisubharmonic potentials on local +embeddings. For a rational line bundle \(L\), analytic nefness means that +\(c_1(L)\) belongs to the closure of the Kähler cone. Equivalently, fix a +Kähler form \(\omega\) and an integer \(r>0\) for which \(rL\) is a line +bundle. For every \(\varepsilon>0\), that line has a smooth Hermitian metric +\(h_\varepsilon\) satisfying +\[ + \frac{\sqrt{-1}}{2\pi r}\Theta_{h_\varepsilon} + \geq-\varepsilon\omega. +\] +Smooth metrics and forms on a singular space are understood through local +embeddings. We call \(L\) semiample if an actual positive integral multiple +is a holomorphic line bundle generated by its global sections. + +\begin{assumption}[Logarithmic Iitaka subadditivity] +\label{asm:log-iitaka} +Let \(f:X\to Y\) be a surjective morphism with connected fibers between +smooth connected projective complex varieties. Let \(D_X\) and \(D_Y\) be +reduced effective simple normal crossing divisors, allowing zero divisors, +such that +\[ + \Supp(f^*D_Y)\subseteq\Supp(D_X). +\] +For a very general smooth fiber \(F\), put \(D_F=D_X|_F\). Then +\[ + \kappa(X,K_X+D_X)\geq + \kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y). +\] +Here \(D_F\) is reduced SNC and +\(K_F+D_F\sim (K_X+D_X)|_F\). The Iitaka dimension is \(-\infty\) when +all positive integral systems are empty, with +\((-\infty)+b=-\infty\) also for \(b=-\infty\); the zero divisor on a +point has Iitaka dimension zero. +\end{assumption} + +\begin{theorem}[Log abundance for compact Kähler spaces]\label{thm:main} +Assume Assumption~\ref{asm:log-iitaka}. Let \(X\) be a normal irreducible +compact Kähler complex analytic space, and let \(\Delta\geq0\) be a rational +divisor such that \((X,\Delta)\) is log canonical and \(K_X+\Delta\) is +rational Cartier. If \(K_X+\Delta\) is analytically nef, then it is +semiample. Explicitly, there is an integer \(m>0\) such that +\(m(K_X+\Delta)\) is Cartier and +\[ + H^0\bigl(X,\OO_X(m(K_X+\Delta))\bigr)\otimes_{\C}\OO_X + \longrightarrow \OO_X(m(K_X+\Delta)) +\] +is surjective at every point of \(X\). +\end{theorem} + +The theorem includes every finite dimension and the zero-boundary case. +Assumption~\ref{asm:log-iitaka} is used through the projective good-model +theorem stated in Proposition~\ref{proj:good-models}. The cited proof of +that projective theorem uses only its zero-boundary case, on a resolved +projective Albanese fibration. + +\subsection{Context and relation to earlier work} + +In the minimal model program, abundance complements the construction of +minimal models: the nef adjoint on a minimal model should determine a +canonical fibration. For minimal projective threefolds, Miyaoka proved the +case of numerical dimension one and Kawamata completed canonical abundance +\cite{Miyaoka1988,Kawamata1992}. Keel, Matsuki, and McKernan established +log abundance for threefolds +\cite{KMM1994,KMM2004}. In arbitrary dimension, Birkar, Cascini, Hacon, and +McKernan established log terminal models for projective klt pairs with big +boundary and pseudo-effective adjoint, and finite generation for big +adjoints \cite{BCHM2010}. + +The compact Kähler history already points to the importance of simple +spaces. Peternell's threefold theorem isolated the possible case of a simple +space not bimeromorphic to a finite quotient of a torus +\cite{Peternell2001}; Demailly and Peternell subsequently proved canonical +nonvanishing for nef terminal Kähler threefolds, including that case +\cite{DemaillyPeternell2003}. Höring and Peternell constructed minimal models +for normal \(\Q\)-factorial terminal compact Kähler threefolds with pseudo-effective canonical class +\cite{HoeringPeternell2016}. Campana, Höring, and Peternell treated canonical abundance +\cite{CHP2016,CHP2023}, and Das and Ou +established log abundance for compact Kähler log canonical threefolds +\cite[Corollary~1.3]{DasOu2026}. + +With \(D_X=D_Y=0\), Assumption~\ref{asm:log-iitaka} is Iitaka's classical +\(C_{n,m}\) inequality +\(\kappa(X)\geq\kappa(F)+\kappa(Y)\) for algebraic fiber spaces +\cite[Equation~(1.0.1)]{CaoPaun2017}. Cao and Păun proved the analogous +logarithmic inequality over an abelian base when the pair on the total space +has an effective rational boundary and is klt +\cite[Theorem~1.1]{CaoPaun2017}. Fujino derives +logarithmic subadditivity from the conjectured equality +\(\kappa_\sigma(X,K_X+D)=\kappa(X,K_X+D)\) for smooth projective +\(X\) with reduced SNC \(D\) +\cite{Fujino2017Subadditivity,Fujino2020Corrigendum}. Hashizume proves the +reduced-SNC subadditivity statement when the log canonical divisor of the +very general fiber is abundant \cite[Theorem~1.2]{Hashizume2020}. Here the +implication is used in the other direction: the projective companion +\cite{LI} derives good models for pseudo-effective projective lc adjoints +from Assumption~\ref{asm:log-iitaka}, and those good models anchor the +Kähler induction. + +Hacon and Xie's cone theorem, adjunction, relative positivity, and +modified-big preparation \cite{HX} provide analytic inputs for our +program constructions. Section~\ref{sec:programs} constructs effective +generalized base data on a higher carrier, with an exact adjoint +pullback identity and a multiplier-ideal inclusion. Fibrewise +$L^2$ positivity and regularization give this presentation; its +Stein-factor consequence supplies the non-klt induction. +The section then proves the restricted semiampleness, proper relative +good-model, and finite-model statements needed here, following the +dimension-induction strategy of \cite{HX}. +The nef data remain globally nef on a fixed carrier, and the total +generalized boundary is globally modified big. Actual rational lines +make the selected ray contractions projective; ordinary projective +analytic results \cite{FujinoAnalyticBCHM} then construct their flips +and the relative models. This produces a Kähler pullback description of +the relevant Bott--Chern class. Generation of the prescribed holomorphic +line requires the additional divisorial decomposition and boundary +arguments. The fibration comparison and period positivity come from +\cite{CI}; the boundary and lifting constructions adapt inputs in +\cite{CB,BL}. The precise imported results are stated at their points +of use. + +\subsection{The proof and its main constructions} + +For a pseudo-effective \((1,1)\)-class \(\alpha\), \(N(\alpha)\) records +the least generic vanishing forced along prime divisors. Given a smooth +compact Kähler manifold \(X\) and a rational SNC boundary \(B\) with +\(J=K_X+B\) pseudo-effective, the induction constructs a modification +\(\mu:Y\to X\) with \(Y\) smooth and +\[ + \mu^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad + R=N(c_1(\mu^*J)), +\] +as an identity of rational holomorphic line bundles, with \(R\) an effective +rational divisor. Section~\ref{sec:inductive-reduction} uses transfer rules, +algebraic reduction, and generically finite covers to reduce the induction +to projective manifolds, nontrivial fibrations, and simple spaces of +algebraic dimension zero. Here simple means that no positive-dimensional +proper compact subvariety passes through a very general point. + +Section~\ref{sec:programs} supplies two program results. For the fibration +case it contracts a known negative part while preserving the prescribed nef +line. For the simple case it proves special termination for a chosen scaling +and reaches a nef model when the pseudo-effective adjoint is rationally +equivalent to a divisor supported on the reduced boundary, allowing negative +coefficients. The termination argument uses lower-dimensional induction to +construct one model on which an interval of perturbed adjoints is nef. On a +common resolution their negative multiplicities are affine, whereas every +nontrivial wall of the restricted program changes a slope. Hence only +finitely many such walls occur. + +Section~\ref{sec:boundary} turns generation on separate strata into generation +on the entire reduced dlt boundary, adapting Fujino's method of admissible +sections \cite[Section~4]{Fujino2000}. When a comparison is needed, a Mori +contraction realizes Kollár's residue comparison at the two coefficient-one +points of a general \(\PP^1\) fiber +\cite[Definition~13 and Proposition~14]{Kollar2013Sources}. On a common +resolution, the pulled-back restrictions of the ambient Kähler class differ +by a real linear combination of Chern classes of line bundles. On a stratum +whose boundary dominates the image of the map +defined by its semiample adjoint, restriction determines sections. For the +remaining strata, we prove finite image for self-comparisons on sufficiently +divisible pluricanonical systems using an invariant integral, period and +lattice arguments, and a uniform cohomological bound for cyclic covers. +Products over these finite images give compatible generating sections, used +in both closing arguments. + +Section~\ref{sec:rank-one} first reduces to fibrations whose very general +fiber has log Kodaira dimension zero. On a prepared fibration \(g:X\to W\), +a relative generating form gives +\[ + K_X+B\sim_{\Q}g^*H+A_*,\qquad H=K_W+T+M, +\] +where \(T\) is a rational SNC boundary and \(M\) is pulled back from a nef +rational line on a projective quotient of \(W\). Fiber induction allows us +to subtract the part of \(A_*\) dominating \(W\) from a positive current for +\(K_X+B\). The resulting metric descends along connected smooth fibers; a +zero among the nonnegative coefficients at primes dominating each base prime +allows extension, proving \(H\) pseudo-effective. If \(a(W)=0\), the +projective quotient is a point, so \(M\sim_{\Q}0\) and lower-dimensional +induction makes \(H\) rationally equivalent to its negative divisor. For +projective \(W\), a reduction via a chosen generalized program derived from +Assumption~\ref{asm:log-iitaka} either lowers the positive base dimension, +handled by secondary induction, or reaches a nef line that is big or torsion. +At a stopping case, an intersection argument identifies the full negative +divisor upstairs. A torsion positive part completes the decomposition +directly. In the big nef case, Section~\ref{sec:programs} contracts that +divisor while preserving the nef line; boundary generation, extension, and +a new log canonical place at a hypothetical base locus prove semiampleness. + +Section~\ref{sec:nonvanishing} proves, independently of +Assumption~\ref{asm:log-iitaka}, that a positive canonical power has a nonzero +meromorphic section on every smooth connected simple compact Kähler manifold +of algebraic dimension zero. Ou's uniruledness and foliation results supply +the cotangent slope control \cite{Ou2025}. A point-threshold bound for big +classes of volume one is contradicted using an auxiliary projective bundle +over \(X\times X\). The diagonal in its square produces a subsheaf occupying +a fixed positive fraction of a symmetric cotangent power of that bundle. +A second construction over the diagonal of \(X\times X\) forces determinant +vanishing which, after restriction to a blowup of \(X\), violates the point +bound. + +Finally, Section~\ref{sec:signed} uses the meromorphic section to obtain, +after resolving and enlarging to a reduced SNC boundary, a pseudo-effective +adjoint with a signed representative supported there. +Section~\ref{sec:programs} gives a nef dlt model, and +Section~\ref{sec:boundary} gives generation on its reduced boundary. +Pseudo-effectivity of the canonical pullback and an intersection argument +isolate positive-dimensional boundary fibers away from components whose +signed coefficients are nonpositive. A local root construction adapted from +\cite[Proposition~3.3]{LI} separates the positive and negative divisor +supports. An SNC Hodge-module calculation extends the lifting method of +\cite[Sections~10--12]{BL} to residual poles and kills every finite-order +obstruction to lifting such a fiber. Douady space and Artin approximation +give compact deformations leaving the boundary; compactness of the cycle-space +components and relative compactness of bounded-volume cycles allow a Baire +argument to produce a covering family. Simplicity +forces the actual nef line to be torsion. The program comparison and +Lemma~\ref{bir:unique-current} then permit subtraction of the added boundary, +yielding the inductive decomposition. diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/02-inductive-reduction.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/02-inductive-reduction.tex index dcedcb1d8326acfc5b8d5a76aa18a287ae928f74..16c9097c10576439dadc9d5f6ff06d682ae6b9fd 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/02-inductive-reduction.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/02-inductive-reduction.tex @@ -1,586 +1,586 @@ -\section{Birational decompositions and the geometric reduction} -\label{sec:inductive-reduction} - -The proof will produce a semiample line bundle on a smooth model, together -with its entire fixed divisorial part. This section makes that statement -precise and reduces the induction to two cases: spaces admitting a -nontrivial fibration, and simple spaces of algebraic dimension zero. - -\subsection{The inductive decomposition and its negative part} - -We use additive notation for rational holomorphic line bundles. Thus -\(L\sim_{\Q}L'\) means that \(mL\) and \(mL'\) are holomorphically -isomorphic for some positive integer \(m\). If a rational divisor occurs in such -an identity, it denotes its associated rational line bundle. In contrast, -\(c_1(L)\) and \(\{D\}=c_1(\OO_X(D))\) denote real Bott--Chern classes. -Canonical comparisons are made with the usual local meromorphic canonical -identifications. In particular, an identity of lines below contains more -information than equality of their Chern classes. - -Let \(X\) be a smooth compact Kähler manifold and let \(\alpha\) be a -pseudo-effective real \((1,1)\)-class. Fix a Kähler form \(\omega\). For a -prime divisor \(D\), its minimal multiplicity is -\[ - \nu_D(\alpha)=\lim_{\varepsilon\downarrow0} - \inf\bigl\{\nu_D(T):T\in\alpha,\ T\geq-\varepsilon\omega\bigr\}. -\] -Here \(T\) ranges over closed currents, and \(\nu_D(T)\) is its generic -Lelong number; one may equivalently use currents with analytic singularities. -The limit is independent of \(\omega\). Boucksom's divisorial decomposition -is -\[ - N(\alpha)=\sum_D\nu_D(\alpha)D, - \qquad Z(\alpha)=\alpha-\{N(\alpha)\}. -\] -The sum is a finite effective real divisor, and \(Z(\alpha)\) is modified -nef: its minimal multiplicity at every prime is zero. We write \(N(L)\) and -\(\nu_D(L)\) for \(N(c_1(L))\) and \(\nu_D(c_1(L))\). These are the -analytic notions, with small Kähler -perturbations, throughout the paper. We use the foundational results in -\cite[Sections~2--3 and~5]{Boucksom2004}. - -The induction asks for a decomposition that retains both the actual line -bundle and this analytic negative divisor. - -\begin{definition}\label{def:good-decomposition} -For \(n\geq0\), let \(\Ggood_n\) be the following assertion. If \(X\) -is a connected smooth compact Kähler manifold of dimension \(n\), \(B\) -is a rational simple normal crossing boundary with coefficients in -\([0,1]\), and \(J=K_X+B\) is pseudo-effective, then there is a smooth -compact Kähler modification \(\mu:Y\to X\) and an actual rational line -identity -\begin{equation}\label{bir:good-decomposition} - \mu^*J\sim_{\Q}P+R, - \qquad P\text{ semiample},\qquad R=N(\mu^*J)\geq0, -\end{equation} -where \(R\) is a rational divisor. -\end{definition} - -The next lemmas make this decomposition stable under further resolutions -and allow exceptional resolution errors to be removed. They also provide -the fixed-section statement needed to descend the case in which the -positive part is torsion. - -\begin{lemma}[Negative-part calculus]\label{bir:calculus} -Let \(\alpha\) be pseudo-effective on a smooth compact Kähler manifold. -\begin{enumerate} -\item Every positive current in \(\alpha\) contains the divisorial current -\([N(\alpha)]\). If \(0\leq F\leq N(\alpha)\), then -\[ - \alpha-\{F\}\text{ is pseudo-effective},\qquad - N(\alpha-\{F\})=N(\alpha)-F. -\] -\item Minimal multiplicities are homogeneous and subadditive. In -particular, if \(\beta\) is nef, then -\(N(\alpha+\beta)\leq N(\alpha)\). -\item If \(\mu:Y\to X\) is a smooth modification and \(D'\) is the strict -transform of a prime \(D\), then -\[ - \nu_{D'}(\mu^*\alpha)=\nu_D(\alpha). -\] -\item If \(\gamma\) is modified nef and \(j:\widetilde D\to X\) is a -resolution of a prime divisor, then \(j^*\gamma\) is pseudo-effective. -\end{enumerate} -\end{lemma} - -\begin{proof} -The first assertion about currents follows from the definition of minimal -multiplicity and Siu decomposition. Homogeneity and subadditivity follow by -scaling and adding testing currents. The subtraction identity is the -corresponding property in the big cone, followed by a small Kähler -perturbation. More explicitly, for \(\alpha_\varepsilon=\alpha+ -\varepsilon\{\omega\}\), subtract -\(F_\varepsilon=\min\{F,N(\alpha_\varepsilon)\}\) coefficientwise. -In the big cone all positive currents contain this divisor, so subtracting -it translates each minimal multiplicity by its coefficient. As -\(\varepsilon\downarrow0\), \(F_\varepsilon\to F\). The class of -\(F-F_\varepsilon\) can be absorbed in a Kähler perturbation tending to -zero (there are only finitely many components). This gives the upper bound -for the asserted identity; subadditivity applied after adding \(F\) gives -the reverse bound. Weak compactness of positive currents gives -pseudo-effectivity of the limit. This is also the subtraction property of -the divisorial decomposition in \cite[Section~3]{Boucksom2004}. - -For the strict-transform formula, pull almost-positive testing currents -to \(Y\). Near the generic point of \(D'\), the map is an isomorphism, -so the generic order is unchanged. This proves one inequality. Conversely, -push a testing current on \(Y\) to \(X\). To control its negative error, -write the error as a small multiple of the fixed positive current -\(\mu_*\omega_Y\), and add a fixed smooth Kähler representative of -\(C\omega_X-\{\mu_*\omega_Y\}\) for \(C\) sufficiently large. The -result is a positive test in a Kähler perturbation tending to zero. The -pushforward of \(\omega_Y\) has no divisorial order at the generic point -of \(D\), where \(\mu\) is an isomorphism. Thus this test has the same -generic order as the original one at \(D'\). Regularization, if needed, -returns to tests with analytic singularities. This proves the other -inequality and the formula. It is the analytic form of the strict-transform -comparison in \cite[Lemma~5.8]{Boucksom2004}. - -Finally, choose analytic-singularity tests for a modified-nef class whose -generic order at the chosen prime tends to zero. Subtract that generic -divisorial order before restricting to a resolution of the prime. The -remaining singular set does not contain its generic point, so restriction -gives an almost-positive current there. The subtracted multiple and the -Kähler error tend to zero. Taking the limit proves the last assertion; see -also \cite[Propositions~2.4 and~3.8]{Boucksom2004}. -\end{proof} - -For a rational line \(J\) on a normal compact Kähler space \(X\) whose pullback -to a smooth resolution is pseudo-effective, we also use minimal -multiplicities for divisorial places. If \(Q\) appears on a smooth -resolution \(p:Y\to X\), set -\[ - \nu_Q(J):=\nu_Q(p^*J). -\] -The strict-transform formula on a common smooth refinement makes this -independent of the chosen resolution. When a real class is already on a -fixed smooth model, \(\nu_Q\) continues to denote its coefficient in the -negative part on that model. - -\begin{lemma}[Exceptional translation]\label{bir:exceptional} -Let \(p:Y\to X\) be a proper bimeromorphic morphism from a smooth compact -Kähler manifold to a normal compact Kähler space. Let \(M\) be a real -\((1,1)\)-class on \(X\) represented by smooth local potentials, and let -\(E\geq0\) be a real \(p\)-exceptional divisor. If -\(\alpha=p^*M+\{E\}\) is pseudo-effective, then \(p^*M\) is -pseudo-effective and -\[ - N(\alpha)=N(p^*M)+E. -\] -\end{lemma} - -\begin{proof} -It suffices by Lemma~\ref{bir:calculus} to prove \(E\leq N(\alpha)\). -Write -\[ - E-N(\alpha)=U-V, - \qquad U,V\geq0, -\] -with no common component, and suppose \(U\neq0\). Its components are -\(p\)-exceptional. Put \(n=\dim Y\) and -\(\ell=\max\{\dim p(U_i):U_i\text{ a component of }U\}\). Normality -gives \(\ell\leq n-2\). Let \(\eta\) be the pullback of a Kähler form -on \(X\), let \(\omega\) be Kähler on \(Y\), and set -\[ - q(\gamma,\delta)= - \int_Y\gamma\,\delta\,\eta^{\ell}\omega^{n-\ell-2}. -\] -The restriction property in Lemma~\ref{bir:calculus} gives -\(q(Z(\alpha),U)\geq0\). The term \(q(p^*M,U)\) is zero: on each -component of \(U\), it contains \(\ell+1\) factors pulled back from an -image of dimension at most \(\ell\). Distinct effective divisors have -nonnegative intersection against the semipositive and Kähler factors in -\(q\). Hence -\[ - 0\leq q(Z(\alpha),U)=q(U,U)-q(V,U)\leq q(U,U). -\] -On the other hand, \(q(U,\eta)=0\) by the same dimension count, whereas -\(q(\eta,\eta)>0\). The mixed Hodge index theorem, applied first with -Kähler factors and then by a semipositive limit, says that \(q\) is -negative semidefinite on \(\eta^\perp\). It follows that \(q(U,U)=0\) -and that \(U\) is in the radical of \(q\): the radical assertion follows -from negative semidefiniteness on \(\eta^\perp\), and then from -\(q(U,\eta)=0\). But -\[ - q(U,\omega)=\sum_i u_i\int_{U_i} - \eta^{\ell}\omega^{n-\ell-1}>0. -\] -Indeed, a component with image dimension \(\ell\) has strictly positive -integrand on a dense open. This is a contradiction. The mixed Hodge index -statement used here is the mixed Hodge--Riemann relation -\cite[Theorems~A and~C]{DinhNguyen}; the limit preserves the assertion that -there is at most one positive direction. -\end{proof} - -\begin{lemma}[Pulling up a known decomposition]\label{bir:pullup} -Let \(\alpha\) be a pseudo-effective real \((1,1)\)-class on a smooth compact -Kähler manifold, and suppose -\[ - \alpha=\beta+\{R\},\qquad \beta\text{ nef},\qquad R=N(\alpha)\geq0. -\] -For every smooth modification \(\mu:Y\to X\), -\[ - N(\mu^*\alpha)=\mu^*R. -\] -In particular, this applies to the Chern classes of an actual rational line -decomposition with a nef rational positive part. -\end{lemma} - -\begin{proof} -Subadditivity and nefness give \(N(\mu^*\alpha)\leq\mu^*R\). By the -strict-transform formula, the difference -\(U=\mu^*R-N(\mu^*\alpha)\) is effective and exceptional. Moreover -\(Z(\mu^*\alpha)=\mu^*\beta+\{U\}\) is modified nef. Apply -Lemma~\ref{bir:exceptional} to this class. It gives -\(U\leq N(Z(\mu^*\alpha))=0\), proving the assertion. -\end{proof} - -\begin{lemma}[Fixed sections]\label{bir:sections} -Let \(L\) be a pseudo-effective rational line on a smooth compact Kähler -manifold. Every section \(s\) of an integral multiple \(mL\) satisfies -\[ - \ord_D(s)\geq m\nu_D(L) -\] -at every prime \(D\). If \(L\sim_{\Q}P+R\), with \(P\) semiample and -\(R=N(L)\) rational, then in every sufficiently divisible degree -multiplication by the canonical section of \(mR\) identifies -\(H^0(X,mP)\) with \(H^0(X,mL)\). -\end{lemma} - -\begin{proof} -The divisor current \([\operatorname{div}(s)]/m\) is a positive current -in \(c_1(L)\), so Lemma~\ref{bir:calculus} gives the order bound. In the -stated decomposition, every section therefore divides holomorphically by -the canonical section of \(mR\). Conversely, multiplication gives a -section of \(mL\). The actual rational line identity makes these operations -inverse after clearing denominators. -\end{proof} - -\begin{lemma}[Uniqueness when the positive part is torsion] -\label{bir:unique-current} -Suppose \(L\sim_{\Q}R\), where \(R=N(L)\) is an effective rational -divisor on a smooth compact Kähler manifold. Then the only positive current -in \(c_1(L)\) is \([R]\). If \(F\geq0\) is a rational divisor and -\(L-F\) is pseudo-effective, then -\[ - F\leq R,\qquad L-F\sim_{\Q}R-F=N(L-F). -\] -The same conclusions hold if \(L\sim_{\Q}P+R\) with \(P\) torsion. -\end{lemma} - -\begin{proof} -Write \(X\) for the manifold. The assertion is immediate if \(\dim X=0\), -so assume \(\dim X>0\). -Every positive current \(T\) in \(c_1(L)\) contains \([R]\). The -residual \(T-[R]\) is positive with zero cohomology class, so its mass -against a Kähler form to the power \(\dim X-1\) is zero. Thus it vanishes. -For the second assertion, add \([F]\) to any positive current in -\(c_1(L-F)\). Uniqueness gives \(F\leq R\), and subtraction in -Lemma~\ref{bir:calculus} gives the asserted negative part. A torsion line -has zero Chern class and becomes trivial after taking a positive multiple, -so the last statement is the same argument. -\end{proof} - -\subsection{Birational transfer of the decomposition} - -All modifications resolving spaces, maps, or ideals can be chosen -projective. A projective modification of a compact Kähler space is Kähler; -graphs between compact Kähler models can likewise be resolved by smooth -compact Kähler manifolds. We use these standard analytic resolution and -flattening results without further mention. - -For log discrepancies our convention is -\[ - a(E;X,\Delta)=1+\ord_E\bigl(K_Y-p^*(K_X+\Delta)\bigr) -\] -on a smooth model \(p:Y\to X\). In particular, on a log resolution of an -lc pair, giving each new exceptional divisor coefficient one produces an -effective exceptional error. The next proposition explains why this -convention is harmless. - -\begin{proposition}[Resolution and descent]\label{bir:resolution-transfer} -Let \((X,\Delta)\) be a normal compact Kähler lc pair with rational -boundary and rational Cartier adjoint \(J\). Let \(p:Y\to X\) be a log -resolution, and put -\[ - B_Y=p_*^{-1}\Delta+\sum_{E\text{ exceptional}}E. -\] -If \(p^*J\) is pseudo-effective and the conclusion of -Definition~\ref{def:good-decomposition} holds for \((Y,B_Y)\), then there -is a smooth compact Kähler modification \(r:V\to X\) with an actual -rational line identity -\[ - r^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad - R=N(r^*J)\geq0, -\] -where \(R\) is a rational divisor. -If \(J\) is analytically nef, it is semiample on \(X\). -\end{proposition} - -\begin{proof} -With compatible canonical choices there is an actual rational divisor -identity -\[ - K_Y+B_Y=p^*J+E_p,\qquad - E_p=\sum_{E\text{ exceptional}}a(E;X,\Delta)E\geq0. -\] -Suppose \(q:V\to Y\) realizes the decomposition for the left side. -Lemma~\ref{bir:exceptional}, applied over \(X\), gives -\[ - N\bigl(q^*(K_Y+B_Y)\bigr) - =N(q^*p^*J)+q^*E_p. -\] -Cancelling \(q^*E_p\) in the actual rational line identities proves the -first assertion. This argument also shows that further resolutions using -the reduced-exceptional convention do not change the assertion -\(\Ggood_n\). - -If \(J\) is nef, its pullback is nef, so its negative part is zero. The -semiample line in the decomposition is therefore \(q^*p^*J\) itself, up -to rational line isomorphism. Choose an actual Cartier multiple generated -on \(V\). Normality gives \((p q)_*\OO_V=\OO_X\), and projection formula -identifies its sections with the sections downstairs. A base point -downstairs would make every pullback section vanish on its nonempty fiber. -There is no such point, so that multiple of \(J\) is globally generated. -\end{proof} - -The second transfer concerns a torsion positive part. Its proof uses the -following local construction for finite maps, which will also be used for -meromorphic sections and for covers of spaces of algebraic dimension zero. - -\begin{lemma}[Finite analytic norms and characteristic polynomials] -\label{bir:finite-norm} -Let \(\nu:Z\to X\) be a finite surjective morphism of normal irreducible -complex spaces, of degree \(d>0\), and let \(L\) be a holomorphic line -bundle on \(X\). A nonzero meromorphic section \(s\) of \(\nu^*L\) -has a nonzero meromorphic norm \(\operatorname{Nm}_\nu(s)\) in -\(L^{\otimes d}\), with -\[ - \operatorname{div}(\operatorname{Nm}_\nu(s)) - =\nu_*\operatorname{div}(s). -\] -If \(s\) is holomorphic, its norm is holomorphic. If, in addition, -\(s\) is nonzero at every point of \(\nu^{-1}(x)\), then its norm is -nonzero at \(x\). - -For a global meromorphic function \(f\) on \(Z\), there is a monic -polynomial \(\chi_f(T)\) of degree \(d\) with global meromorphic -coefficient functions on \(X\) such that \(\chi_f(f)=0\) after pulling -the coefficients to \(Z\). -\end{lemma} - -\begin{proof} -At \(x\in X\), choose a holomorphic frame of \(L\) and put -\[ - R=\OO_{X,x},\qquad K=\operatorname{Frac}(R),\qquad - A=(\nu_*\OO_Z)_x. -\] -The generic algebra \(A\otimes_RK\) is a product of finite field -extensions of \(K\), of total dimension \(d\). The local meromorphic -coefficient \(a\) of \(s\) belongs to this algebra. Since \(Z\) is -irreducible and \(s\) is not identically zero, its restriction is nonzero -on every generic local component. Thus \(a\) is nonzero in every field -factor, and the determinant of multiplication by \(a\) is nonzero. -Under a change of frame by a unit \(g\), this determinant changes by -\(g^{-d}\). The determinants therefore glue to the asserted -meromorphic section. The valuation formula for a norm over a discrete -valuation ring gives the displayed divisor identity at every prime of -the normal space \(X\). - -If \(s\) is holomorphic, then \(a\in A\) is integral over \(R\). -Its norm is integral over \(R\) and belongs to \(K\), hence belongs to -\(R\) by normality. This does not require \(A\) to be locally free. -If \(s\) is nonzero at every point over \(x\), then \(a\) is a unit -in the finite semilocal algebra \(A\); applying the same argument to -\(a^{-1}\) shows that its norm is a unit in \(R\). - -For a meromorphic function \(f\), use instead the determinant of -\(T\) minus multiplication by its local coefficient. These monic -polynomials agree on overlaps, and Cayley--Hamilton shows that they -annihilate \(f\). All these constructions use fractions of local analytic -germs, so they remain available when the global meromorphic functions on -\(X\) are constant. -\end{proof} - -We now descend a decomposition whose positive part is torsion, the form -needed for a covering family on a space of algebraic dimension zero. - -\begin{proposition}[Descent of a purely negative decomposition] -\label{bir:finite-descent} -Let \(e:Y\to X\) be a proper generically finite surjective morphism -between connected smooth compact Kähler manifolds. Let \(L\) be a -pseudo-effective rational line on \(X\), and let \(F\geq0\) be a -rational divisor on \(Y\). Suppose that on a smooth model over \(Y\), -the line \(e^*L+F\) is rationally linearly equivalent to its rational -negative part. Then there is an effective rational divisor \(D\) on -\(X\) such that -\[ - L\sim_{\Q}D=N(L). -\] -\end{proposition} - -\begin{proof} -Replace \(Y\) by the smooth model in the hypothesis and pull \(F\) back. -Write \(e^*L+F\sim_{\Q}R=N(e^*L+F)\). Pull back a positive current in -\(c_1(L)\) and add \([F]\). Lemma~\ref{bir:unique-current} gives -\(F\leq R\). Put \(D_Y=R-F\). Subtraction in -Lemma~\ref{bir:calculus} gives -\[ - e^*L\sim_{\Q}D_Y=N(e^*L)\geq0. -\] -In particular \([D_Y]\) is the unique positive current in \(c_1(e^*L)\). - -Choose \(m\) so that \(mL\) and \(mD_Y\) are integral and the latter is -the divisor of a holomorphic section \(s\) of \(e^*(mL)\). Factor \(e\) -through its normal Stein space: -\[ - Y\xrightarrow{h}Z\xrightarrow{\nu}X, - \qquad \nu\text{ finite of degree }d=\deg e. -\] -The connected birational map \(h\) descends \(s\) to a holomorphic -section \(s_Z\) of \(\nu^*(mL)\), by normality and projection formula. -Lemma~\ref{bir:finite-norm} gives a holomorphic section of \(mdL\) whose -divisor is -\[ - \operatorname{div}(\operatorname{Nm}_\nu(s_Z)) - =\nu_*\operatorname{div}(s_Z)=m e_*D_Y. -\] -Thus \(D=d^{-1}e_*D_Y\) is effective and satisfies \(L\sim_{\Q}D\). - -Both \(e^*D\) and \(D_Y\) are positive divisor currents in -\(c_1(e^*L)\), so uniqueness gives \(e^*D=D_Y\). We already know -\(N(L)\leq D\). If this inequality were strict at a prime \(A\subset X\), -choose analytic-singularity tests for \(L\) whose generic orders at \(A\) -tend to \(\nu_A(L)\), and pull them to \(Y\). For any prime \(A'\) -dominating \(A\), the generic orders are multiplied by -\(\operatorname{mult}_{A'}e^*A\). The pulled-back Kähler errors can be -enlarged to Kähler errors upstairs tending to zero. Consequently -\[ - \nu_{A'}(e^*L) - \leq \operatorname{mult}_{A'}(e^*A)\,\nu_A(L) - <\operatorname{mult}_{A'}(e^*A)\,\operatorname{coeff}_A(D), -\] -contrary to \(N(e^*L)=D_Y=e^*D\). This proves \(D=N(L)\). -\end{proof} - -We will also use Proposition~\ref{bir:finite-descent} when \(e\) is a -modification and \(F=0\). Once the positive part of a decomposition is -known to be torsion, it gives \(L\sim_{\Q}N(L)\) on the original smooth -space; Lemma~\ref{bir:pullup} then identifies the negative divisor upstairs -with the pullback of \(N(L)\). - -\subsection{The projective anchor and the two geometric cases} - -\begin{proposition}[Projective good models]\label{proj:good-models} -Under Assumption~\ref{asm:log-iitaka}, every projective lc pair over -\(\C\) with rational boundary and pseudo-effective rational Cartier -adjoint has a good log minimal model. In particular, \(\Ggood_n\) holds -for projective manifolds in every dimension, and every nef rational lc -adjoint on a projective variety is semiample as an actual rational line. -\end{proposition} - -\begin{proof} -The good-model assertion is the rational-boundary consequence of the full -argument in \cite[Proposition~2.5, Theorem~9.6, and Section~10]{LI}. That -argument establishes the additional inductive and nonvanishing premises of -its Proposition~2.5 before applying it, and proves a stronger real-boundary -induction. Its only use of Assumption~\ref{asm:log-iitaka} is in its -Lemma~6.1, on a resolved projective Albanese fibration with both boundaries -zero. Thus its premise is precisely available here. - -For clarity, its good-model conclusion gives on a common smooth projective -resolution -\[ - u^*(K_X+B)=v^*(K_{X_m}+B_m)+F, - \qquad F\geq0\text{ and }v\text{-exceptional}, -\] -as an actual rational divisor identity; the line on \(X_m\) is semiample. -This is the comparison in \cite[Lemma~2.2]{LI}. Algebraic nefness of a -rational line on a projective variety implies analytic nefness, by adding -arbitrarily small ample rational classes. Lemma~\ref{bir:exceptional} -therefore identifies \(F\) with the analytic negative part of the left -side. On a smooth projective variety, analytic pseudo-effectivity of a -divisor class agrees with algebraic pseudo-effectivity -\cite[Proposition~1.2]{BDPP}; hence the good-model assertion applies to -every projective instance of \(\Ggood_n\). This proves the stated form of -\(\Ggood_n\). If the original adjoint -is nef, the same comparison and normal descent give its semiampleness. -\end{proof} - -The remainder of the paper proves the following two propositions. They -are stated here so that the global induction can be completed before its -technical components are developed. A compact space is \emph{simple} if -no positive-dimensional proper compact analytic subvariety passes through -a very general point. We write \(a(X)\) for its algebraic dimension. - -\begin{proposition}[The fibration case]\label{rank:fibration-case} -Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume -\(\Ggood_j\) for \(j0\) and assume \(\Ggood_j\) for \(j0\). - -If \(a(X)=n\), the Kähler Moishezon theorem makes \(X\) projective, and -Proposition~\ref{proj:good-models} applies. If \(00\). The mixed Hodge index theorem, applied first with +Kähler factors and then by a semipositive limit, says that \(q\) is +negative semidefinite on \(\eta^\perp\). It follows that \(q(U,U)=0\) +and that \(U\) is in the radical of \(q\): the radical assertion follows +from negative semidefiniteness on \(\eta^\perp\), and then from +\(q(U,\eta)=0\). But +\[ + q(U,\omega)=\sum_i u_i\int_{U_i} + \eta^{\ell}\omega^{n-\ell-1}>0. +\] +Indeed, a component with image dimension \(\ell\) has strictly positive +integrand on a dense open. This is a contradiction. The mixed Hodge index +statement used here is the mixed Hodge--Riemann relation +\cite[Theorems~A and~C]{DinhNguyen}; the limit preserves the assertion that +there is at most one positive direction. +\end{proof} + +\begin{lemma}[Pulling up a known decomposition]\label{bir:pullup} +Let \(\alpha\) be a pseudo-effective real \((1,1)\)-class on a smooth compact +Kähler manifold, and suppose +\[ + \alpha=\beta+\{R\},\qquad \beta\text{ nef},\qquad R=N(\alpha)\geq0. +\] +For every smooth modification \(\mu:Y\to X\), +\[ + N(\mu^*\alpha)=\mu^*R. +\] +In particular, this applies to the Chern classes of an actual rational line +decomposition with a nef rational positive part. +\end{lemma} + +\begin{proof} +Subadditivity and nefness give \(N(\mu^*\alpha)\leq\mu^*R\). By the +strict-transform formula, the difference +\(U=\mu^*R-N(\mu^*\alpha)\) is effective and exceptional. Moreover +\(Z(\mu^*\alpha)=\mu^*\beta+\{U\}\) is modified nef. Apply +Lemma~\ref{bir:exceptional} to this class. It gives +\(U\leq N(Z(\mu^*\alpha))=0\), proving the assertion. +\end{proof} + +\begin{lemma}[Fixed sections]\label{bir:sections} +Let \(L\) be a pseudo-effective rational line on a smooth compact Kähler +manifold. Every section \(s\) of an integral multiple \(mL\) satisfies +\[ + \ord_D(s)\geq m\nu_D(L) +\] +at every prime \(D\). If \(L\sim_{\Q}P+R\), with \(P\) semiample and +\(R=N(L)\) rational, then in every sufficiently divisible degree +multiplication by the canonical section of \(mR\) identifies +\(H^0(X,mP)\) with \(H^0(X,mL)\). +\end{lemma} + +\begin{proof} +The divisor current \([\operatorname{div}(s)]/m\) is a positive current +in \(c_1(L)\), so Lemma~\ref{bir:calculus} gives the order bound. In the +stated decomposition, every section therefore divides holomorphically by +the canonical section of \(mR\). Conversely, multiplication gives a +section of \(mL\). The actual rational line identity makes these operations +inverse after clearing denominators. +\end{proof} + +\begin{lemma}[Uniqueness when the positive part is torsion] +\label{bir:unique-current} +Suppose \(L\sim_{\Q}R\), where \(R=N(L)\) is an effective rational +divisor on a smooth compact Kähler manifold. Then the only positive current +in \(c_1(L)\) is \([R]\). If \(F\geq0\) is a rational divisor and +\(L-F\) is pseudo-effective, then +\[ + F\leq R,\qquad L-F\sim_{\Q}R-F=N(L-F). +\] +The same conclusions hold if \(L\sim_{\Q}P+R\) with \(P\) torsion. +\end{lemma} + +\begin{proof} +Write \(X\) for the manifold. The assertion is immediate if \(\dim X=0\), +so assume \(\dim X>0\). +Every positive current \(T\) in \(c_1(L)\) contains \([R]\). The +residual \(T-[R]\) is positive with zero cohomology class, so its mass +against a Kähler form to the power \(\dim X-1\) is zero. Thus it vanishes. +For the second assertion, add \([F]\) to any positive current in +\(c_1(L-F)\). Uniqueness gives \(F\leq R\), and subtraction in +Lemma~\ref{bir:calculus} gives the asserted negative part. A torsion line +has zero Chern class and becomes trivial after taking a positive multiple, +so the last statement is the same argument. +\end{proof} + +\subsection{Birational transfer of the decomposition} + +All modifications resolving spaces, maps, or ideals can be chosen +projective. A projective modification of a compact Kähler space is Kähler; +graphs between compact Kähler models can likewise be resolved by smooth +compact Kähler manifolds. We use these standard analytic resolution and +flattening results without further mention. + +For log discrepancies our convention is +\[ + a(E;X,\Delta)=1+\ord_E\bigl(K_Y-p^*(K_X+\Delta)\bigr) +\] +on a smooth model \(p:Y\to X\). In particular, on a log resolution of an +lc pair, giving each new exceptional divisor coefficient one produces an +effective exceptional error. The next proposition explains why this +convention is harmless. + +\begin{proposition}[Resolution and descent]\label{bir:resolution-transfer} +Let \((X,\Delta)\) be a normal compact Kähler lc pair with rational +boundary and rational Cartier adjoint \(J\). Let \(p:Y\to X\) be a log +resolution, and put +\[ + B_Y=p_*^{-1}\Delta+\sum_{E\text{ exceptional}}E. +\] +If \(p^*J\) is pseudo-effective and the conclusion of +Definition~\ref{def:good-decomposition} holds for \((Y,B_Y)\), then there +is a smooth compact Kähler modification \(r:V\to X\) with an actual +rational line identity +\[ + r^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad + R=N(r^*J)\geq0, +\] +where \(R\) is a rational divisor. +If \(J\) is analytically nef, it is semiample on \(X\). +\end{proposition} + +\begin{proof} +With compatible canonical choices there is an actual rational divisor +identity +\[ + K_Y+B_Y=p^*J+E_p,\qquad + E_p=\sum_{E\text{ exceptional}}a(E;X,\Delta)E\geq0. +\] +Suppose \(q:V\to Y\) realizes the decomposition for the left side. +Lemma~\ref{bir:exceptional}, applied over \(X\), gives +\[ + N\bigl(q^*(K_Y+B_Y)\bigr) + =N(q^*p^*J)+q^*E_p. +\] +Cancelling \(q^*E_p\) in the actual rational line identities proves the +first assertion. This argument also shows that further resolutions using +the reduced-exceptional convention do not change the assertion +\(\Ggood_n\). + +If \(J\) is nef, its pullback is nef, so its negative part is zero. The +semiample line in the decomposition is therefore \(q^*p^*J\) itself, up +to rational line isomorphism. Choose an actual Cartier multiple generated +on \(V\). Normality gives \((p q)_*\OO_V=\OO_X\), and projection formula +identifies its sections with the sections downstairs. A base point +downstairs would make every pullback section vanish on its nonempty fiber. +There is no such point, so that multiple of \(J\) is globally generated. +\end{proof} + +The second transfer concerns a torsion positive part. Its proof uses the +following local construction for finite maps, which will also be used for +meromorphic sections and for covers of spaces of algebraic dimension zero. + +\begin{lemma}[Finite analytic norms and characteristic polynomials] +\label{bir:finite-norm} +Let \(\nu:Z\to X\) be a finite surjective morphism of normal irreducible +complex spaces, of degree \(d>0\), and let \(L\) be a holomorphic line +bundle on \(X\). A nonzero meromorphic section \(s\) of \(\nu^*L\) +has a nonzero meromorphic norm \(\operatorname{Nm}_\nu(s)\) in +\(L^{\otimes d}\), with +\[ + \operatorname{div}(\operatorname{Nm}_\nu(s)) + =\nu_*\operatorname{div}(s). +\] +If \(s\) is holomorphic, its norm is holomorphic. If, in addition, +\(s\) is nonzero at every point of \(\nu^{-1}(x)\), then its norm is +nonzero at \(x\). + +For a global meromorphic function \(f\) on \(Z\), there is a monic +polynomial \(\chi_f(T)\) of degree \(d\) with global meromorphic +coefficient functions on \(X\) such that \(\chi_f(f)=0\) after pulling +the coefficients to \(Z\). +\end{lemma} + +\begin{proof} +At \(x\in X\), choose a holomorphic frame of \(L\) and put +\[ + R=\OO_{X,x},\qquad K=\operatorname{Frac}(R),\qquad + A=(\nu_*\OO_Z)_x. +\] +The generic algebra \(A\otimes_RK\) is a product of finite field +extensions of \(K\), of total dimension \(d\). The local meromorphic +coefficient \(a\) of \(s\) belongs to this algebra. Since \(Z\) is +irreducible and \(s\) is not identically zero, its restriction is nonzero +on every generic local component. Thus \(a\) is nonzero in every field +factor, and the determinant of multiplication by \(a\) is nonzero. +Under a change of frame by a unit \(g\), this determinant changes by +\(g^{-d}\). The determinants therefore glue to the asserted +meromorphic section. The valuation formula for a norm over a discrete +valuation ring gives the displayed divisor identity at every prime of +the normal space \(X\). + +If \(s\) is holomorphic, then \(a\in A\) is integral over \(R\). +Its norm is integral over \(R\) and belongs to \(K\), hence belongs to +\(R\) by normality. This does not require \(A\) to be locally free. +If \(s\) is nonzero at every point over \(x\), then \(a\) is a unit +in the finite semilocal algebra \(A\); applying the same argument to +\(a^{-1}\) shows that its norm is a unit in \(R\). + +For a meromorphic function \(f\), use instead the determinant of +\(T\) minus multiplication by its local coefficient. These monic +polynomials agree on overlaps, and Cayley--Hamilton shows that they +annihilate \(f\). All these constructions use fractions of local analytic +germs, so they remain available when the global meromorphic functions on +\(X\) are constant. +\end{proof} + +We now descend a decomposition whose positive part is torsion, the form +needed for a covering family on a space of algebraic dimension zero. + +\begin{proposition}[Descent of a purely negative decomposition] +\label{bir:finite-descent} +Let \(e:Y\to X\) be a proper generically finite surjective morphism +between connected smooth compact Kähler manifolds. Let \(L\) be a +pseudo-effective rational line on \(X\), and let \(F\geq0\) be a +rational divisor on \(Y\). Suppose that on a smooth model over \(Y\), +the line \(e^*L+F\) is rationally linearly equivalent to its rational +negative part. Then there is an effective rational divisor \(D\) on +\(X\) such that +\[ + L\sim_{\Q}D=N(L). +\] +\end{proposition} + +\begin{proof} +Replace \(Y\) by the smooth model in the hypothesis and pull \(F\) back. +Write \(e^*L+F\sim_{\Q}R=N(e^*L+F)\). Pull back a positive current in +\(c_1(L)\) and add \([F]\). Lemma~\ref{bir:unique-current} gives +\(F\leq R\). Put \(D_Y=R-F\). Subtraction in +Lemma~\ref{bir:calculus} gives +\[ + e^*L\sim_{\Q}D_Y=N(e^*L)\geq0. +\] +In particular \([D_Y]\) is the unique positive current in \(c_1(e^*L)\). + +Choose \(m\) so that \(mL\) and \(mD_Y\) are integral and the latter is +the divisor of a holomorphic section \(s\) of \(e^*(mL)\). Factor \(e\) +through its normal Stein space: +\[ + Y\xrightarrow{h}Z\xrightarrow{\nu}X, + \qquad \nu\text{ finite of degree }d=\deg e. +\] +The connected birational map \(h\) descends \(s\) to a holomorphic +section \(s_Z\) of \(\nu^*(mL)\), by normality and projection formula. +Lemma~\ref{bir:finite-norm} gives a holomorphic section of \(mdL\) whose +divisor is +\[ + \operatorname{div}(\operatorname{Nm}_\nu(s_Z)) + =\nu_*\operatorname{div}(s_Z)=m e_*D_Y. +\] +Thus \(D=d^{-1}e_*D_Y\) is effective and satisfies \(L\sim_{\Q}D\). + +Both \(e^*D\) and \(D_Y\) are positive divisor currents in +\(c_1(e^*L)\), so uniqueness gives \(e^*D=D_Y\). We already know +\(N(L)\leq D\). If this inequality were strict at a prime \(A\subset X\), +choose analytic-singularity tests for \(L\) whose generic orders at \(A\) +tend to \(\nu_A(L)\), and pull them to \(Y\). For any prime \(A'\) +dominating \(A\), the generic orders are multiplied by +\(\operatorname{mult}_{A'}e^*A\). The pulled-back Kähler errors can be +enlarged to Kähler errors upstairs tending to zero. Consequently +\[ + \nu_{A'}(e^*L) + \leq \operatorname{mult}_{A'}(e^*A)\,\nu_A(L) + <\operatorname{mult}_{A'}(e^*A)\,\operatorname{coeff}_A(D), +\] +contrary to \(N(e^*L)=D_Y=e^*D\). This proves \(D=N(L)\). +\end{proof} + +We will also use Proposition~\ref{bir:finite-descent} when \(e\) is a +modification and \(F=0\). Once the positive part of a decomposition is +known to be torsion, it gives \(L\sim_{\Q}N(L)\) on the original smooth +space; Lemma~\ref{bir:pullup} then identifies the negative divisor upstairs +with the pullback of \(N(L)\). + +\subsection{The projective anchor and the two geometric cases} + +\begin{proposition}[Projective good models]\label{proj:good-models} +Under Assumption~\ref{asm:log-iitaka}, every projective lc pair over +\(\C\) with rational boundary and pseudo-effective rational Cartier +adjoint has a good log minimal model. In particular, \(\Ggood_n\) holds +for projective manifolds in every dimension, and every nef rational lc +adjoint on a projective variety is semiample as an actual rational line. +\end{proposition} + +\begin{proof} +The good-model assertion is the rational-boundary consequence of the full +argument in \cite[Proposition~2.5, Theorem~9.6, and Section~10]{LI}. That +argument establishes the additional inductive and nonvanishing premises of +its Proposition~2.5 before applying it, and proves a stronger real-boundary +induction. Its only use of Assumption~\ref{asm:log-iitaka} is in its +Lemma~6.1, on a resolved projective Albanese fibration with both boundaries +zero. Thus its premise is precisely available here. + +For clarity, its good-model conclusion gives on a common smooth projective +resolution +\[ + u^*(K_X+B)=v^*(K_{X_m}+B_m)+F, + \qquad F\geq0\text{ and }v\text{-exceptional}, +\] +as an actual rational divisor identity; the line on \(X_m\) is semiample. +This is the comparison in \cite[Lemma~2.2]{LI}. Algebraic nefness of a +rational line on a projective variety implies analytic nefness, by adding +arbitrarily small ample rational classes. Lemma~\ref{bir:exceptional} +therefore identifies \(F\) with the analytic negative part of the left +side. On a smooth projective variety, analytic pseudo-effectivity of a +divisor class agrees with algebraic pseudo-effectivity +\cite[Proposition~1.2]{BDPP}; hence the good-model assertion applies to +every projective instance of \(\Ggood_n\). This proves the stated form of +\(\Ggood_n\). If the original adjoint +is nef, the same comparison and normal descent give its semiampleness. +\end{proof} + +The remainder of the paper proves the following two propositions. They +are stated here so that the global induction can be completed before its +technical components are developed. A compact space is \emph{simple} if +no positive-dimensional proper compact analytic subvariety passes through +a very general point. We write \(a(X)\) for its algebraic dimension. + +\begin{proposition}[The fibration case]\label{rank:fibration-case} +Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume +\(\Ggood_j\) for \(j0\) and assume \(\Ggood_j\) for \(j0\). + +If \(a(X)=n\), the Kähler Moishezon theorem makes \(X\) projective, and +Proposition~\ref{proj:good-models} applies. If \(00\) and the evaluation ideal -\[ - \operatorname{im}\bigl(f^*f_*\OO_T(rJ)\longrightarrow\OO_T(rJ)\bigr) - =\mathcal I\otimes\OO_T(rJ). -\] -Outside the exceptional locus of \(f\), evaluation is an isomorphism. -On a positive-dimensional projective fiber, every section vanishes: -\(rJ\) has negative degree on every fiber curve, and curves cover -every positive-dimensional fiber component. Connectedness gives the -same vanishing on the whole nontrivial fiber. Hence the zero set of -\(\mathcal I\) is exactly the exceptional locus. On a common -resolution principalizing \(\mathcal I\), relative generation on -the positive side identifies \(rF\) in -Equation~\eqref{prog:comparison} with the divisor of -\(\mathcal I\OO_W\). Thus \(\Supp F\) is the full inverse image -of that locus. On a common resolution of a longer negative program, -discrepancy monotonicity makes the cumulative comparison dominate -this first-step divisor. These are support statements for the actual -comparison, stronger than effectiveness and exceptionality alone. - -We next state the restricted model inputs used in this section. A nef -b-class is a Bott--Chern class that is globally nef on one fixed smooth -compact Kähler carrier, with its linear traces on the other models. The -generalized adjoint on \(T\) is -\(\alpha=\{K_T+B\}+\beta_T\). Generalized klt, abbreviated gklt, -means that all generalized log discrepancies, computed on that carrier, -are positive. The modified-bigness hypothesis below concerns the whole -boundary-plus-nef trace \(\{B\}+\beta_T\). We use the convention of -\cite[Definition~2.8]{HX}: a modified-big trace is the pushforward of -a big Bott--Chern class on a modification. The trace itself may be -a current; the generalized adjoint is required to define a -Bott--Chern class. -For a proper map $f:T\to S$, let $q_f$ be the quotient map to -$H^{1,1}_{\BC}(T,\R)/f^*H^{1,1}_{\BC}(S,\R)$. Relative -pseudo-effectivity means -$q_f(\alpha)\in\overline{q_f(\operatorname{Psef}(T))}$, where -$\operatorname{Psef}(T)$ is the absolute pseudo-effective cone. -Equivalently, for Kähler classes $\omega_T,\omega_S$, the class -$\alpha$ is pseudo-effective over $S$ if for every $\epsilon>0$ -some $c_\epsilon\geq0$ makes -$\alpha+\epsilon\omega_T+c_\epsilon f^*\omega_S$ big. -This imposes no projectivity hypothesis on $f$. For the actual -relative line classes of an already projective map, this is the -usual relative pseudo-effective cone. - -\begin{proposition}[Restricted Kähler model inputs]\label{prog:mori} -Let \((T,B+\boldsymbol\beta)\) be an effective compact Kähler gklt -pair of dimension \(d\), with \(T\) globally strongly -\(\Q\)-factorial. Suppose that its nef b-data are globally nef on a -fixed smooth carrier and that \(\{B\}+\beta_T\) is globally -modified big. Write \(\alpha=\{K_T+B\}+\beta_T\). -\begin{enumerate} -\item If \(\alpha\) is nef, there is a proper morphism with connected -fibers \(g:T\to Z\) to a normal compact Kähler space and a Kähler -class \(\omega_Z\) such that \(\alpha=g^*\omega_Z\). -\item For a proper morphism \(\pi:T\to V\) to a normal compact -Kähler space, if \(\alpha\) is pseudo-effective over \(V\), there -is a chosen good log terminal model over \(V\). It is nonextracting, -compact Kähler and globally strongly \(\Q\)-factorial. Its adjoint -is nef over \(V\), and on a common projective resolution its comparison -with \(\alpha\) is effective and exceptional over the new model, -with strictly positive coefficient at every prime contracted from \(T\). -The construction retains compatible forward Bott--Chern traces of the -specified data on a common smooth carrier. -\item A compact polytope of these data on one fixed carrier has -finitely many marked relative canonical models and relative weak log -canonical models with normal compact Kähler targets. -\end{enumerate} -The relative assertion applies to proper morphisms, without assuming -that \(\pi\) is projective. It is an assertion about a chosen model, -not termination of every generalized flip sequence. -\end{proposition} - -\begin{proof} -These are the semiampleness, proper relative model and finite-model -assertions of Theorem~\ref{prog:restricted-package}, proved below -by dimension induction independently of \(\Ggood\). -Finite marked weak-model geography is recorded in -Lemma~\ref{prog:finite-geography}. Projective analytic model -constructions used in that induction are supplied by -Proposition~\ref{prog:projective-good-model}. None of these statements -promotes an undetected generalized-ray contraction to a projective map. -\end{proof} - -We use separately the analytic cone theorem -\cite[Theorem~1.3]{HX}: an \(\alpha\)-negative analytic extremal ray -of an effective gklt adjoint has a rational curve generator \(C\) with -\(0<-\alpha\cdot C\leq2d\). Existence of the supporting contraction -in the applications below follows from the nef assertion above. -When an ordinary rational adjoint is negative on that ray, -Lemma~\ref{prog:detected-projectivity} and -Proposition~\ref{prog:detected-flip} give its projective contraction and ordinary -flip. We call these \emph{detected ordinary steps}. - -The Bott--Chern transform through such a birational step has a concrete -description which does not require the source to be smooth. Write the -step as \(T\xrightarrow{f}Z\xleftarrow{f^+}T^+\), with -\(f^+=\id\) for a divisorial contraction, and let \(J\) be its -negative rational adjoint. For \(\gamma\in H^{1,1}_{\BC}(T,\R)\), -choose the unique real number \(c\) for which -\(\gamma-c\{J\}\) annihilates the contracted analytic ray. -Every contracted curve is on that ray. Lemma~\ref{prog:bc-descent} -therefore gives a unique class \(\delta\) on \(Z\) with -\(\gamma-c\{J\}=f^*\delta\), and we put -\[ - \gamma^+=(f^+)^*\delta+c\{J^+\}. -\] -The rational-singularity and vanishing hypotheses of that lemma hold: -slightly lowering the rational Cartier floor gives a klt adjoint still -antiample over \(Z\), and relative vanishing gives rational -singularities on the base. On a common resolution, the pullback -difference is \(c\) times the adjoint comparison in -Equation~\eqref{prog:comparison}; it is exceptional over the new -model, and over both models in a flip. This defines a linear transform. -For the Chern class of a rational line it agrees with its reflexive trace, -by the coherent line comparison \cite[Lemma~7.2]{CB} and exceptional -negativity. The already-projective relative program below only needs a -ray for degrees of global rational lines; this Bott--Chern construction -is used for its subsequent detected analytic-ray programs. - -When such a class difference lies in an exceptional divisor span, we write -it as \(\{F\}\) for the representing real exceptional divisor \(F\). -This divisor is unique: an exceptional divisor with zero class has zero -degree on every contracted curve, and exceptional negativity applied to -both signs makes it zero. - -We also use local projective analytic results. For a projective -analytic morphism over a Stein neighborhood of a compactum with -Fujino's property (P), relative klt base point freeness says that, if -\(L\) is a relatively nef Cartier line and -\(aL-(K+\Gamma)\) is relatively ample for some positive integer -\(a\), then all sufficiently high powers of \(L\) are relatively -generated after shrinking -\cite[Theorems~6.2 and 6.5]{FujinoAnalyticBCHM}. -We use the finite negative-ray truncation in the relative cone theorem -in the same setting. Finally, the multigraded adjoint finite-generation -theorem applies to a projective morphism from a smooth space, for simultaneous -effective SNC klt boundaries with a common relatively ample rational -summand \cite[Theorem~3.1]{DHP}. Finitely many smaller Stein -neighborhoods suffice over a compact base. - -\begin{lemma}[Descent without changing a Cartier exponent] -\label{prog:integral-descent} -Let \(f:T\to Z\) be a projective bimeromorphic contraction with normal -target. Suppose an ordinary rational klt adjoint is \(f\)-antiample. -If a Cartier line \(L\) has degree zero on every contracted curve, -then \(f_*L\) is an invertible sheaf and evaluation is an isomorphism -\[ - f^*(f_*L)\simeq L. -\] -The same conclusion holds for an ordinary dlt adjoint whose floor is -rational Cartier and which is \(f\)-antiample. In a flip, a line with -zero contracted degree therefore has the same Cartier exponent on -both sides, and the two lines are pullbacks of a line on the common -base. -\end{lemma} - -\begin{proof} -For dlt input, decrease the floor by a sufficiently small positive -rational multiple. The resulting pair is klt and remains -\(f\)-antiample. For this klt pair the base point free hypothesis -holds for \(L\), because \(L\) has zero fiber degrees and the negative -adjoint is relatively ample. Over a smaller base neighborhood every -sufficiently high power of \(L\) is therefore generated. The -associated morphism is constant on each connected fiber: its -tautological line has degree zero on every fiber curve, and every -positive-dimensional projective image contains a curve. Its graph -then factors through \(Z\); the graph projection is finite -bimeromorphic and \(Z\) is normal. Descend two consecutive generated -powers \(L^k,L^{k+1}\) to lines \(M_k,M_{k+1}\). The line -\(M_{k+1}\otimes M_k^{-1}\) pulls back to \(L\). Projection formula -identifies this quotient with -\(f_*L\), so these local descents and their evaluation maps agree on -overlaps. Pull the descended line to the positive side of a flip. -This is the argument of \cite[Lemma~7.4]{CB}. -\end{proof} - -\subsection{Relative klt programs and small models} - -The local finite-generation theorem also supplies an ordinary klt program -over a compact base. We use it to obtain globally strongly -\(\Q\)-factorial small models of strata. The construction keeps track of -degrees of global rational lines, which are the degrees needed for exact -Cartier descent. - -\begin{proposition}[Relative klt programs over a compact base] -\label{prog:relative-klt} -Let \(\pi:T\to V\) be projective bimeromorphic, where \(V\) is a -normal compact Kähler space and \(T\) is normal, compact Kähler, and -globally strongly \(\Q\)-factorial. Let \((T,B)\) be an effective -rational klt pair with rational-line adjoint \(J=K_T+B\). -There is a finite sequence of ordinary \(J\)-negative steps over -\(V\), all projective over \(V\), whose last adjoint is curve-nef -over \(V\), meaning that it has nonnegative degree on every curve -contracted over \(V\). Every working space is compact Kähler and globally -strongly \(\Q\)-factorial. On a common resolution the comparison -from the first adjoint to the last is effective and exceptional over -the last space. -\end{proposition} - -\begin{proof} -We give the reduction to the local finite-generation input, including why -it yields a single finite program over the compact base. Let -\(\mathcal N_{\R}\) be the finite-dimensional space of degrees of global -rational lines on curves over \(V\), and put \(r=\dim\mathcal N_{\R}\). -If \(r=0\), then \(J\) is already curve-nef. Assume \(r>0\). Choose -rational lines \(H_1,\ldots,H_r\) whose degree classes form a basis and -such that both \(H_i\) and \(J+H_i\) are relatively ample. Such a basis -exists because the relatively ample classes whose sum with \(J\) is also -relatively ample form a nonempty open set containing sufficiently positive -classes. Define the rational simplex of adjoints -\[ - \Pi=\operatorname{conv}\{J,J+H_1,\ldots,J+H_r\}. -\] -We will choose a general direction \(H=\sum_i\theta_iH_i\) with -\(\theta_i>0\) and \(\sum_i\theta_i=1\). Its entire scaling segment lies -in this simplex: -\[ - J+tH=(1-t)J+\sum_i t\theta_i(J+H_i)\in\Pi - \qquad(0\leq t\leq1), -\] -and it lies in the relative interior for \(0t\) for which \(J+sH\) is -relatively ample. On the zero face of the normalized compact curve slice, -\[ - J=-\frac{t}{s-t}(J+sH) -\] -in degrees. Thus \(J\) is uniformly negative on that face. Choose the -positive ample truncation so that its compact remainder \(K\) misses the -face. The finite rational polyhedral test and the general choice of \(H\) -make the face a single ray \(R\). The wall class has a positive minimum on -\(K\) and is positive on every other truncated curve generator. Hence a -sufficiently small open neighborhood of \(w\) inside the annihilator -\(R^\perp\) consists of nef classes with zero face exactly \(R\). - -Suppose first that \(R^\perp\) has positive dimension. It is a rational -hyperplane because \(R\) is represented by an integral curve. Choose a -rational simplex of support classes in that neighborhood containing the -degree class of \(w\) in its relative interior, and represent each vertex -by a rational line \(N\). For each such \(N\), compactness gives an -integer \(a>0\) for which \(aN-J\) is relatively ample. Relative base -point freeness contracts exactly \(R\) and descends \(N\) to a relatively -ample rational line on the contraction base. These bases agree because -the contracted curves agree. The difference between \(w\) and the -corresponding convex combination of these rational lines has zero -relative degree. In the finite rational span of the lines involved, the -degree map has rational kernel, since rational lines have rational degrees -on integral curves. The difference is therefore a real combination of -rational lines of zero relative degree. Lemma~\ref{prog:integral-descent} -descends these lines, and lifting curves from the base shows that their -descents still have zero degree over \(V\). They do not affect relative -ampleness. Convexity now shows that \(w\) descends to a relatively ample -real line class over \(V\). If \(R^\perp=0\), the same zero-degree descent -applies to \(w\); the contraction base has no curves over \(V\), hence is -finite over \(V\), and relative ampleness is automatic. - -After a flip, let \(w^+\) be the pullback of this wall class to the positive -side. For sufficiently small \(\delta>0\), -\[ - J^++(t-\delta)H^+ - =\left(1-\frac{\delta}{t}\right)w^+ - +\frac{\delta}{t}J^+ -\] -is relatively ample over \(V\): the wall is pulled back from a relatively -ample class on the contraction base, and \(J^+\) is relatively ample over -that base. After a divisorial contraction, openness of the ample cone on -the base gives the same nonempty interval of relative ampleness. - -Each working model is consequently one of the marked ample models -already counted. Indeed, for a fixed finite prefix and an interior -parameter \(00\) for which -\((S,\Delta-\epsilon D)\) is effective and klt. On a projective log -resolution \(r:W\to S\), use the strict lowered boundary and give -every exceptional prime a coefficient strictly between its crepant -coefficient and \(1\), and at least zero. The resulting klt adjoint -has the actual form -\[ - K_W+\Gamma\sim_{\Q}r^*(K_S+\Delta-\epsilon D)+F, - \qquad F\geq0, -\] -where every \(r\)-exceptional prime has positive coefficient in -\(F\). Apply Proposition~\ref{prog:relative-klt} over \(S\). -On its endpoint the trace \(F_Y\) is effective, exceptional over -\(S\), and relatively nef. Negativity makes \(F_Y=0\). -Since all initially exceptional primes had positive coefficient, -\(Y\to S\) is small and the lowered adjoint is crepant. -Restoring \(\epsilon D\) gives the full crepant identity. - -A projective small morphism to a smooth germ is an isomorphism: -transport a relatively ample line to the smooth germ, where it is -Cartier; smallness makes the original line its pullback, contradicting -relative ampleness on a nontrivial fiber. Thus \(h\) is an isomorphism -over an SNC open meeting all lc centers of the dlt pair. The full -crepant comparison then proves dlt on \(Y\). -\end{proof} - -\subsection{Scaling toward zero and keeping a nef rational line} - -The next construction runs every positive truncation of an ordinary -dlt scaling. It also explains why a sufficiently large multiple of -a prescribed nef rational line forces every step to be trivial for -that line. The integer that clears the line will be the same at -every step. - -For the detected ordinary scalings constructed below, let -\(\alpha_0(\lambda)=\{H\}+\lambda\omega\) on the first space -\(T_0\), with initial parameter \(1\), and let \(\alpha_i(\lambda)\) -be its linear transform on a working model \(T_i\). Put -\(\lambda_{-1}=1\). If the next step occurs at threshold -\(\lambda_i\), its ray is annihilated by \(\alpha_i(\lambda_i)\) -and is negative for \(\alpha_i(0)\). The trace \(\alpha_i(\lambda)\) -is nef on \([\lambda_i,\lambda_{i-1}]\), which we call its -\emph{working interval}. If \(\alpha_i(0)\) is nef, we set -\(\lambda_i=0\) and stop. Working intervals are allowed to be -degenerate; strict decrease of positive thresholds will be arranged only -in the proof of special termination. - -\begin{lemma}[Positive truncations and a fixed nef line] -\label{prog:nef-trivial} -Let \((T,B)\) be an effective rational dlt pair on a globally strongly -\(\Q\)-factorial compact Kähler \(d\)-fold, and put \(J=K_T+B\). -\begin{enumerate} -\item For a sufficiently large Kähler class \(\omega\), there is an -ordinary \(J\)-program with scaling of \(\omega\), starting with -\(\{J\}+\omega\) Kähler. Every positive truncation is finite. -If \(J\) is pseudo-effective, the program either reaches a nef adjoint -or has positive thresholds tending to zero. If \(J\) is not -pseudo-effective, the program is finite and ends with a Mori fiber -contraction. -\item Let \(P\) be a nef rational line with \(mP\) Cartier, and choose -\(b\in\Q\) with \(b>4dm\). The same alternatives hold for -\(H=J+bP\), according to pseudo-effectivity of \(H\), with -\(\omega\) enlarged if necessary. Every birational step is an ordinary -\(J\)-negative step on which \(P\) is trivial, and the final Mori -ray, if present, is also \(P\)-trivial. The same line \(mP\) descends -through every birational contraction and pulls back to its positive -side. Its traces remain analytically nef; if \(P\) is semiample, its -same generated multiple and associated morphism are preserved, also -through the final Mori contraction. -\end{enumerate} -\end{lemma} - -\begin{proof} -Write \(H=J\) in the first case and \(H=J+bP\) in the second. -Fix \(\omega\) with \(\{H\}+\omega\) Kähler. For each -\(0<\eta<1\), choose a positive rational \(\epsilon\) so small that, -with \(F=\lfloor B\rfloor\), -\[ - (T,B-\epsilon F)\ \text{is klt},\qquad - \eta\omega+\epsilon\{F\}\ \text{is Kähler}. -\] -For \(\lambda\in[\eta,1]\) use the presentations -\[ - \{H\}+\lambda\omega - =\{K_T+B-\epsilon F\} - +\bigl(b\{P\}+\lambda\omega+\epsilon\{F\}\bigr), -\] -omitting \(b\{P\}\) in the first case. Their b-data are globally -nef on the fixed initial carrier, and their boundary-plus-nef traces -are globally modified big. These are effective gklt data. - -We first construct a step whenever the unscaled working trace is not -nef. Choose a positive shift strictly below the current wall and use the -gklt presentation of its current trace \(A\). The ray at the wall is -\(A\)-negative. On a compact slice of the analytic curve cone, the -cone theorem makes the \(A\)-negative part locally polyhedral after -a sufficiently small positive Kähler truncation. Separating the -chosen extremal ray from the other finitely many rays in that -truncation gives a Kähler class \(\omega_R\) such that -\(A+\omega_R\) is nef and its zero face is precisely this ray; -see the supporting construction in Proposition~\ref{prog:ordinary-step}. -Add the pullback of this new Kähler class to the fixed nef carrier. -The resulting adjoint still has globally nef data and modified-big -boundary-plus-nef trace, so the nef assertion of -Proposition~\ref{prog:mori} supplies its supporting contraction. -This uses a new supporting direction on the current space; it does -not require the trace of the original scaling direction to be Kähler. -We verify a rational ordinary detector before asserting projectivity -or constructing a flip. -In the second case nefness of \(P\) makes \(J\) negative on the ray; -the first case has this property directly. Lower the floor by a small -rational amount, retaining negativity of \(J_\epsilon\), so that -\(-J\cdot R\leq2(-J_\epsilon\cdot R)\). The ordinary klt cone -length bound gives a rational curve generator \(C\) with -\[ - 0<-J\cdot C\leq4d. -\] -In the second case, if \(P\cdot C>0\), integrality of \(mP\) gives -\(P\cdot C\geq1/m\), contradicting -\[ - H\cdot C\geq-4d+b/m>0. -\] -Thus in that case the ray is \(P\)-trivial. The lowered ordinary adjoint is a -global rational detector. The detected projectivity and flip -constructions in Lemma~\ref{prog:detected-projectivity} -and Proposition~\ref{prog:detected-flip} supply the projective ordinary step. -The difference of \(J\) and a positive rational multiple of the -lowered adjoint has zero contracted degree; exact descent in -Lemma~\ref{prog:integral-descent} shows that restoring the floor -preserves the positive-side sign. This is the ordinary replacement -of Proposition~\ref{prog:ordinary-replacement}, and the full pair -remains dlt by Equation~\eqref{prog:comparison}. - -In the second case, Lemma~\ref{prog:integral-descent} descends the actual Cartier line -\(mP\) without changing \(m\). Analytic nefness descends and pulls -back under these projective contractions \cite[Lemma~2.44]{HX}. -A generated multiple descends as a generated line by projection -formula and pulls back as one. Thus the same length estimate and the -same \(b\) apply at the next step. The corresponding assertion on -a final Mori ray follows from exactly the same estimate. For a -semiample \(P\), the map of its fixed generated multiple is constant -on each connected projective contracted fiber: otherwise its image -contains a curve of positive degree. The map therefore factors through -the normal contraction base, including in the fiber-type case. - -The traces used in this construction are linear by the degree-zero -descent description preceding Lemma~\ref{prog:integral-descent}. -At a wall the scaled class is pulled back from its base. The comparison -for an earlier parameter is a nonnegative multiple of the ordinary -adjoint comparison. Consequently the gklt presentations on any fixed -positive segment remain gklt along its program, and every working -model is a weak model for a member of that segment. The finite marked -weak-model assertion in Proposition~\ref{prog:mori} applies. -A marked model cannot recur: its boundary trace would be the same, -whereas a nontrivial intervening ordinary step strictly increases a -discrepancy. There are therefore only finitely many steps with -threshold at least \(\eta\). - -For decreasing cutoffs, any already-constructed finite prefix remains -valid. The floor perturbations cancel in the displayed classes and -continue to cancel under their linear transforms; their nonpositive -comparisons preserve the new gklt data. Starting at its last nef wall -therefore extends the prefix to the smaller cutoff. This gives one -compatible scaling program. A positive limiting threshold would lie -in a finite positive truncation, so an infinite program has limit zero. -Pseudo-effectivity excludes a negative Mori endpoint. If \(H\) is -not pseudo-effective, its pseudo-effective threshold in the initial -Kähler direction is positive. Every nef working trace, together with -its effective exceptional comparison, makes the initial scaled class -pseudo-effective, so the scaling thresholds are bounded below by -that positive number. The program is therefore finite. Its endpoint -cannot be nef, since that comparison would make \(H\) -pseudo-effective; it is the stated Mori fiber contraction. -\end{proof} - -We call these programs \emph{detected ordinary scalings}. Their steps -are ordinary negative steps of one analytic extremal ray, with the -linear Bott--Chern transform described above. -The next lemma connects their nef working traces -to the analytic negative part, including its limit at parameter zero. - -\begin{lemma}[Negative parts along a detected ordinary scaling] -\label{prog:negative-along-scaling} -In a scaling from Lemma~\ref{prog:nef-trivial}, write \(H=J\) in its first -case and \(H=J+bP\) in its second, and use the notation -\(\alpha_0(\lambda)=\{H\}+\lambda\omega\) above. For a parameter -\(\lambda\) in the working interval of \(T_i\), a common smooth -resolution \(p:W\to T_0\), \(q:W\to T_i\) gives -\begin{equation} - p^*\alpha_0(\lambda)=q^*\alpha_i(\lambda)+\{E_i(\lambda)\}, - \qquad E_i(\lambda)\geq0,\qquad q_*E_i(\lambda)=0. - \label{prog:working-negative} -\end{equation} -Here \(E_i(\lambda)\) is an actual real exceptional divisor, and -\[ - N(p^*\alpha_0(\lambda))=E_i(\lambda). -\] -For the limit statement, let \(H\) be any rational line on a normal compact -Kähler space \(T\) whose pullback to a smooth resolution is -pseudo-effective, and let \(A\) be a Kähler class on \(T\). On any fixed -smooth resolution \(r:U\to T\) and for every prime \(Q\subset U\), -\[ - \lim_{\lambda\downarrow0}\nu_Q(r^*(\{H\}+\lambda A)) - =\nu_Q(r^*\{H\}). -\] -Thus the same convergence holds for every divisorial place, after choosing -a resolution on which it appears. -\end{lemma} - -\begin{proof} -For \(\lambda\) in the working interval of \(T_i\), every earlier -threshold is at least \(\lambda\). Each earlier step is therefore negative -or trivial for this parameter. Composing its comparisons gives -Equation~\eqref{prog:working-negative}. Since \(\alpha_i(\lambda)\) is -nef, Lemma~\ref{bir:exceptional} identifies the exceptional divisor with -the negative part. - -On the fixed resolution \(U\), choose a Kähler class \(\Omega\) and -\(c>0\) such that \(c\Omega-r^*A\) is Kähler. Monotonicity of each -minimal multiplicity under addition of a nef class gives -\[ - \nu_Q(r^*\{H\}+\lambda c\Omega) - \leq \nu_Q(r^*(\{H\}+\lambda A)) - \leq \nu_Q(r^*\{H\}). -\] -The left side tends to the right side by the definition using small Kähler -perturbations. This proves the convergence on \(U\), and the same argument -applies on a resolution representing any chosen divisorial place. -\end{proof} - -\begin{proposition}[Contracting a known negative part] -\label{prog:contract-negative} -Let \((T,B)\) be an effective rational dlt pair on a globally -strongly \(\Q\)-factorial compact Kähler space, and put \(J=K_T+B\). -Suppose there is an actual rational-line identity -\[ - J\sim_{\Q}P+E, -\] -where \(P\) is analytically nef, \(E\) is an effective rational -Cartier divisor, and, on one smooth compact Kähler resolution -\(\pi:W\to T\) projective over \(T\), -\[ - N(\pi^*\{J\})=\pi^*E. -\] -There is a finite ordinary \(J\)-negative program to a nef model -\(T'\) which contracts precisely the prime components of \(E\) -among the primes of \(T\). Every step is \(P\)-trivial, with a -fixed Cartier exponent as in Lemma~\ref{prog:nef-trivial}, and -\[ - J_{T'}\sim_{\Q}P_{T'}. -\] -If \(P\) is semiample, then so is this actual last adjoint. -\end{proposition} - -\begin{proof} -Lemma~\ref{bir:pullup} makes the stated equality valid on any -higher resolution. Choose \(b\) as in -Lemma~\ref{prog:nef-trivial}. Adding \(bP\) leaves the same -negative part. Indeed, on a smooth resolution write -\(\alpha=\pi^*\{J\}\), \(p=\pi^*\{P\}\), and \(e=\pi^*E=N(\alpha)\). -Subadditivity and homogeneity of negative multiplicities give -\begin{align*} - N(\alpha+bp)&\leq e,\\ - (1+b)e=N((1+b)\alpha) - &\leq N(\alpha+bp)+bN(\{e\}) - \leq N(\alpha+bp)+be. -\end{align*} -Thus \(N(\alpha+bp)=e\). - -Run the scaling for \(H=J+bP\). -Let \(Q\) be the strict transform on a fixed resolution of a prime -component of \(E\). Its multiplicity in \(N(\pi^*\{H\})\) is -positive. The limit in Lemma~\ref{prog:negative-along-scaling} makes its multiplicity -positive for every sufficiently small positive parameter. If -that prime still survived on the corresponding working model, -Equation~\eqref{prog:working-negative} would make its multiplicity zero, -because the comparison there is exceptional over that model. -It must therefore be contracted after finitely many steps. The -same conclusion holds if a nef endpoint is reached before taking -the limit. There are only finitely many components of \(E\), so -after a finite prefix all have disappeared. - -Throughout the program \(P\) descends and remains nef. The -actual identity \(J_i\sim_{\Q}P_i+E_i\), with \(E_i\) the -codimension-one pushforward of \(E\), follows from the reflexive -trace convention. Once \(E_i=0\), we have \(J_i\sim_{\Q}P_i\); since -\(P_i\) is nef, the program stops. Conversely, a divisorial negative step can -contract only a prime of \(E_i\): its ray is \(P_i\)-trivial, -so the effective divisor \(E_i\) has negative degree on every -contracted curve, and those curves are contained in its support. -This proves the exact assertion about contracted primes. -\end{proof} - -The next lemma records the cohomology carried by a detected ordinary scaling -that begins on a smooth space. Its surjectivity will let us choose one -generic initial scaling direction for all later models. Its exceptional -splitting will also carry the span of line classes to the terminal models -used in Section~\ref{sec:boundary}. - -\begin{lemma}[Cohomology through ordinary steps] -\label{prog:cohomology-transport} -Let a finite prefix of one of the detected ordinary scalings constructed -in Lemma~\ref{prog:nef-trivial} start on a smooth compact Kähler -space, and let \(V\) be any working space. -For every smooth compact Kähler resolution \(p:U\to V\) -projective over \(V\), -\begin{align} - H^2(U,\R) - &=p^*H^2(V,\R)\ \oplus\ - \bigoplus_{Q\ {\rm exceptional\ for}\ p}\R\{Q\}, \label{prog:H2}\\ - H^{1,1}(U,\R) - &=p^*H^{1,1}_{\BC}(V,\R)\ \oplus\ - \bigoplus_{Q\ {\rm exceptional\ for}\ p}\R\{Q\}. - \label{prog:H11} -\end{align} -The linear transform from the first space is surjective onto -\(H^2(V,\R)\) and \(H^{1,1}_{\BC}(V,\R)\), respectively. For a small correspondence between -working spaces, the difference of the pulled-back classes and -their transforms on a common resolution lies in the span of the -primes exceptional over both spaces. -Under the displayed decompositions, the Chern class of a global -holomorphic line on \(U\) is the sum of the pullback of a rational -holomorphic line class on \(V\) and an exceptional rational divisor -class. In particular, passage to the exceptional quotient preserves -the real span of line Chern classes. -\end{lemma} - -\begin{proof} -For a modification of a smooth compact Kähler manifold, the -degree-two modification theorem gives these decompositions, and -the exceptional classes have type \((1,1)\). Suppose they hold -at one step. Its contraction base and both working spaces have -rational singularities: slight lowering of the rational Cartier -floor makes the ordinary pair klt, and the detected contraction has -a base with rational singularities. These contractions are -bimeromorphic, so a common resolution and Leray give -\(R^if_*\OO=0\) for \(i>0\); this verifies the additional -hypothesis in Lemma~\ref{prog:bc-descent}. - -The ray of this detected contraction is one ray for Bott--Chern degrees. -This also suffices for degree-two classes. Indeed, pull a class -\(\gamma\in H^2(V,\R)\) to a smooth resolution \(p:U\to V\) -and take its Hodge decomposition. By the decomposition already proved for -this working model in Equation~\eqref{prog:H11}, its \((1,1)\)-part is \(p^*\beta+\{E\}\) for a -Bott--Chern class \(\beta\) and an exceptional real divisor -\(E\). The other Hodge components have zero degree on curves. -On every \(p\)-contracted curve the degree of \(E\) is therefore -zero, so both signs of negativity make \(E=0\). Lifting any -curve of \(V\) through the projective resolution now shows that -\(\gamma\) and \(\beta\) have the same curve degrees. Thus -subtracting a multiple of \(\{J\}\) from either a degree-two or -a Bott--Chern class can annihilate every curve of the contracted -fiber. By -Lemma~\ref{prog:bc-descent}, a degree-two class, or a Bott--Chern class, -with these zero degrees is pulled back from the base. -Subtract a multiple of \(\{J\}\) to make any given class -annihilate that ray, descend the remainder, and use the trace -\(\{J^+\}\) to define its transform. On a common resolution the -difference is a multiple of the exceptional adjoint comparison in -Equation~\eqref{prog:comparison}. For a flip the two exceptional prime -sets on that resolution are the same. For a divisorial -contraction the target resolution has just the one additional -exceptional prime of the step. The old decomposition therefore -spans the new one and proves surjectivity of the transform. - -The sums are direct. If the pullback of a class from \(V\) is -an exceptional divisor class, that divisor has zero degree on -all curves over \(V\). Applying exceptional negativity to both -signs makes the divisor zero; injectivity of pullback gives that -the class is zero. The pullback injectivity in degree two and in -Bott--Chern cohomology is also -Lemma~\ref{prog:bc-descent}, applied to the resolution. Passing to -higher smooth resolutions proves the assertion for every \(p\). - -For the line assertion, let \(\mathscr H\) be a holomorphic line on -\(U\), and put \(\mathscr Q=(p_*\mathscr H)^{**}\). Global strong -\(\Q\)-factoriality gives an invertible sheaf -\(\mathscr M=\mathscr Q^{[m]}\) for some \(m>0\). The coherent -evaluation comparison of \cite[Lemma~7.2]{CB} gives -\[ - \mathscr H^{\otimes m}\simeq p^*\mathscr M\otimes\OO_U(E), - \qquad E\ \text{an integral \(p\)-exceptional divisor}. -\] -Indeed \((p_*\mathscr H^{\otimes m})^{**}=\mathscr M\), since the -two sheaves agree at the general points of all divisors of \(V\). -Local generators of \(p_*\mathscr H^{\otimes m}\) in a frame -of \(\mathscr M\) compare their evaluations with the pulled-back -coefficients by meromorphic quotients. Those quotients agree on the -isomorphism locus and hence glue; their zeros and poles are exceptional. -Taking Chern classes proves the assertion without choosing a meromorphic -section of \(\mathscr H\). -\end{proof} - -Before returning to special termination, we record the consequence used in -the boundary argument: a lower-dimensional manifold of Kodaira dimension -zero has a terminal torsion model carrying the preceding exceptional -splitting. - -\begin{corollary}[Terminal models in Kodaira dimension zero] -\label{prog:terminal-kappa-zero} -Let \(X\) be a smooth compact Kähler \(d\)-fold with \(d0\), and transformed classes -\(\{J_V\},A_V\) such that -\begin{equation} - \{J_V\}+tA_V\ \text{is nef for every }0\leq t\leq\delta. - \label{prog:whole-nef-interval} -\end{equation} -On a common smooth resolution of \(S\) and \(V\), the -differences of pullbacks of \(J,A\) and these transforms lie -in the common exceptional span. -\end{lemma} - -\begin{proof} -Let \(P_M=J_M\) denote the semiample rational line from -Lemma~\ref{prog:stratum-model}, and use its transform \(A_M\). -On a common resolution \(p:W\to S,\ q:W\to M\), let \(F_A\) be the -real divisor in the common exceptional span defined by -\[ - q^*A_M-p^*A=\{F_A\}. -\] -It is effective: \(-F_A\) is \(q\)-nef, because \(p^*A\) is -nef and the other term is pulled back from \(M\); apply -exceptional negativity. Its support is disjoint from \(p^{-1}(U)\). -In particular \(A_M\) is pseudo-effective, and its negative -multiplicity at every prime of \(M\) is zero. Indeed, -\(q^*A_M=p^*A+\{F_A\}\), and -Lemma~\ref{bir:exceptional} applied over \(S\) gives -\(N(q^*A_M)=F_A\). - -Let \(D_M\) be the rational Cartier trace of \(D\). -Choose \(\tau>0\) so small that \(A+\tau\{D\}\) is Kähler. -For a fixed sufficiently small \(t>0\), consider on \(M\) the -boundary \(\Delta_M-t\tau D_M\) and the nef b-class represented on -\(W\) by \(t p^*(A+\tau\{D\})\). Its trace on \(M\) is -\(t(A_M+\tau\{D_M\})\), so its generalized adjoint is -\begin{equation} - \{P_M\}+tA_M. - \label{prog:interval-gadjoint} -\end{equation} -These are effective gklt data. Over \(U\) the b-class -descends to \(M\), so its generalized discrepancy -contribution is zero there, and subtracting \(t\tau D_M\) -raises every zero log discrepancy of the dlt pair. -Away from the floor the ordinary pair is klt; on one fixed -log resolution its finitely many subunit coefficients stay -subunit for small \(t\). This proves gklt. The generalized -boundary is big. On a common resolution the pullback of the -trace of \(A+\tau\{D\}\) minus \(p^*(A+\tau\{D\})\) is effective exceptional -by the same negativity argument as for \(F_A\). The latter -pullback is nef and big, so the trace is big; the remaining -boundary is effective. - -Choose an integer \(m>0\) such that \(mP_M\) is generated, and -let \(g:M\to Z\) be the Stein factor of its morphism. Then \(Z\) -is normal projective and \(mP_M=g^*H\) for an ample generated Cartier line -\(H\) on \(Z\). Set \(L=\{P_M\}+tA_M\). The preceding -preparation makes \(L\) an effective gklt adjoint with globally nef -b-data and globally modified-big boundary-plus-nef trace. It is -pseudo-effective because \(P_M\) is nef and \(A_M\) is modified -nef. Apply the proper relative assertion of -Proposition~\ref{prog:mori} over \(Z\). This produces a chosen -nonextracting good log terminal model \(\phi:M\dashrightarrow V\) -and a morphism \(g_V:V\to Z\), with \(L_V\) nef over \(Z\). -The hypothesis here is properness: the semiample morphism \(M\to Z\) -need not be projective. Put -\[ - mP_V=g_V^*H,\qquad A_V=(L_V-\{P_V\})/t. -\] -The relative construction preserves the actual line pulled back from -\(Z\), and its forward Bott--Chern transport preserves the displayed -linear relation; these are the transforms of \(P_M,A_M\). - -The map \(\phi\) is small. On a common projective resolution -\(a:W'\to M\), \(c:W'\to V\), its comparison is -\[ - a^*L=c^*L_V+\{E\},\qquad E\geq0, -\] -where \(E\) is exceptional over \(V\) and has positive coefficient -at the strict transform of each prime contracted from \(M\). -The forward trace \(L_V\) is pseudo-effective, so exceptional -translation gives -\[ - N(a^*L)=N(c^*L_V)+E\geq E. -\] -But \(L\) is modified nef on \(M\); the coefficient of -\(N(a^*L)\) at a strict prime of \(M\) is zero. Thus no prime of -\(M\) is contracted. Nonextraction then proves smallness. - -Fix \(b>2dm\). We claim that \(L_V+b\{P_V\}\) is absolutely -nef. Otherwise the cone theorem for the effective gklt adjoint -\(L_V\) gives an \(L_V\)-negative rational extremal generator -\(C\) such that -\[ - (L_V+b\{P_V\})\cdot C<0,\qquad 0<-L_V\cdot C\leq2d. -\] -Indeed the cone summand on which \(L_V\) is nonnegative also has -nonnegative \(P_V\)-degree, so a negative ray summand must account -for any failure of nefness. If \(P_V\cdot C=0\), ampleness of -\(H\) makes \(C\) vertical over \(Z\), contradicting relative -nefness of \(L_V\). Otherwise Cartier integrality gives -\(P_V\cdot C\geq1/m\), contradicting -\[ - (L_V+b\{P_V\})\cdot C\geq-2d+b/m>0. -\] -This proves the claim. Both \(\{P_V\}\) and -\((1+b)\{P_V\}+tA_V\) are nef. By convexity, -\[ - \{P_V\}+sA_V\ \text{is nef for }0\leq s\leq\frac{t}{1+b}. -\] -Thus \(\delta=t/(1+b)\) works. The actual line \(P_V\) has the -same generated multiple as \(P_M\), since both are pulled back from -\(H/m\). Its class is the transform of \(J\). -The relative comparison and linear transport give exceptional-divisor -comparisons for \(P_M,A_M\). Since \(S,M,V\) agree in codimension -one, their final comparison divisors are exceptional over both \(S\) -and \(V\), as required. -\end{proof} - -\subsection{Special termination} - -\begin{theorem}[Special termination for a chosen dlt scaling] -\label{prog:special-termination} -Assume \(\Ggood_d\) for every \(d0\) on this working space, above the remaining -walls, and set -\[ - A=\{J\}+\lambda_0H. -\] -It is Kähler, as the restriction of the interior ambient -Kähler class. The exact change of parameter is -\begin{equation} - \{J\}+tA=(1+t)(\{J\}+\lambda H),\qquad - \lambda=\frac{\lambda_0t}{1+t},\qquad - t=\frac{\lambda}{\lambda_0-\lambda}. - \label{prog:reparameterization} -\end{equation} -It applies for \(0\leq\lambda<\lambda_0\). The new walls -\(t_i\) strictly decrease toward zero, and the traces -\(\{J_i\}+tA_i\) are nef on their successive restricted -working intervals. - -On a common resolution of a nontrivial small stratum step, let \(D_i\) -be the real exceptional divisor defined by -\[ - p_i^*\{J_i\}-q_i^*\{J_{i+1}\}=\{D_i\}. -\] -The strict adjoint comparison makes \(D_i\) an effective -nonzero divisor exceptional over both strata. Equality of -the scaled pullbacks at its wall gives the exact -Bott--Chern equality -\begin{equation} - p_i^*(\{J_i\}+tA_i)-q_i^*(\{J_{i+1}\}+tA_{i+1}) - =\left(1-\frac{t}{t_i}\right)\{D_i\}. - \label{prog:restricted-comparison} -\end{equation} -For \(t\) in a later working interval all preceding factors -on the right are nonnegative. Telescoping on a common -resolution expresses the pullback of \(\{J\}+tA\) as the -pullback of its nef working trace plus an effective divisor -exceptional over that working stratum. -Lemma~\ref{bir:exceptional} identifies the latter divisor -with the negative part. - -Push the resulting positive currents to one fixed smooth -resolution of \(S\) and let \(t\downarrow0\). Closedness of -the pseudo-effective cone proves that \(J\) is -pseudo-effective. The negative multiplicities converge -valuation by valuation by the limit statement in -Lemma~\ref{prog:negative-along-scaling}. The comparison divisors -have no strict prime of \(S\), since the maps are small. -Their coefficients also vanish at every place whose center -meets the floor: each finite composition is unchanged on a -neighborhood of the floor by the induction on smaller -centers. Passing to the limit proves -Equation~\eqref{prog:away-floor}. - -The stratum has dimension less than \(n\), and the divisor \(D_S\) -constructed above is rational Cartier with support equal to its floor. -Lemmas~\ref{prog:stratum-model} and -\ref{prog:nef-interval} give one small model \(V\) on -which all transformed classes of \(\{J\}+tA\) are nef -for \(0\leq t\leq\delta\). On a common smooth -resolution \(p:W\to S,\ q:W\to V\), define the real divisor \(F(t)\) -in the common exceptional span by -\begin{equation} - p^*(\{J\}+tA)-q^*(\{J_V\}+tA_V)=\{F(t)\}. - \label{prog:affine-comparison} -\end{equation} -Uniqueness of the representing exceptional divisor makes \(F(t)\) -coefficientwise affine in \(t\). It is effective on -\([0,\delta]\): \(-F(t)\) is \(p\)-nef because the -class pulled back from \(V\) is nef, so exceptional -negativity applies. It is exceptional over \(V\) as -well. Lemma~\ref{bir:exceptional} gives -\[ - N\bigl(p^*(\{J\}+tA)\bigr)=F(t) - \qquad(0\leq t\leq\delta). -\] -Every divisorial negative multiplicity is thus affine on -this whole interval. This remains true on higher -resolutions by Lemma~\ref{bir:pullup}. - -Choose a nontrivial stratum wall \(t_i\in(0,\delta)\). -On a common resolution of \(S,V\) and the adjacent -strata, their nef working models compute the same -negative part on their respective open intervals. Their -two affine expressions agree at \(t_i\). But -Equation~\eqref{prog:restricted-comparison} says that, at a prime -with coefficient \(d_i>0\) in \(D_i\), their difference -is -\[ - \left(1-\frac{t}{t_i}\right)d_i. -\] -Their slopes differ by \(-d_i/t_i\). A single affine -function on \([0,\delta]\) cannot agree with both on -two open intervals. This contradiction excludes every -such wall. Infinitely many walls would tend to zero, so -the nontrivial diagrams on this stratum are finite. - -There are finitely many surviving strata. Induction on -their dimensions makes all sufficiently late ambient -flipping and flipped loci disjoint from the floor. This -proves the theorem. -\end{proof} - -\begin{corollary}[A signed adjoint supported on the floor] -\label{prog:signed-nef} -Assume \(\Ggood_d\) for every \(d0\) and the evaluation ideal +\[ + \operatorname{im}\bigl(f^*f_*\OO_T(rJ)\longrightarrow\OO_T(rJ)\bigr) + =\mathcal I\otimes\OO_T(rJ). +\] +Outside the exceptional locus of \(f\), evaluation is an isomorphism. +On a positive-dimensional projective fiber, every section vanishes: +\(rJ\) has negative degree on every fiber curve, and curves cover +every positive-dimensional fiber component. Connectedness gives the +same vanishing on the whole nontrivial fiber. Hence the zero set of +\(\mathcal I\) is exactly the exceptional locus. On a common +resolution principalizing \(\mathcal I\), relative generation on +the positive side identifies \(rF\) in +Equation~\eqref{prog:comparison} with the divisor of +\(\mathcal I\OO_W\). Thus \(\Supp F\) is the full inverse image +of that locus. On a common resolution of a longer negative program, +discrepancy monotonicity makes the cumulative comparison dominate +this first-step divisor. These are support statements for the actual +comparison, stronger than effectiveness and exceptionality alone. + +We next state the restricted model inputs used in this section. A nef +b-class is a Bott--Chern class that is globally nef on one fixed smooth +compact Kähler carrier, with its linear traces on the other models. The +generalized adjoint on \(T\) is +\(\alpha=\{K_T+B\}+\beta_T\). Generalized klt, abbreviated gklt, +means that all generalized log discrepancies, computed on that carrier, +are positive. The modified-bigness hypothesis below concerns the whole +boundary-plus-nef trace \(\{B\}+\beta_T\). We use the convention of +\cite[Definition~2.8]{HX}: a modified-big trace is the pushforward of +a big Bott--Chern class on a modification. The trace itself may be +a current; the generalized adjoint is required to define a +Bott--Chern class. +For a proper map $f:T\to S$, let $q_f$ be the quotient map to +$H^{1,1}_{\BC}(T,\R)/f^*H^{1,1}_{\BC}(S,\R)$. Relative +pseudo-effectivity means +$q_f(\alpha)\in\overline{q_f(\operatorname{Psef}(T))}$, where +$\operatorname{Psef}(T)$ is the absolute pseudo-effective cone. +Equivalently, for Kähler classes $\omega_T,\omega_S$, the class +$\alpha$ is pseudo-effective over $S$ if for every $\epsilon>0$ +some $c_\epsilon\geq0$ makes +$\alpha+\epsilon\omega_T+c_\epsilon f^*\omega_S$ big. +This imposes no projectivity hypothesis on $f$. For the actual +relative line classes of an already projective map, this is the +usual relative pseudo-effective cone. + +\begin{proposition}[Restricted Kähler model inputs]\label{prog:mori} +Let \((T,B+\boldsymbol\beta)\) be an effective compact Kähler gklt +pair of dimension \(d\), with \(T\) globally strongly +\(\Q\)-factorial. Suppose that its nef b-data are globally nef on a +fixed smooth carrier and that \(\{B\}+\beta_T\) is globally +modified big. Write \(\alpha=\{K_T+B\}+\beta_T\). +\begin{enumerate} +\item If \(\alpha\) is nef, there is a proper morphism with connected +fibers \(g:T\to Z\) to a normal compact Kähler space and a Kähler +class \(\omega_Z\) such that \(\alpha=g^*\omega_Z\). +\item For a proper morphism \(\pi:T\to V\) to a normal compact +Kähler space, if \(\alpha\) is pseudo-effective over \(V\), there +is a chosen good log terminal model over \(V\). It is nonextracting, +compact Kähler and globally strongly \(\Q\)-factorial. Its adjoint +is nef over \(V\), and on a common projective resolution its comparison +with \(\alpha\) is effective and exceptional over the new model, +with strictly positive coefficient at every prime contracted from \(T\). +The construction retains compatible forward Bott--Chern traces of the +specified data on a common smooth carrier. +\item A compact polytope of these data on one fixed carrier has +finitely many marked relative canonical models and relative weak log +canonical models with normal compact Kähler targets. +\end{enumerate} +The relative assertion applies to proper morphisms, without assuming +that \(\pi\) is projective. It is an assertion about a chosen model, +not termination of every generalized flip sequence. +\end{proposition} + +\begin{proof} +These are the semiampleness, proper relative model and finite-model +assertions of Theorem~\ref{prog:restricted-package}, proved below +by dimension induction independently of \(\Ggood\). +Finite marked weak-model geography is recorded in +Lemma~\ref{prog:finite-geography}. Projective analytic model +constructions used in that induction are supplied by +Proposition~\ref{prog:projective-good-model}. None of these statements +promotes an undetected generalized-ray contraction to a projective map. +\end{proof} + +We use separately the analytic cone theorem +\cite[Theorem~1.3]{HX}: an \(\alpha\)-negative analytic extremal ray +of an effective gklt adjoint has a rational curve generator \(C\) with +\(0<-\alpha\cdot C\leq2d\). Existence of the supporting contraction +in the applications below follows from the nef assertion above. +When an ordinary rational adjoint is negative on that ray, +Lemma~\ref{prog:detected-projectivity} and +Proposition~\ref{prog:detected-flip} give its projective contraction and ordinary +flip. We call these \emph{detected ordinary steps}. + +The Bott--Chern transform through such a birational step has a concrete +description which does not require the source to be smooth. Write the +step as \(T\xrightarrow{f}Z\xleftarrow{f^+}T^+\), with +\(f^+=\id\) for a divisorial contraction, and let \(J\) be its +negative rational adjoint. For \(\gamma\in H^{1,1}_{\BC}(T,\R)\), +choose the unique real number \(c\) for which +\(\gamma-c\{J\}\) annihilates the contracted analytic ray. +Every contracted curve is on that ray. Lemma~\ref{prog:bc-descent} +therefore gives a unique class \(\delta\) on \(Z\) with +\(\gamma-c\{J\}=f^*\delta\), and we put +\[ + \gamma^+=(f^+)^*\delta+c\{J^+\}. +\] +The rational-singularity and vanishing hypotheses of that lemma hold: +slightly lowering the rational Cartier floor gives a klt adjoint still +antiample over \(Z\), and relative vanishing gives rational +singularities on the base. On a common resolution, the pullback +difference is \(c\) times the adjoint comparison in +Equation~\eqref{prog:comparison}; it is exceptional over the new +model, and over both models in a flip. This defines a linear transform. +For the Chern class of a rational line it agrees with its reflexive trace, +by the coherent line comparison \cite[Lemma~7.2]{CB} and exceptional +negativity. The already-projective relative program below only needs a +ray for degrees of global rational lines; this Bott--Chern construction +is used for its subsequent detected analytic-ray programs. + +When such a class difference lies in an exceptional divisor span, we write +it as \(\{F\}\) for the representing real exceptional divisor \(F\). +This divisor is unique: an exceptional divisor with zero class has zero +degree on every contracted curve, and exceptional negativity applied to +both signs makes it zero. + +We also use local projective analytic results. For a projective +analytic morphism over a Stein neighborhood of a compactum with +Fujino's property (P), relative klt base point freeness says that, if +\(L\) is a relatively nef Cartier line and +\(aL-(K+\Gamma)\) is relatively ample for some positive integer +\(a\), then all sufficiently high powers of \(L\) are relatively +generated after shrinking +\cite[Theorems~6.2 and 6.5]{FujinoAnalyticBCHM}. +We use the finite negative-ray truncation in the relative cone theorem +in the same setting. Finally, the multigraded adjoint finite-generation +theorem applies to a projective morphism from a smooth space, for simultaneous +effective SNC klt boundaries with a common relatively ample rational +summand \cite[Theorem~3.1]{DHP}. Finitely many smaller Stein +neighborhoods suffice over a compact base. + +\begin{lemma}[Descent without changing a Cartier exponent] +\label{prog:integral-descent} +Let \(f:T\to Z\) be a projective bimeromorphic contraction with normal +target. Suppose an ordinary rational klt adjoint is \(f\)-antiample. +If a Cartier line \(L\) has degree zero on every contracted curve, +then \(f_*L\) is an invertible sheaf and evaluation is an isomorphism +\[ + f^*(f_*L)\simeq L. +\] +The same conclusion holds for an ordinary dlt adjoint whose floor is +rational Cartier and which is \(f\)-antiample. In a flip, a line with +zero contracted degree therefore has the same Cartier exponent on +both sides, and the two lines are pullbacks of a line on the common +base. +\end{lemma} + +\begin{proof} +For dlt input, decrease the floor by a sufficiently small positive +rational multiple. The resulting pair is klt and remains +\(f\)-antiample. For this klt pair the base point free hypothesis +holds for \(L\), because \(L\) has zero fiber degrees and the negative +adjoint is relatively ample. Over a smaller base neighborhood every +sufficiently high power of \(L\) is therefore generated. The +associated morphism is constant on each connected fiber: its +tautological line has degree zero on every fiber curve, and every +positive-dimensional projective image contains a curve. Its graph +then factors through \(Z\); the graph projection is finite +bimeromorphic and \(Z\) is normal. Descend two consecutive generated +powers \(L^k,L^{k+1}\) to lines \(M_k,M_{k+1}\). The line +\(M_{k+1}\otimes M_k^{-1}\) pulls back to \(L\). Projection formula +identifies this quotient with +\(f_*L\), so these local descents and their evaluation maps agree on +overlaps. Pull the descended line to the positive side of a flip. +This is the argument of \cite[Lemma~7.4]{CB}. +\end{proof} + +\subsection{Relative klt programs and small models} + +The local finite-generation theorem also supplies an ordinary klt program +over a compact base. We use it to obtain globally strongly +\(\Q\)-factorial small models of strata. The construction keeps track of +degrees of global rational lines, which are the degrees needed for exact +Cartier descent. + +\begin{proposition}[Relative klt programs over a compact base] +\label{prog:relative-klt} +Let \(\pi:T\to V\) be projective bimeromorphic, where \(V\) is a +normal compact Kähler space and \(T\) is normal, compact Kähler, and +globally strongly \(\Q\)-factorial. Let \((T,B)\) be an effective +rational klt pair with rational-line adjoint \(J=K_T+B\). +There is a finite sequence of ordinary \(J\)-negative steps over +\(V\), all projective over \(V\), whose last adjoint is curve-nef +over \(V\), meaning that it has nonnegative degree on every curve +contracted over \(V\). Every working space is compact Kähler and globally +strongly \(\Q\)-factorial. On a common resolution the comparison +from the first adjoint to the last is effective and exceptional over +the last space. +\end{proposition} + +\begin{proof} +We give the reduction to the local finite-generation input, including why +it yields a single finite program over the compact base. Let +\(\mathcal N_{\R}\) be the finite-dimensional space of degrees of global +rational lines on curves over \(V\), and put \(r=\dim\mathcal N_{\R}\). +If \(r=0\), then \(J\) is already curve-nef. Assume \(r>0\). Choose +rational lines \(H_1,\ldots,H_r\) whose degree classes form a basis and +such that both \(H_i\) and \(J+H_i\) are relatively ample. Such a basis +exists because the relatively ample classes whose sum with \(J\) is also +relatively ample form a nonempty open set containing sufficiently positive +classes. Define the rational simplex of adjoints +\[ + \Pi=\operatorname{conv}\{J,J+H_1,\ldots,J+H_r\}. +\] +We will choose a general direction \(H=\sum_i\theta_iH_i\) with +\(\theta_i>0\) and \(\sum_i\theta_i=1\). Its entire scaling segment lies +in this simplex: +\[ + J+tH=(1-t)J+\sum_i t\theta_i(J+H_i)\in\Pi + \qquad(0\leq t\leq1), +\] +and it lies in the relative interior for \(0t\) for which \(J+sH\) is +relatively ample. On the zero face of the normalized compact curve slice, +\[ + J=-\frac{t}{s-t}(J+sH) +\] +in degrees. Thus \(J\) is uniformly negative on that face. Choose the +positive ample truncation so that its compact remainder \(K\) misses the +face. The finite rational polyhedral test and the general choice of \(H\) +make the face a single ray \(R\). The wall class has a positive minimum on +\(K\) and is positive on every other truncated curve generator. Hence a +sufficiently small open neighborhood of \(w\) inside the annihilator +\(R^\perp\) consists of nef classes with zero face exactly \(R\). + +Suppose first that \(R^\perp\) has positive dimension. It is a rational +hyperplane because \(R\) is represented by an integral curve. Choose a +rational simplex of support classes in that neighborhood containing the +degree class of \(w\) in its relative interior, and represent each vertex +by a rational line \(N\). For each such \(N\), compactness gives an +integer \(a>0\) for which \(aN-J\) is relatively ample. Relative base +point freeness contracts exactly \(R\) and descends \(N\) to a relatively +ample rational line on the contraction base. These bases agree because +the contracted curves agree. The difference between \(w\) and the +corresponding convex combination of these rational lines has zero +relative degree. In the finite rational span of the lines involved, the +degree map has rational kernel, since rational lines have rational degrees +on integral curves. The difference is therefore a real combination of +rational lines of zero relative degree. Lemma~\ref{prog:integral-descent} +descends these lines, and lifting curves from the base shows that their +descents still have zero degree over \(V\). They do not affect relative +ampleness. Convexity now shows that \(w\) descends to a relatively ample +real line class over \(V\). If \(R^\perp=0\), the same zero-degree descent +applies to \(w\); the contraction base has no curves over \(V\), hence is +finite over \(V\), and relative ampleness is automatic. + +After a flip, let \(w^+\) be the pullback of this wall class to the positive +side. For sufficiently small \(\delta>0\), +\[ + J^++(t-\delta)H^+ + =\left(1-\frac{\delta}{t}\right)w^+ + +\frac{\delta}{t}J^+ +\] +is relatively ample over \(V\): the wall is pulled back from a relatively +ample class on the contraction base, and \(J^+\) is relatively ample over +that base. After a divisorial contraction, openness of the ample cone on +the base gives the same nonempty interval of relative ampleness. + +Each working model is consequently one of the marked ample models +already counted. Indeed, for a fixed finite prefix and an interior +parameter \(00\) for which +\((S,\Delta-\epsilon D)\) is effective and klt. On a projective log +resolution \(r:W\to S\), use the strict lowered boundary and give +every exceptional prime a coefficient strictly between its crepant +coefficient and \(1\), and at least zero. The resulting klt adjoint +has the actual form +\[ + K_W+\Gamma\sim_{\Q}r^*(K_S+\Delta-\epsilon D)+F, + \qquad F\geq0, +\] +where every \(r\)-exceptional prime has positive coefficient in +\(F\). Apply Proposition~\ref{prog:relative-klt} over \(S\). +On its endpoint the trace \(F_Y\) is effective, exceptional over +\(S\), and relatively nef. Negativity makes \(F_Y=0\). +Since all initially exceptional primes had positive coefficient, +\(Y\to S\) is small and the lowered adjoint is crepant. +Restoring \(\epsilon D\) gives the full crepant identity. + +A projective small morphism to a smooth germ is an isomorphism: +transport a relatively ample line to the smooth germ, where it is +Cartier; smallness makes the original line its pullback, contradicting +relative ampleness on a nontrivial fiber. Thus \(h\) is an isomorphism +over an SNC open meeting all lc centers of the dlt pair. The full +crepant comparison then proves dlt on \(Y\). +\end{proof} + +\subsection{Scaling toward zero and keeping a nef rational line} + +The next construction runs every positive truncation of an ordinary +dlt scaling. It also explains why a sufficiently large multiple of +a prescribed nef rational line forces every step to be trivial for +that line. The integer that clears the line will be the same at +every step. + +For the detected ordinary scalings constructed below, let +\(\alpha_0(\lambda)=\{H\}+\lambda\omega\) on the first space +\(T_0\), with initial parameter \(1\), and let \(\alpha_i(\lambda)\) +be its linear transform on a working model \(T_i\). Put +\(\lambda_{-1}=1\). If the next step occurs at threshold +\(\lambda_i\), its ray is annihilated by \(\alpha_i(\lambda_i)\) +and is negative for \(\alpha_i(0)\). The trace \(\alpha_i(\lambda)\) +is nef on \([\lambda_i,\lambda_{i-1}]\), which we call its +\emph{working interval}. If \(\alpha_i(0)\) is nef, we set +\(\lambda_i=0\) and stop. Working intervals are allowed to be +degenerate; strict decrease of positive thresholds will be arranged only +in the proof of special termination. + +\begin{lemma}[Positive truncations and a fixed nef line] +\label{prog:nef-trivial} +Let \((T,B)\) be an effective rational dlt pair on a globally strongly +\(\Q\)-factorial compact Kähler \(d\)-fold, and put \(J=K_T+B\). +\begin{enumerate} +\item For a sufficiently large Kähler class \(\omega\), there is an +ordinary \(J\)-program with scaling of \(\omega\), starting with +\(\{J\}+\omega\) Kähler. Every positive truncation is finite. +If \(J\) is pseudo-effective, the program either reaches a nef adjoint +or has positive thresholds tending to zero. If \(J\) is not +pseudo-effective, the program is finite and ends with a Mori fiber +contraction. +\item Let \(P\) be a nef rational line with \(mP\) Cartier, and choose +\(b\in\Q\) with \(b>4dm\). The same alternatives hold for +\(H=J+bP\), according to pseudo-effectivity of \(H\), with +\(\omega\) enlarged if necessary. Every birational step is an ordinary +\(J\)-negative step on which \(P\) is trivial, and the final Mori +ray, if present, is also \(P\)-trivial. The same line \(mP\) descends +through every birational contraction and pulls back to its positive +side. Its traces remain analytically nef; if \(P\) is semiample, its +same generated multiple and associated morphism are preserved, also +through the final Mori contraction. +\end{enumerate} +\end{lemma} + +\begin{proof} +Write \(H=J\) in the first case and \(H=J+bP\) in the second. +Fix \(\omega\) with \(\{H\}+\omega\) Kähler. For each +\(0<\eta<1\), choose a positive rational \(\epsilon\) so small that, +with \(F=\lfloor B\rfloor\), +\[ + (T,B-\epsilon F)\ \text{is klt},\qquad + \eta\omega+\epsilon\{F\}\ \text{is Kähler}. +\] +For \(\lambda\in[\eta,1]\) use the presentations +\[ + \{H\}+\lambda\omega + =\{K_T+B-\epsilon F\} + +\bigl(b\{P\}+\lambda\omega+\epsilon\{F\}\bigr), +\] +omitting \(b\{P\}\) in the first case. Their b-data are globally +nef on the fixed initial carrier, and their boundary-plus-nef traces +are globally modified big. These are effective gklt data. + +We first construct a step whenever the unscaled working trace is not +nef. Choose a positive shift strictly below the current wall and use the +gklt presentation of its current trace \(A\). The ray at the wall is +\(A\)-negative. On a compact slice of the analytic curve cone, the +cone theorem makes the \(A\)-negative part locally polyhedral after +a sufficiently small positive Kähler truncation. Separating the +chosen extremal ray from the other finitely many rays in that +truncation gives a Kähler class \(\omega_R\) such that +\(A+\omega_R\) is nef and its zero face is precisely this ray; +see the supporting construction in Proposition~\ref{prog:ordinary-step}. +Add the pullback of this new Kähler class to the fixed nef carrier. +The resulting adjoint still has globally nef data and modified-big +boundary-plus-nef trace, so the nef assertion of +Proposition~\ref{prog:mori} supplies its supporting contraction. +This uses a new supporting direction on the current space; it does +not require the trace of the original scaling direction to be Kähler. +We verify a rational ordinary detector before asserting projectivity +or constructing a flip. +In the second case nefness of \(P\) makes \(J\) negative on the ray; +the first case has this property directly. Lower the floor by a small +rational amount, retaining negativity of \(J_\epsilon\), so that +\(-J\cdot R\leq2(-J_\epsilon\cdot R)\). The ordinary klt cone +length bound gives a rational curve generator \(C\) with +\[ + 0<-J\cdot C\leq4d. +\] +In the second case, if \(P\cdot C>0\), integrality of \(mP\) gives +\(P\cdot C\geq1/m\), contradicting +\[ + H\cdot C\geq-4d+b/m>0. +\] +Thus in that case the ray is \(P\)-trivial. The lowered ordinary adjoint is a +global rational detector. The detected projectivity and flip +constructions in Lemma~\ref{prog:detected-projectivity} +and Proposition~\ref{prog:detected-flip} supply the projective ordinary step. +The difference of \(J\) and a positive rational multiple of the +lowered adjoint has zero contracted degree; exact descent in +Lemma~\ref{prog:integral-descent} shows that restoring the floor +preserves the positive-side sign. This is the ordinary replacement +of Proposition~\ref{prog:ordinary-replacement}, and the full pair +remains dlt by Equation~\eqref{prog:comparison}. + +In the second case, Lemma~\ref{prog:integral-descent} descends the actual Cartier line +\(mP\) without changing \(m\). Analytic nefness descends and pulls +back under these projective contractions \cite[Lemma~2.44]{HX}. +A generated multiple descends as a generated line by projection +formula and pulls back as one. Thus the same length estimate and the +same \(b\) apply at the next step. The corresponding assertion on +a final Mori ray follows from exactly the same estimate. For a +semiample \(P\), the map of its fixed generated multiple is constant +on each connected projective contracted fiber: otherwise its image +contains a curve of positive degree. The map therefore factors through +the normal contraction base, including in the fiber-type case. + +The traces used in this construction are linear by the degree-zero +descent description preceding Lemma~\ref{prog:integral-descent}. +At a wall the scaled class is pulled back from its base. The comparison +for an earlier parameter is a nonnegative multiple of the ordinary +adjoint comparison. Consequently the gklt presentations on any fixed +positive segment remain gklt along its program, and every working +model is a weak model for a member of that segment. The finite marked +weak-model assertion in Proposition~\ref{prog:mori} applies. +A marked model cannot recur: its boundary trace would be the same, +whereas a nontrivial intervening ordinary step strictly increases a +discrepancy. There are therefore only finitely many steps with +threshold at least \(\eta\). + +For decreasing cutoffs, any already-constructed finite prefix remains +valid. The floor perturbations cancel in the displayed classes and +continue to cancel under their linear transforms; their nonpositive +comparisons preserve the new gklt data. Starting at its last nef wall +therefore extends the prefix to the smaller cutoff. This gives one +compatible scaling program. A positive limiting threshold would lie +in a finite positive truncation, so an infinite program has limit zero. +Pseudo-effectivity excludes a negative Mori endpoint. If \(H\) is +not pseudo-effective, its pseudo-effective threshold in the initial +Kähler direction is positive. Every nef working trace, together with +its effective exceptional comparison, makes the initial scaled class +pseudo-effective, so the scaling thresholds are bounded below by +that positive number. The program is therefore finite. Its endpoint +cannot be nef, since that comparison would make \(H\) +pseudo-effective; it is the stated Mori fiber contraction. +\end{proof} + +We call these programs \emph{detected ordinary scalings}. Their steps +are ordinary negative steps of one analytic extremal ray, with the +linear Bott--Chern transform described above. +The next lemma connects their nef working traces +to the analytic negative part, including its limit at parameter zero. + +\begin{lemma}[Negative parts along a detected ordinary scaling] +\label{prog:negative-along-scaling} +In a scaling from Lemma~\ref{prog:nef-trivial}, write \(H=J\) in its first +case and \(H=J+bP\) in its second, and use the notation +\(\alpha_0(\lambda)=\{H\}+\lambda\omega\) above. For a parameter +\(\lambda\) in the working interval of \(T_i\), a common smooth +resolution \(p:W\to T_0\), \(q:W\to T_i\) gives +\begin{equation} + p^*\alpha_0(\lambda)=q^*\alpha_i(\lambda)+\{E_i(\lambda)\}, + \qquad E_i(\lambda)\geq0,\qquad q_*E_i(\lambda)=0. + \label{prog:working-negative} +\end{equation} +Here \(E_i(\lambda)\) is an actual real exceptional divisor, and +\[ + N(p^*\alpha_0(\lambda))=E_i(\lambda). +\] +For the limit statement, let \(H\) be any rational line on a normal compact +Kähler space \(T\) whose pullback to a smooth resolution is +pseudo-effective, and let \(A\) be a Kähler class on \(T\). On any fixed +smooth resolution \(r:U\to T\) and for every prime \(Q\subset U\), +\[ + \lim_{\lambda\downarrow0}\nu_Q(r^*(\{H\}+\lambda A)) + =\nu_Q(r^*\{H\}). +\] +Thus the same convergence holds for every divisorial place, after choosing +a resolution on which it appears. +\end{lemma} + +\begin{proof} +For \(\lambda\) in the working interval of \(T_i\), every earlier +threshold is at least \(\lambda\). Each earlier step is therefore negative +or trivial for this parameter. Composing its comparisons gives +Equation~\eqref{prog:working-negative}. Since \(\alpha_i(\lambda)\) is +nef, Lemma~\ref{bir:exceptional} identifies the exceptional divisor with +the negative part. + +On the fixed resolution \(U\), choose a Kähler class \(\Omega\) and +\(c>0\) such that \(c\Omega-r^*A\) is Kähler. Monotonicity of each +minimal multiplicity under addition of a nef class gives +\[ + \nu_Q(r^*\{H\}+\lambda c\Omega) + \leq \nu_Q(r^*(\{H\}+\lambda A)) + \leq \nu_Q(r^*\{H\}). +\] +The left side tends to the right side by the definition using small Kähler +perturbations. This proves the convergence on \(U\), and the same argument +applies on a resolution representing any chosen divisorial place. +\end{proof} + +\begin{proposition}[Contracting a known negative part] +\label{prog:contract-negative} +Let \((T,B)\) be an effective rational dlt pair on a globally +strongly \(\Q\)-factorial compact Kähler space, and put \(J=K_T+B\). +Suppose there is an actual rational-line identity +\[ + J\sim_{\Q}P+E, +\] +where \(P\) is analytically nef, \(E\) is an effective rational +Cartier divisor, and, on one smooth compact Kähler resolution +\(\pi:W\to T\) projective over \(T\), +\[ + N(\pi^*\{J\})=\pi^*E. +\] +There is a finite ordinary \(J\)-negative program to a nef model +\(T'\) which contracts precisely the prime components of \(E\) +among the primes of \(T\). Every step is \(P\)-trivial, with a +fixed Cartier exponent as in Lemma~\ref{prog:nef-trivial}, and +\[ + J_{T'}\sim_{\Q}P_{T'}. +\] +If \(P\) is semiample, then so is this actual last adjoint. +\end{proposition} + +\begin{proof} +Lemma~\ref{bir:pullup} makes the stated equality valid on any +higher resolution. Choose \(b\) as in +Lemma~\ref{prog:nef-trivial}. Adding \(bP\) leaves the same +negative part. Indeed, on a smooth resolution write +\(\alpha=\pi^*\{J\}\), \(p=\pi^*\{P\}\), and \(e=\pi^*E=N(\alpha)\). +Subadditivity and homogeneity of negative multiplicities give +\begin{align*} + N(\alpha+bp)&\leq e,\\ + (1+b)e=N((1+b)\alpha) + &\leq N(\alpha+bp)+bN(\{e\}) + \leq N(\alpha+bp)+be. +\end{align*} +Thus \(N(\alpha+bp)=e\). + +Run the scaling for \(H=J+bP\). +Let \(Q\) be the strict transform on a fixed resolution of a prime +component of \(E\). Its multiplicity in \(N(\pi^*\{H\})\) is +positive. The limit in Lemma~\ref{prog:negative-along-scaling} makes its multiplicity +positive for every sufficiently small positive parameter. If +that prime still survived on the corresponding working model, +Equation~\eqref{prog:working-negative} would make its multiplicity zero, +because the comparison there is exceptional over that model. +It must therefore be contracted after finitely many steps. The +same conclusion holds if a nef endpoint is reached before taking +the limit. There are only finitely many components of \(E\), so +after a finite prefix all have disappeared. + +Throughout the program \(P\) descends and remains nef. The +actual identity \(J_i\sim_{\Q}P_i+E_i\), with \(E_i\) the +codimension-one pushforward of \(E\), follows from the reflexive +trace convention. Once \(E_i=0\), we have \(J_i\sim_{\Q}P_i\); since +\(P_i\) is nef, the program stops. Conversely, a divisorial negative step can +contract only a prime of \(E_i\): its ray is \(P_i\)-trivial, +so the effective divisor \(E_i\) has negative degree on every +contracted curve, and those curves are contained in its support. +This proves the exact assertion about contracted primes. +\end{proof} + +The next lemma records the cohomology carried by a detected ordinary scaling +that begins on a smooth space. Its surjectivity will let us choose one +generic initial scaling direction for all later models. Its exceptional +splitting will also carry the span of line classes to the terminal models +used in Section~\ref{sec:boundary}. + +\begin{lemma}[Cohomology through ordinary steps] +\label{prog:cohomology-transport} +Let a finite prefix of one of the detected ordinary scalings constructed +in Lemma~\ref{prog:nef-trivial} start on a smooth compact Kähler +space, and let \(V\) be any working space. +For every smooth compact Kähler resolution \(p:U\to V\) +projective over \(V\), +\begin{align} + H^2(U,\R) + &=p^*H^2(V,\R)\ \oplus\ + \bigoplus_{Q\ {\rm exceptional\ for}\ p}\R\{Q\}, \label{prog:H2}\\ + H^{1,1}(U,\R) + &=p^*H^{1,1}_{\BC}(V,\R)\ \oplus\ + \bigoplus_{Q\ {\rm exceptional\ for}\ p}\R\{Q\}. + \label{prog:H11} +\end{align} +The linear transform from the first space is surjective onto +\(H^2(V,\R)\) and \(H^{1,1}_{\BC}(V,\R)\), respectively. For a small correspondence between +working spaces, the difference of the pulled-back classes and +their transforms on a common resolution lies in the span of the +primes exceptional over both spaces. +Under the displayed decompositions, the Chern class of a global +holomorphic line on \(U\) is the sum of the pullback of a rational +holomorphic line class on \(V\) and an exceptional rational divisor +class. In particular, passage to the exceptional quotient preserves +the real span of line Chern classes. +\end{lemma} + +\begin{proof} +For a modification of a smooth compact Kähler manifold, the +degree-two modification theorem gives these decompositions, and +the exceptional classes have type \((1,1)\). Suppose they hold +at one step. Its contraction base and both working spaces have +rational singularities: slight lowering of the rational Cartier +floor makes the ordinary pair klt, and the detected contraction has +a base with rational singularities. These contractions are +bimeromorphic, so a common resolution and Leray give +\(R^if_*\OO=0\) for \(i>0\); this verifies the additional +hypothesis in Lemma~\ref{prog:bc-descent}. + +The ray of this detected contraction is one ray for Bott--Chern degrees. +This also suffices for degree-two classes. Indeed, pull a class +\(\gamma\in H^2(V,\R)\) to a smooth resolution \(p:U\to V\) +and take its Hodge decomposition. By the decomposition already proved for +this working model in Equation~\eqref{prog:H11}, its \((1,1)\)-part is \(p^*\beta+\{E\}\) for a +Bott--Chern class \(\beta\) and an exceptional real divisor +\(E\). The other Hodge components have zero degree on curves. +On every \(p\)-contracted curve the degree of \(E\) is therefore +zero, so both signs of negativity make \(E=0\). Lifting any +curve of \(V\) through the projective resolution now shows that +\(\gamma\) and \(\beta\) have the same curve degrees. Thus +subtracting a multiple of \(\{J\}\) from either a degree-two or +a Bott--Chern class can annihilate every curve of the contracted +fiber. By +Lemma~\ref{prog:bc-descent}, a degree-two class, or a Bott--Chern class, +with these zero degrees is pulled back from the base. +Subtract a multiple of \(\{J\}\) to make any given class +annihilate that ray, descend the remainder, and use the trace +\(\{J^+\}\) to define its transform. On a common resolution the +difference is a multiple of the exceptional adjoint comparison in +Equation~\eqref{prog:comparison}. For a flip the two exceptional prime +sets on that resolution are the same. For a divisorial +contraction the target resolution has just the one additional +exceptional prime of the step. The old decomposition therefore +spans the new one and proves surjectivity of the transform. + +The sums are direct. If the pullback of a class from \(V\) is +an exceptional divisor class, that divisor has zero degree on +all curves over \(V\). Applying exceptional negativity to both +signs makes the divisor zero; injectivity of pullback gives that +the class is zero. The pullback injectivity in degree two and in +Bott--Chern cohomology is also +Lemma~\ref{prog:bc-descent}, applied to the resolution. Passing to +higher smooth resolutions proves the assertion for every \(p\). + +For the line assertion, let \(\mathscr H\) be a holomorphic line on +\(U\), and put \(\mathscr Q=(p_*\mathscr H)^{**}\). Global strong +\(\Q\)-factoriality gives an invertible sheaf +\(\mathscr M=\mathscr Q^{[m]}\) for some \(m>0\). The coherent +evaluation comparison of \cite[Lemma~7.2]{CB} gives +\[ + \mathscr H^{\otimes m}\simeq p^*\mathscr M\otimes\OO_U(E), + \qquad E\ \text{an integral \(p\)-exceptional divisor}. +\] +Indeed \((p_*\mathscr H^{\otimes m})^{**}=\mathscr M\), since the +two sheaves agree at the general points of all divisors of \(V\). +Local generators of \(p_*\mathscr H^{\otimes m}\) in a frame +of \(\mathscr M\) compare their evaluations with the pulled-back +coefficients by meromorphic quotients. Those quotients agree on the +isomorphism locus and hence glue; their zeros and poles are exceptional. +Taking Chern classes proves the assertion without choosing a meromorphic +section of \(\mathscr H\). +\end{proof} + +Before returning to special termination, we record the consequence used in +the boundary argument: a lower-dimensional manifold of Kodaira dimension +zero has a terminal torsion model carrying the preceding exceptional +splitting. + +\begin{corollary}[Terminal models in Kodaira dimension zero] +\label{prog:terminal-kappa-zero} +Let \(X\) be a smooth compact Kähler \(d\)-fold with \(d0\), and transformed classes +\(\{J_V\},A_V\) such that +\begin{equation} + \{J_V\}+tA_V\ \text{is nef for every }0\leq t\leq\delta. + \label{prog:whole-nef-interval} +\end{equation} +On a common smooth resolution of \(S\) and \(V\), the +differences of pullbacks of \(J,A\) and these transforms lie +in the common exceptional span. +\end{lemma} + +\begin{proof} +Let \(P_M=J_M\) denote the semiample rational line from +Lemma~\ref{prog:stratum-model}, and use its transform \(A_M\). +On a common resolution \(p:W\to S,\ q:W\to M\), let \(F_A\) be the +real divisor in the common exceptional span defined by +\[ + q^*A_M-p^*A=\{F_A\}. +\] +It is effective: \(-F_A\) is \(q\)-nef, because \(p^*A\) is +nef and the other term is pulled back from \(M\); apply +exceptional negativity. Its support is disjoint from \(p^{-1}(U)\). +In particular \(A_M\) is pseudo-effective, and its negative +multiplicity at every prime of \(M\) is zero. Indeed, +\(q^*A_M=p^*A+\{F_A\}\), and +Lemma~\ref{bir:exceptional} applied over \(S\) gives +\(N(q^*A_M)=F_A\). + +Let \(D_M\) be the rational Cartier trace of \(D\). +Choose \(\tau>0\) so small that \(A+\tau\{D\}\) is Kähler. +For a fixed sufficiently small \(t>0\), consider on \(M\) the +boundary \(\Delta_M-t\tau D_M\) and the nef b-class represented on +\(W\) by \(t p^*(A+\tau\{D\})\). Its trace on \(M\) is +\(t(A_M+\tau\{D_M\})\), so its generalized adjoint is +\begin{equation} + \{P_M\}+tA_M. + \label{prog:interval-gadjoint} +\end{equation} +These are effective gklt data. Over \(U\) the b-class +descends to \(M\), so its generalized discrepancy +contribution is zero there, and subtracting \(t\tau D_M\) +raises every zero log discrepancy of the dlt pair. +Away from the floor the ordinary pair is klt; on one fixed +log resolution its finitely many subunit coefficients stay +subunit for small \(t\). This proves gklt. The generalized +boundary is big. On a common resolution the pullback of the +trace of \(A+\tau\{D\}\) minus \(p^*(A+\tau\{D\})\) is effective exceptional +by the same negativity argument as for \(F_A\). The latter +pullback is nef and big, so the trace is big; the remaining +boundary is effective. + +Choose an integer \(m>0\) such that \(mP_M\) is generated, and +let \(g:M\to Z\) be the Stein factor of its morphism. Then \(Z\) +is normal projective and \(mP_M=g^*H\) for an ample generated Cartier line +\(H\) on \(Z\). Set \(L=\{P_M\}+tA_M\). The preceding +preparation makes \(L\) an effective gklt adjoint with globally nef +b-data and globally modified-big boundary-plus-nef trace. It is +pseudo-effective because \(P_M\) is nef and \(A_M\) is modified +nef. Apply the proper relative assertion of +Proposition~\ref{prog:mori} over \(Z\). This produces a chosen +nonextracting good log terminal model \(\phi:M\dashrightarrow V\) +and a morphism \(g_V:V\to Z\), with \(L_V\) nef over \(Z\). +The hypothesis here is properness: the semiample morphism \(M\to Z\) +need not be projective. Put +\[ + mP_V=g_V^*H,\qquad A_V=(L_V-\{P_V\})/t. +\] +The relative construction preserves the actual line pulled back from +\(Z\), and its forward Bott--Chern transport preserves the displayed +linear relation; these are the transforms of \(P_M,A_M\). + +The map \(\phi\) is small. On a common projective resolution +\(a:W'\to M\), \(c:W'\to V\), its comparison is +\[ + a^*L=c^*L_V+\{E\},\qquad E\geq0, +\] +where \(E\) is exceptional over \(V\) and has positive coefficient +at the strict transform of each prime contracted from \(M\). +The forward trace \(L_V\) is pseudo-effective, so exceptional +translation gives +\[ + N(a^*L)=N(c^*L_V)+E\geq E. +\] +But \(L\) is modified nef on \(M\); the coefficient of +\(N(a^*L)\) at a strict prime of \(M\) is zero. Thus no prime of +\(M\) is contracted. Nonextraction then proves smallness. + +Fix \(b>2dm\). We claim that \(L_V+b\{P_V\}\) is absolutely +nef. Otherwise the cone theorem for the effective gklt adjoint +\(L_V\) gives an \(L_V\)-negative rational extremal generator +\(C\) such that +\[ + (L_V+b\{P_V\})\cdot C<0,\qquad 0<-L_V\cdot C\leq2d. +\] +Indeed the cone summand on which \(L_V\) is nonnegative also has +nonnegative \(P_V\)-degree, so a negative ray summand must account +for any failure of nefness. If \(P_V\cdot C=0\), ampleness of +\(H\) makes \(C\) vertical over \(Z\), contradicting relative +nefness of \(L_V\). Otherwise Cartier integrality gives +\(P_V\cdot C\geq1/m\), contradicting +\[ + (L_V+b\{P_V\})\cdot C\geq-2d+b/m>0. +\] +This proves the claim. Both \(\{P_V\}\) and +\((1+b)\{P_V\}+tA_V\) are nef. By convexity, +\[ + \{P_V\}+sA_V\ \text{is nef for }0\leq s\leq\frac{t}{1+b}. +\] +Thus \(\delta=t/(1+b)\) works. The actual line \(P_V\) has the +same generated multiple as \(P_M\), since both are pulled back from +\(H/m\). Its class is the transform of \(J\). +The relative comparison and linear transport give exceptional-divisor +comparisons for \(P_M,A_M\). Since \(S,M,V\) agree in codimension +one, their final comparison divisors are exceptional over both \(S\) +and \(V\), as required. +\end{proof} + +\subsection{Special termination} + +\begin{theorem}[Special termination for a chosen dlt scaling] +\label{prog:special-termination} +Assume \(\Ggood_d\) for every \(d0\) on this working space, above the remaining +walls, and set +\[ + A=\{J\}+\lambda_0H. +\] +It is Kähler, as the restriction of the interior ambient +Kähler class. The exact change of parameter is +\begin{equation} + \{J\}+tA=(1+t)(\{J\}+\lambda H),\qquad + \lambda=\frac{\lambda_0t}{1+t},\qquad + t=\frac{\lambda}{\lambda_0-\lambda}. + \label{prog:reparameterization} +\end{equation} +It applies for \(0\leq\lambda<\lambda_0\). The new walls +\(t_i\) strictly decrease toward zero, and the traces +\(\{J_i\}+tA_i\) are nef on their successive restricted +working intervals. + +On a common resolution of a nontrivial small stratum step, let \(D_i\) +be the real exceptional divisor defined by +\[ + p_i^*\{J_i\}-q_i^*\{J_{i+1}\}=\{D_i\}. +\] +The strict adjoint comparison makes \(D_i\) an effective +nonzero divisor exceptional over both strata. Equality of +the scaled pullbacks at its wall gives the exact +Bott--Chern equality +\begin{equation} + p_i^*(\{J_i\}+tA_i)-q_i^*(\{J_{i+1}\}+tA_{i+1}) + =\left(1-\frac{t}{t_i}\right)\{D_i\}. + \label{prog:restricted-comparison} +\end{equation} +For \(t\) in a later working interval all preceding factors +on the right are nonnegative. Telescoping on a common +resolution expresses the pullback of \(\{J\}+tA\) as the +pullback of its nef working trace plus an effective divisor +exceptional over that working stratum. +Lemma~\ref{bir:exceptional} identifies the latter divisor +with the negative part. + +Push the resulting positive currents to one fixed smooth +resolution of \(S\) and let \(t\downarrow0\). Closedness of +the pseudo-effective cone proves that \(J\) is +pseudo-effective. The negative multiplicities converge +valuation by valuation by the limit statement in +Lemma~\ref{prog:negative-along-scaling}. The comparison divisors +have no strict prime of \(S\), since the maps are small. +Their coefficients also vanish at every place whose center +meets the floor: each finite composition is unchanged on a +neighborhood of the floor by the induction on smaller +centers. Passing to the limit proves +Equation~\eqref{prog:away-floor}. + +The stratum has dimension less than \(n\), and the divisor \(D_S\) +constructed above is rational Cartier with support equal to its floor. +Lemmas~\ref{prog:stratum-model} and +\ref{prog:nef-interval} give one small model \(V\) on +which all transformed classes of \(\{J\}+tA\) are nef +for \(0\leq t\leq\delta\). On a common smooth +resolution \(p:W\to S,\ q:W\to V\), define the real divisor \(F(t)\) +in the common exceptional span by +\begin{equation} + p^*(\{J\}+tA)-q^*(\{J_V\}+tA_V)=\{F(t)\}. + \label{prog:affine-comparison} +\end{equation} +Uniqueness of the representing exceptional divisor makes \(F(t)\) +coefficientwise affine in \(t\). It is effective on +\([0,\delta]\): \(-F(t)\) is \(p\)-nef because the +class pulled back from \(V\) is nef, so exceptional +negativity applies. It is exceptional over \(V\) as +well. Lemma~\ref{bir:exceptional} gives +\[ + N\bigl(p^*(\{J\}+tA)\bigr)=F(t) + \qquad(0\leq t\leq\delta). +\] +Every divisorial negative multiplicity is thus affine on +this whole interval. This remains true on higher +resolutions by Lemma~\ref{bir:pullup}. + +Choose a nontrivial stratum wall \(t_i\in(0,\delta)\). +On a common resolution of \(S,V\) and the adjacent +strata, their nef working models compute the same +negative part on their respective open intervals. Their +two affine expressions agree at \(t_i\). But +Equation~\eqref{prog:restricted-comparison} says that, at a prime +with coefficient \(d_i>0\) in \(D_i\), their difference +is +\[ + \left(1-\frac{t}{t_i}\right)d_i. +\] +Their slopes differ by \(-d_i/t_i\). A single affine +function on \([0,\delta]\) cannot agree with both on +two open intervals. This contradiction excludes every +such wall. Infinitely many walls would tend to zero, so +the nontrivial diagrams on this stratum are finite. + +There are finitely many surviving strata. Induction on +their dimensions makes all sufficiently late ambient +flipping and flipped loci disjoint from the floor. This +proves the theorem. +\end{proof} + +\begin{corollary}[A signed adjoint supported on the floor] +\label{prog:signed-nef} +Assume \(\Ggood_d\) for every \(d0\) -\cite[Theorem~5.2]{FujinoAnalyticBCHM}. Composing a resolution of \(T\) -with \(f\), the Leray spectral sequence proves that \(Z\) has rational -singularities. The same reasoning works when the klt boundary is -chosen separately on finitely many Stein base neighborhoods. For -fiber-type contractions in case~(ii), rationality of the base follows -from \cite[Lemma~8.8(i)]{DHP}; thus that application also precedes -the construction of generalized data on the base. - -\begin{lemma}[Projectivity from a rational line detecting the ray] -\label{prog:detected-projectivity} -Let $f:T\to Z$ be a proper contraction between normal compact -K\"ahler spaces, with $T$ having rational singularities. Suppose -$\gamma$ is K\"ahler on $Z$ and -\[ - \alpha=f^*\gamma,\qquad - \overline{\mathrm{NA}}(T)\cap\alpha^\perp=R -\] -is a nonzero ray. If a global rational line bundle $D$ has -$D\cdot R<0$, then $-D$ is relatively ample and $f$ is projective. -\end{lemma} -\begin{proof} -Fix a K\"ahler class $\omega$ on $T$ and normalize $\overline{\mathrm{NA}}(T)$ by -$\omega\cdot z=1$. The resulting slice $S$ is compact. Its unique -point on $R$ has negative $D$-degree, so $d=c_1(D)$ is negative on -a neighborhood of that point in $S$. On the complementary compact -set, $\alpha$ has a strictly positive minimum, while $d$ is bounded. -For all sufficiently small $s>0$, the class $\alpha-sd$ is therefore -strictly positive on $S$ and is K\"ahler by cone duality. - -Choose an integer $m>0$ making $mD$ integral. Then -\[ - c_1(-mD)+f^*((m/s)\gamma)=(m/s)(\alpha-sd) -\] -is K\"ahler. On a neighborhood in the base with a potential for -$\gamma$, absorb that potential in a smooth metric on the same global -line bundle $-mD$. Its curvature is positive on the fibres. The analytic -positive-line-bundle criterion and the fibrewise relative-ampleness -criterion make $-mD$ relatively ample. The line bundle was global -throughout, so no gluing of unrelated local polarizations is required. -\end{proof} - - -An analytic contraction supplied by a supporting class therefore becomes -projective as soon as one rational line has negative degree on its ray. -For a small contraction, that same line will define the flip globally. -The local ordinary adjoints used to prove finite generation need not -glue; their section algebras will be Veronese subalgebras of one algebra. - -\begin{lemma}[Divisor representatives over a Stein base] -\label{prog:stein-representatives} -Let $\pi:Y\to U$ be a projective surjective morphism with connected fibres -between normal irreducible complex spaces, where $U$ is Stein, and let -$\mathcal L$ be a holomorphic line bundle on $Y$. -If $\pi$ is bimeromorphic, $\mathcal L$ has a holomorphic section -which is not identically zero, and hence an effective Cartier divisor -representative. More generally, after shrinking $U$ around any specified -point, $\mathcal L$ has a nonzero meromorphic section. -\end{lemma} - -\begin{proof} -Suppose first that $\pi$ is bimeromorphic. Grauert's proper direct-image -theorem makes $\mathcal F=\pi_*\mathcal L$ coherent. On the nonempty -open subset $U^\circ$ where $\pi$ is an isomorphism, it is a line bundle. -Choose $u_*\in U^\circ$. Cartan's theorem~A says that global sections -generate $\mathcal F_{u_*}$, so one global section has nonzero image in -$\mathcal F_{u_*}/\mathfrak m_{u_*}\mathcal F_{u_*}$. -Under -\[ - H^0(U,\pi_*\mathcal L)=H^0(Y,\mathcal L) -\] -it gives a section $s$ nonzero at the unique point above $u_*$. -Its zero divisor is effective Cartier and represents $\mathcal L$. -The section may vanish elsewhere; only its meromorphic inverse is -being used to obtain a meromorphic trivialization. - -For the general case, fix $u\in U$ and $y\in\pi^{-1}(u)$, and choose -a $\pi$-ample line bundle $\mathcal H$. Relative Serre generation, -after shrinking to a Stein neighbourhood of $u$, gives an integer -$n$ for which both $\mathcal L\otimes\mathcal H^n$ and -$\mathcal H^n$ are relatively generated. Their proper direct images -are coherent. Cartan's theorem~A and the evaluation maps supply -sections $s$ and $t$ of these two bundles which are both nonzero at -$y$. The quotient $s/t$ is the required nonzero meromorphic section -of $\mathcal L$. At no point is $Y$ assumed Stein. -\end{proof} - -\begin{proposition}[A global algebra for a detected flip] -\label{prog:detected-flip} -Let $(T,B+\mathbf M)$ be a gklt pair on a normal compact K\"ahler -space which is globally strongly $\Q$-factorial, and let $A\in H^{1,1}_{\BC}(T,\R)$ -be its adjoint class. Let $f:T\to Z$ be a projective small contraction -onto a normal compact K\"ahler space. Suppose that the classes of all -$f$-vertical curves lie on one ray $R\subset\overline{\mathrm{NA}}(T)$, that $A\cdot R<0$, -and that a global line bundle $L$ satisfies $L\cdot R<0$. - -Then $\bigoplus_{m\geq0}f_*L^m$ is locally finitely generated. The -relative Proj of a sufficiently divisible Veronese is a projective -small contraction $f^+:T^+\to Z$, where $T^+$ is compact K\"ahler and -globally strongly $\Q$-factorial. The reflexive transform $L^+$ is a -rational line bundle, with a positive power equal to the relatively -ample tautological bundle. The transformed boundary and the same -b-nef datum define a gklt pair on $T^+$, with relatively K\"ahler -adjoint $A^+$. This is the generalized flip, with the usual -strict discrepancy increase at centres in either exceptional locus. - -Moreover, if $t=(A\cdot C)/(L\cdot C)>0$ for an $f$-vertical curve $C$, -then there is a unique $\eta\in H^{1,1}_{\BC}(Z,\R)$ such that -\[ - A=t\,c_1(L)+f^*\eta,\qquad - A^+=t\,c_1(L^+)+(f^+)^*\eta. -\] -Every class on $T$ has the corresponding forward transport with an -actual exceptional-divisor correction. No assertion of surjectivity -on Bott--Chern groups is required. -\end{proposition} - -\begin{proof} -Since $f$ has a global relatively ample bundle and all its vertical -curves lie on $R$, numerical invariance of ampleness on the projective -fibres shows that $-L$ is $f$-ample. - -\smallskip\noindent -\emph{A local rational ordinary adjoint.} -Fix $z\in Z$ and a relatively compact Stein neighbourhood $U$. -Take a projective log resolution $p:V\to T_U$ carrying $\mathbf M$, -and put $\rho=f p$. Write its actual structure boundary as $B_V$, so -\[ - [K_V+B_V]+[\mathbf M_V]=p^*A. -\] -Every coefficient of $B_V$ is less than one. Apply -Lemma~\ref{prog:stein-representatives} to obtain a Cartier -representative $D_L$ of $L$ on $T_U$. The same birational direct-image -argument applies to the coherent rank-one reflexive canonical sheaf: -it supplies a canonical form nonzero at a chosen point of the common -smooth isomorphism locus. Use this meromorphic form on $T_U$ and its -pullback to $V$ to choose compatible canonical representatives, and set -\[ - N=t p^*D_L-K_V-B_V. -\] -For every $\rho$-vertical curve, its degree equals that of $\mathbf M_V$. -Thus $N$ is $\rho$-nef. - -The relative bigness of $N$ uses the additional fact that -$\rho=f\circ p$ is \emph{bimeromorphic}: both the small contraction -$f$ and the resolution $p$ are bimeromorphic. It does not follow from -relative nefness alone. In fact every real Cartier divisor is relatively -big for this projective bimeromorphic map over the Stein neighbourhood. -Here is the required ample-plus-effective decomposition explicitly. -Write $N=\sum_j r_jN_j$, where $r_j>0$, $\sum_jr_j=1$, and the $N_j$ -are rational Cartier divisors in its finite Cartier span. Choose a -$\rho$-ample line bundle and a Cartier representative -$P$ for it using Lemma~\ref{prog:stein-representatives}. -If $q_jN_j$ is integral, apply the birational case of that lemma to -$\mathcal O_V(q_jN_j-P)$. Its nonzero holomorphic section has an -effective Cartier zero divisor $E_j$, giving the actual relation -$q_jN_j\sim P+E_j$. This use depends on $\rho$ being bimeromorphic; -the section is allowed to vanish along exceptional fibres. Therefore -\[ - N\sim_{\R} - \underbrace{\left(\sum_j\frac{r_j}{q_j}\right)P}_{H} - +\underbrace{\sum_j\frac{r_j}{q_j}E_j}_{E}. -\] -Here $H$ is $\rho$-ample and $E\geq0$, proving $\rho$-bigness. -For a map with positive-dimensional general fibres the direct image -used above can be zero; no such bigness claim is made for that setting. -Make the finitely many section choices on a slightly larger Stein -neighbourhood, and now shrink $U$ so its closure is compact in that -neighbourhood. Properness makes the inverse image of this closure -compact. A log resolution of the finitely many resulting divisors -therefore gives a uniform positive bound for the following choice. -Choose $\epsilon>0$ small enough that $(V,B_V+\epsilon E)$ is sub-klt. -The divisor $(1-\epsilon)N+\epsilon H$ is relatively ample. Express it -as a positive combination of rational ample divisors and take general -members of sufficiently high multiples. Relative Bertini -\cite[Theorem~2.20]{DHP}, applied on a log resolution of $B_V+E$, -gives an effective $\Theta\sim_{\R}N$ such that $(V,B_V+\Theta)$ is -sub-klt. The high multiples make the added coefficients arbitrarily -small. All choices are made near the compact fibre; shrink $U$ once -to retain them. - -Put $\Delta=p_*(B_V+\Theta)\geq0$. Write the actual linear equivalence -as $\Theta=N+\sum_j a_j\operatorname{div}_V(g_j)$. -The bimeromorphic map $p$ identifies meromorphic function fields, so -$g_j=p^*h_j$. Pushing down gives -$K_{T_U}+\Delta=tD_L+\sum_j a_j\operatorname{div}_{T_U}(h_j)$; -in particular this divisor is real Cartier. Pulling back this same -identity gives -\[ - K_{T_U}+\Delta\sim_{\R}tD_L, - \qquad p^*(K_{T_U}+\Delta)=K_V+B_V+\Theta. -\] -Thus $(T_U,\Delta)$ is klt. Any finitely many base line bundles in a -relative linear equivalence can first be trivialized by shrinking $U$. -Record the resulting equality as -\begin{equation}\label{prog:detector-rationalization} - K_{T_U}+\sum_i d_iD_i=tD_L+\sum_j b_j\operatorname{div}(g_j), - \qquad d_i>0, -\end{equation} -using the positive support of $\Delta$. After a further relatively -compact shrinking, the displayed divisors have finitely many prime -components: their locally finite supports meet a compact inverse image. -Equality of coefficients in \eqref{prog:detector-rationalization} is a -finite rational affine system in $(d_i,t,b_j)$. The klt condition is -open within this system, as seen on one log resolution; the pullbacks -of its adjoints vary linearly by the displayed identity. A nearby -rational solution therefore gives a rational effective klt boundary -$\Delta_q$ and $t_q\in\Q_{>0}$ with -\begin{equation}\label{prog:detector-local-adjoint} - K_{T_U}+\Delta_q\sim_{\Q}t_qD_L. -\end{equation} -The individual local primes $D_i$ need not be $\Q$-Cartier: the identity -ensures that the total adjoint is $\Q$-Cartier. - -\smallskip\noindent -\emph{One global flip algebra.} -The adjoint in \eqref{prog:detector-local-adjoint} is $f_U$-antiample. -Its ordinary flip and local finite generation follow from -\cite[Theorems~1.14 and~1.18]{FujinoAnalyticBCHM}. Clearing the displayed -linear equivalence identifies a Veronese of its canonical algebra -with a Veronese of $\mathcal R:=\bigoplus_{m\geq0}f_*L^m$ over $U$. -Hence $\mathcal R$ is locally finitely generated by -\cite[Lemma~2.26]{FujinoAnalyticBCHM}. The graded pieces are coherent. Over a connected normal base open -the inverse image is irreducible; products of nonzero sections of line -bundles are nonzero there. Thus its graded section algebra is an -integral domain, without choosing any global meromorphic frame of $L$. -Compactness supplies one sufficiently divisible Veronese -$\mathcal R^{(r)}$ generated in degree one. Its relative Proj $T^+$ -restricts to the ordinary flip over every such $U$. It is therefore -normal, locally klt, and small over $Z$, and has the global relatively -ample bundle -$\mathcal O_{T^+}(1)$. On the common big open this bundle is $L^r$. -Reflexivity gives -\[ - (L^+)^{[r]}\simeq\mathcal O_{T^+}(1). -\] -This proves that $L^+$ is a rational line bundle before any assertion -of factoriality. Relative positivity over the compact K\"ahler base -makes $T^+$ compact K\"ahler. - -For any global rank-one reflexive sheaf $\mathcal F^+$ on $T^+$, -take its coherent reflexive transform $\mathcal F$ on $T$ using a -common resolution. Some $M=\mathcal F^{[m]}$ is a line bundle. -Choose $c\in\Q$ so that $(M+cL)\cdot C=0$, and clear its denominator -to obtain an integral bundle $J=n(M+cL)$ numerically trivial over $f$. -The local rational klt pairs \eqref{prog:detector-local-adjoint} have -antiample adjoint, so Lemma~\ref{prog:integral-descent} yields -one global line bundle $J_Z$ with $J=f^*J_Z$. -On $T^+$, pull back $J_Z$, undo the rational twist $ncL^+$, and compare -on the common big open. After clearing the already established index -of $L^+$, reflexivity gives an invertible positive reflexive power of -$\mathcal F^+$. This proves global strong factoriality. - -\smallskip\noindent -\emph{The original generalized pair.} -The local log-Fano pairs above give $R^if_*\mathcal O_T=0$ for $i>0$ -by \cite[Theorem~5.2]{FujinoAnalyticBCHM}. They make $T$ locally klt, hence rational. -Leray for a projective resolution then shows that $Z$ is rational. -Lemma~\ref{prog:bc-descent}(i) therefore gives -$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$, with injective pullback. Thus $A=t c_1(L)+f^*\eta$ for a unique $\eta$. -Define $A^+=t c_1(L^+)+(f^+)^*\eta$; it is relatively K\"ahler. - -To construct its actual discrepancy divisor, let $\mathcal I$ be the -ideal defined by -\[ - \operatorname{im}(f^*f_*L^r\longrightarrow L^r) - =\mathcal I\otimes L^r. -\] -Principalize $\mathcal I$ on a common resolution $p:W\to T$, -$q:W\to T^+$ carrying $\mathbf M$. If -$\mathcal I\mathcal O_W=\mathcal O_W(-F_L)$, the tautological quotient -gives the actual bundle identity -\[ - q^*\mathcal O_{T^+}(1)=p^*L^r\otimes\mathcal O_W(-F_L). -\] -Hence $E_L=F_L/r$ is effective and exceptional over both sides, and -$[E_L]=p^*c_1(L)-q^*c_1(L^+)$. If $B_W$ is the original structure -boundary, put $B_W^+=B_W-tE_L$. Then -\[ - [K_W+B_W^+]+[\mathbf M_W]=q^*A^+. -\] -Its pushforward is $B^+$, and its discrepancies do not decrease. -The local ordinary comparison in \eqref{prog:detector-local-adjoint} -has discrepancy divisor $t_qE_L$, so its strictness proves the stated -strictness for $tE_L$ as well. Thus the original pair remains gklt. - -Finally, write any $\theta\in H^{1,1}_{\BC}(T,\R)$ uniquely as -$\theta=f^*\eta_\theta+s c_1(L)$ and set -$\theta^+=(f^+)^*\eta_\theta+s c_1(L^+)$. Its correction is the actual -divisor $sE_L$. For a rational line $M$, the number $s$ is rational. Applying -Lemma~\ref{prog:integral-descent} to a Cartier multiple of $M-sL$ -gives $M=sL+f^*M_Z$ as actual rational lines. Reflexive extension -on the common big open gives $M^+=sL^++(f^+)^*M_Z$. -Consequently the cohomological transport agrees with the reflexive -transform on Chern classes. -\end{proof} - - -\subsubsection{Real boundaries and their linear transport} - -The rational ordinary step constructed below also determines the step -for every sufficiently close real boundary. The relevant comparison is -an identity of actual rational lines over the contraction base. - -\begin{proposition}[Ordinary real boundaries] -\label{prog:ordinary-replacement} -Let $T$ be normal compact K\"ahler and globally Weil $\Q$-factorial, -with canonical sheaf a rational line bundle. Let $B\geq0$ be a real -boundary with $(T,B)$ klt. Assume the rational ordinary-step result -of Proposition~\ref{prog:ordinary-step} for $T$ and its subsequent models. Then every -$(K_T+B)$-negative extremal analytic ray has an ordinary step with -projective contractions, compact K\"ahler models, and the expected -opposite relative ample signs. Global Weil $\Q$-factoriality is preserved; -global strong $\Q$-factoriality is preserved when imposed initially. -\end{proposition} -\begin{proof} -Put $D=K_T+B$ and fix a $D$-negative extremal ray $R$. On the finite -positive support of $B$, effectiveness, the klt condition and negativity -on $R$ persist under small coefficient changes. Klt openness is checked -on one log resolution of this support. Choose nearby effective rational -klt boundaries $B_0,\ldots,B_m$ and positive real numbers $u_j$ with -\[ - B=\sum_j u_jB_j,\qquad \sum_j u_j=1, - \qquad D_j\cdot R<0,\quad D_j=K_T+B_j. -\] -For $B=0$ take just $B_0=0$. Every $D_j$ is a global rational line -bundle. The rational-step theorem for $(T,B_0)$ provides a projective -contraction $f:T\to Z$ onto a compact K\"ahler base, contracting -exactly $R$, with $-D_0$ relatively ample. The base is rational by -\cite[Lemma~8.8]{DHP}, so Lemma~\ref{prog:bc-descent} gives -$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$; use case~(ii) when $f$ is of -fiber type. -Let $C$ be a rational curve spanning $R$, and set -$a_j=(D_j\cdot C)/(D_0\cdot C)\in\Q_{>0}$. On every projective -fibre, $-D_j$ is numerically equivalent to the positive multiple -$-a_jD_0$. Numerical invariance of ampleness on that fibre, followed -by the relative-ampleness criterion, makes $-D_j$ relatively ample. -The positive combination $-D$ is relatively ample as a real line -bundle. This also handles a fibre-type contraction. - -Suppose henceforth that $f$ is birational. A Cartier multiple -$M_j=n_j(D_j-a_jD_0)$ is numerically trivial over $Z$ and -$M_j-D_0$ is relatively ample. Lemma~\ref{prog:integral-descent}, -applied to the rational klt pair $(T,B_0)$ gives a rational line bundle -$A_j$ on $Z$ with the actual identity -\begin{equation}\label{prog:real-line-descent} - D_j=a_jD_0+f^*A_j. -\end{equation} -Here and below an identity of rational line bundles means an isomorphism -after a common integral multiple. The descent lemma is used only for -birational $f$. - -For a small $f$, let $f^+:T^+\to Z$ be its rational $D_0$-flip. -The rational theorem supplies a global relatively ample adjoint -$D_0^+$ and preserves the stated factoriality and canonical-sheaf -conditions. The other transformed adjoints $D_j^+$ are therefore -rational line bundles. Transform \eqref{prog:real-line-descent} on the common -big open and extend by reflexivity to obtain -\[ - D_j^+=a_jD_0^++(f^+)^*A_j. -\] -All $D_j^+$ are relatively ample, as is -$D^+=\sum_j u_jD_j^+$. Thus this same small map is the required -ordinary real-boundary flip. The real ordinary discrepancy comparison -proves that $(T^+,B^+)$ is klt and gives the strict increases at -exceptional centres. - -For a divisorial contraction, write -$D_0=f^*D_0'+e_0E$ with $e_0>0$. Transforming -\eqref{prog:real-line-descent} on the target and summing yields -\[ - D-f^*D'=\left(\sum_j u_ja_j\right)e_0E. -\] -This coefficient is positive. The same discrepancy comparison proves -the required assertions. The ambient category was already preserved -by the rational step in both cases. -\end{proof} - - -For later use, let \(D=K_T+B\) drive one of the birational steps just -constructed, and write \(f':T'\to Z\) for its positive morphism -(or the identity for a divisorial contraction). Lemma~\ref{prog:bc-descent} -gives the unique decomposition and forward transform -\begin{equation} - \theta=f^*\eta+s\{D\},\qquad - \theta'=(f')^*\eta+s\{D'\}. - \label{prog:ordinary-forward-transport} -\end{equation} -On a common resolution \(p:W\to T\), \(q:W\to T'\), the ordinary -comparison \(p^*D-q^*D'=F\) is an actual effective real divisor, -exceptional over \(T'\) and over both sides for a flip. Hence -\(p^*\theta-q^*\theta'=s\{F\}\). On Chern classes this is the -reflexive transform, by the line identities in the proof above and -Lemma~\ref{prog:integral-descent}. This construction gives forward -transport without assuming surjectivity onto the next Bott--Chern group. - -\subsubsection{Actual real lines and relative numerical spaces} - -A second polarization argument will be used for maps whose fibers are -Moishezon. It applies to a real combination of global line bundles, -provided that its degrees on the fibers agree with a relatively Kähler -class. - -\begin{lemma}[Polarization by an actual real line bundle] -\label{prog:actual-line-polarization} -Let $f:T\to Z$ be a proper Moishezon map, where $T$ is a compact -K\"ahler space. Suppose that $L\in\operatorname{Pic}(T)\otimes\R$ -and a relatively K\"ahler class $\omega$ satisfy -\[ - L\cdot C=\omega\cdot C - \quad\text{for every compact curve $C$ contracted by $f$.} -\] -Then $L$ is relatively ample and $f$ is projective. -\end{lemma} - -\begin{proof} -Let $V$ be an irreducible positive-dimensional subspace of a reduced -fibre. It is Moishezon. Normalize $V$ and take a smooth projective -modification $\pi:\widetilde V\to V^{\mathrm n}$, chosen so that -an effective exceptional divisor $E$ has $-E$ relatively ample. -The pullback of the K\"ahler class from $V$ to its normalization is -K\"ahler. Consequently -$\pi^*\omega-\delta[E]$ is K\"ahler for sufficiently small -$\delta>0$. On the projective manifold $\widetilde V$ the actual -real line bundle $\pi^*L-\delta E$ has the same curve degrees as -that K\"ahler class, and is therefore ample. For example, subtract -a sufficiently small positive multiple of a fixed ample class from -the K\"ahler class; the corresponding real line bundle is nef by -the projective numerical criterion. Also $\pi^*L$ is nef, and the -decomposition -\[ - \pi^*L=(\pi^*L-\delta E)+\delta E -\] -shows that it is big. Thus -\[ - L^{\dim V}\cdot V=(\pi^*L)^{\dim V}>0. -\] -The real Nakai--Moishezon criterion for proper algebraic spaces -\cite[Theorem~1.6]{FujinoMiyamoto2020}, applied to the algebraizations -of the Moishezon fibres, proves that $L$ is ample on every reduced -fibre. The criterion is insensitive to nilpotents and reducible -components. In the coefficient space of the finitely many global -line bundles occurring in $L$, choose, for each fibre, a rational -simplex containing $L$ in its interior whose vertices restrict to -ample rational lines on that fibre. Openness of fibrewise ampleness -for each of these finitely many rational lines gives a base -neighbourhood on which the entire simplex is relatively ample. -Choose finitely many such neighborhoods covering the base and -intersect their coefficient neighborhoods of $L$. Every rational -point of the resulting neighborhood is relatively ample globally. -A rational simplex in this intersection also expresses $L$ as a -positive combination of relatively ample rational lines. Clearing -the denominator of one rational point proves projectivity. -\end{proof} - -\begin{remark} -The proof uses only positivity of the top self-intersections of -$L$. Equality of curve degrees with $\omega$ does not assert -equality of their higher intersection numbers or of their -Bott--Chern classes. -\end{remark} - - - - -The projective morphism needed for the relative program can be chosen -on a smooth model of the rational quotient. Here the Hodge-theoretic -polarization has an elementary construction, independent of any -birational projectivity-descent assertion. - -\begin{lemma}[Smooth Hodge-theoretic polarization] -\label{prog:smooth-polarization} -Let \(g:W\to S\) be a surjective morphism with connected fibers -between smooth compact Kähler manifolds. If -\(g^*:H^0(S,\Omega_S^2)\to H^0(W,\Omega_W^2)\) is an isomorphism, -then \(g\) is projective. This applies when its general fiber is -rationally connected. -\end{lemma} - -\begin{proof} -We use the smooth argument of \cite[Theorem~3.1, Step~1]{CH}. -Let \(d=\dim W-\dim S\). Choose a rational class \(u\in H^2(W,\Q)\) -close to a Kähler class. Write its Hodge decomposition as -\(u=b+g^*v\), where \(b\) is Kähler and \(v\) has types \((2,0)\) -and \((0,2)\). Hodge type and the projection formula give -\[ - c=g_*(u^d)=g_*(b^d)>0,\qquad - g_*(u^{d+1})=g_*(b^{d+1})+(d+1)c\,v. -\] -Since pushforward is rational, the class -\[ - \ell=u-g^*\frac{g_*(u^{d+1})}{(d+1)c} - =b-g^*\frac{g_*(b^{d+1})}{(d+1)c} -\] -is rational and of type \((1,1)\). Lefschetz's theorem makes it the -Chern class of a rational line. The displayed equality supplies a -metric with positive curvature locally over \(S\), by adding a -potential for the base class. Thus this line is relatively ample. -For rationally connected general fibers, holomorphic two-forms are -pulled back from the smooth base, so the hypothesis holds. -\end{proof} - -\begin{lemma}[A rational Hodge correction over a smooth base] -\label{prog:relative-hodge} -Let $f:T\to S$ be an already projective surjective morphism of normal -compact K\"ahler spaces. Assume that $S$ is smooth, $T$ has globally -strongly $\Q$-factorial klt singularities, and, for a projective resolution -$r:W\to T$, the morphism $g=f\circ r$ satisfies -\[ - g^*:H^0(S,\Omega_S^2)\xrightarrow{\ \simeq\ } - H^0(W,\Omega_W^2). -\] -Then every $\alpha\in H^{1,1}_{\BC}(T,\R)$ can be written -\[ - \alpha=c_1(L)+f^*\gamma, -\] -where $L\in\operatorname{Pic}(T)\otimes_{\mathbb Z}\mathbb R$ is a -finite real combination of global line bundles and -$\gamma\in H^{1,1}(S,\mathbb R)$. -\end{lemma} - -\begin{proof} -The composition $g$ is projective, since all the spaces are compact. -Choose a $g$-ample line bundle on $W$, with integral Chern class $H$. -Set $d=\dim W-\dim S$ and -\[ - c=g_*(H^d)>0. -\] -Thus $c$ is the positive degree of $H^d$ on a general fibre. The -cohomological pushforward here is the usual pushforward between the -smooth compact manifolds $W$ and $S$, defined over $\mathbb Q$ and of -Hodge bidegree $(-d,-d)$. - -Define a rational linear operator on degree-two cohomology by -\[ - \Pi(\eta)=\eta-g^*\!\left(\frac{g_*(\eta\smile H^d)}{c}\right). -\] -Every class of type $(2,0)$ on $W$ is pulled back from $S$, and the -same holds for type $(0,2)$. The projection formula therefore shows -that $\Pi$ kills both types. It preserves type $(1,1)$. Consequently -\[ - \Pi(H^2(W,\mathbb Q))\subset - H^2(W,\mathbb Q)\cap H^{1,1}(W). -\] -By the Lefschetz $(1,1)$ theorem the image consists of rational Chern -classes of global line bundles. Applying $\Pi$ to $r^*\alpha$ gives -\[ - r^*\alpha=c_1(L_W)+g^*\gamma, - \qquad L_W\in\operatorname{Pic}(W)\otimes\mathbb R, - \qquad - \gamma=\frac{g_*(r^*\alpha\smile H^d)}{c}\in H^{1,1}(S,\mathbb R). -\] -For example, express $r^*\alpha$ in a rational basis of $H^2(W,\mathbb Q)$ -and apply $\Pi$ to each basis vector. This gives the required finite -real combination $L_W$. - -Write $L_W=\sum_j a_jL_j$ with actual line bundles $L_j$ on $W$. -For each $j$, let -\[ - \mathcal F_j=(r_*L_j)^{**}. -\] -It is a rank-one reflexive coherent sheaf. Strong -$\mathbb Q$-factoriality gives an integer $m_j>0$ for which -$M_j=\mathcal F_j^{[m_j]}$ is a line bundle on $T$. -The canonical identification over the isomorphism locus of $r$ gives -\[ - L_j^{\otimes m_j}\simeq r^*M_j\otimes\mathcal O_W(E_j) -\] -for an integral $r$-exceptional divisor $E_j$. -One may obtain this comparison directly from the evaluation map for -$r_*L_j$: the two coherent rank-one sheaves agree away from the -exceptional locus, so their invertible transforms differ by an -exceptional divisor. No global meromorphic frame of $L_j$ or of the -canonical sheaf is required. - -Set $L=\sum_j(a_j/m_j)M_j$ and -$E=\sum_j(a_j/m_j)E_j$. The preceding identities yield -\[ - r^*(\alpha-c_1(L)-f^*\gamma)=[E]. -\] -The divisor $E$ is $r$-exceptional and is numerically trivial on every -$r$-contracted curve. The exceptional negativity lemma applied in both -signs gives $E=0$. Injectivity of pullback by a resolution, or pushforward -of the equality of currents, now gives -$\alpha=c_1(L)+f^*\gamma$. -\end{proof} - - -The hypothesis of Lemma~\ref{prog:relative-hodge} persists on each -projective globally strongly \(\Q\)-factorial klt model over this fixed -smooth base: on a common smooth resolution, holomorphic two-forms are -invariant under modifications. Thus the lemma applies separately on -every working model. It supplies real combinations of global lines; -Lemma~\ref{prog:stein-representatives} supplies their Cartier-divisor -representatives on sufficiently small Stein base neighborhoods. - -\begin{lemma}[The projective relative cone in Bott--Chern cohomology] -\label{prog:projective-cone} -Let \(f:T\to S\) be projective between normal compact Kähler spaces -with rational singularities, and assume -\[ - H^{1,1}_{\BC}(T,\R) - =c_1\bigl(\Pic(T)\otimes\R\bigr) - +f^*H^{1,1}_{\BC}(S,\R). -\] -Let \(N_1^{\mathrm{line}}(T/S)\) denote the space of real combinations -of \(f\)-contracted curves modulo degrees of global line bundles, -and let \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) be their -closed effective cone. Their analytic classes give an isomorphism -onto the span of the face -\[ - \mathcal F_f= - \overline{\mathrm{NA}}(T)\cap(f^*\omega_S)^\perp, -\] -where \(\omega_S\) is any Kähler class on \(S\); this isomorphism -identifies \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) with -\(\mathcal F_f\). In particular, relative extremal rays in the line -space are extremal rays of the full analytic cone. -\end{lemma} - -\begin{proof} -The assumed decomposition makes analytic equivalence and line-degree -equivalence identical on combinations of vertical curves. Their map -into the analytic numerical space is therefore well-defined and -injective. We must show that its closed effective cone is the entire -face, including its classes represented by positive currents. - -Fix \(z\in\mathcal F_f\). For every \(\gamma\in -H^{1,1}_{\BC}(S,\R)\), both \(a\omega_S+\gamma\) and -\(a\omega_S-\gamma\) are Kähler for large \(a\). Positivity then -shows that \(z\) annihilates \(f^*\gamma\). Fix a global relatively -ample line \(H\). If a real line \(D\) has nonnegative degree on -every vertical curve, the projective relative numerical criterion -makes \(D+\epsilon H\) relatively ample for every \(\epsilon>0\). -Its Chern class becomes Kähler after adding a sufficiently large -multiple of \(f^*\omega_S\). Therefore -\[ - (c_1(D)+\epsilon c_1(H))\cdot z\geq0, - \qquad c_1(D)\cdot z\geq0. -\] -Applying this to both signs of a relatively numerically trivial line -shows that pairing with \(z\) factors through the finite-dimensional -space of relative line degrees. It is nonnegative on its nef cone. -By duality for this projective relative space, it is represented by -an element of \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\). -The decomposition in the statement makes its analytic image equal -to \(z\). Conversely every vertical effective curve lies in -\(\mathcal F_f\), and finite-dimensional injectivity preserves -closedness. This proves the cone equality and the extremality claim. -\end{proof} +\subsection{Polarizations, ordinary replacements, and descent} +\label{prog:foundations} + +The program constructions below use the analytic cone +\(\overline{\mathrm{NA}}(T)\), dual to the nef cone in +\(H^{1,1}_{\BC}(T,\R)\) for spaces with rational singularities +\cite[Proposition~2.4]{DHP}. Degrees of global line bundles form a +possibly smaller numerical space. We pass from the analytic cone to +projective geometry by producing an actual relatively ample line. +The ensuing descent statements apply on singular working models as +well as on smooth initial spaces. + +\begin{lemma}[Degree-zero Bott--Chern descent] +\label{prog:bc-descent} +Let \(f:T\to Z\) be a proper surjective morphism with connected fibers +between normal compact complex spaces with rational singularities. +Suppose either +\begin{enumerate} +\item \(f\) is bimeromorphic and both spaces belong to Fujiki's class +\(\mathcal C\); or +\item \(f\) is projective and an effective rational boundary +\(\Delta\) makes \((T,\Delta)\) klt with +\(-(K_T+\Delta)\) relatively nef and big. +\end{enumerate} +Pullback is injective on both \(H^2(-,\R)\) and +\(H^{1,1}_{\BC}(-,\R)\). In either group its image consists exactly +of the classes with degree zero on every curve contracted by \(f\). +\end{lemma} + +\begin{proof} +These are, respectively, the two cases of +\cite[Lemma~2.6(1),(2)]{DH2024}. The first case is birational descent +between spaces with rational singularities and does not require a +projectivity criterion. The second case uses relative vanishing for +the ordinary rational klt pair and the projective relative curve +space. In particular, it can be applied to a projective log-Fano +contraction before constructing generalized data on its base. +\end{proof} + +In applications to an ordinary birational negative contraction, +the rationality hypotheses are automatic. The source is locally klt. +For a dlt pair, first decrease its rational Cartier floor slightly; +relative antiampleness persists. Relative vanishing gives +\(R^if_*\OO_T=0\) for \(i>0\) +\cite[Theorem~5.2]{FujinoAnalyticBCHM}. Composing a resolution of \(T\) +with \(f\), the Leray spectral sequence proves that \(Z\) has rational +singularities. The same reasoning works when the klt boundary is +chosen separately on finitely many Stein base neighborhoods. For +fiber-type contractions in case~(ii), rationality of the base follows +from \cite[Lemma~8.8(i)]{DHP}; thus that application also precedes +the construction of generalized data on the base. + +\begin{lemma}[Projectivity from a rational line detecting the ray] +\label{prog:detected-projectivity} +Let $f:T\to Z$ be a proper contraction between normal compact +K\"ahler spaces, with $T$ having rational singularities. Suppose +$\gamma$ is K\"ahler on $Z$ and +\[ + \alpha=f^*\gamma,\qquad + \overline{\mathrm{NA}}(T)\cap\alpha^\perp=R +\] +is a nonzero ray. If a global rational line bundle $D$ has +$D\cdot R<0$, then $-D$ is relatively ample and $f$ is projective. +\end{lemma} +\begin{proof} +Fix a K\"ahler class $\omega$ on $T$ and normalize $\overline{\mathrm{NA}}(T)$ by +$\omega\cdot z=1$. The resulting slice $S$ is compact. Its unique +point on $R$ has negative $D$-degree, so $d=c_1(D)$ is negative on +a neighborhood of that point in $S$. On the complementary compact +set, $\alpha$ has a strictly positive minimum, while $d$ is bounded. +For all sufficiently small $s>0$, the class $\alpha-sd$ is therefore +strictly positive on $S$ and is K\"ahler by cone duality. + +Choose an integer $m>0$ making $mD$ integral. Then +\[ + c_1(-mD)+f^*((m/s)\gamma)=(m/s)(\alpha-sd) +\] +is K\"ahler. On a neighborhood in the base with a potential for +$\gamma$, absorb that potential in a smooth metric on the same global +line bundle $-mD$. Its curvature is positive on the fibres. The analytic +positive-line-bundle criterion and the fibrewise relative-ampleness +criterion make $-mD$ relatively ample. The line bundle was global +throughout, so no gluing of unrelated local polarizations is required. +\end{proof} + + +An analytic contraction supplied by a supporting class therefore becomes +projective as soon as one rational line has negative degree on its ray. +For a small contraction, that same line will define the flip globally. +The local ordinary adjoints used to prove finite generation need not +glue; their section algebras will be Veronese subalgebras of one algebra. + +\begin{lemma}[Divisor representatives over a Stein base] +\label{prog:stein-representatives} +Let $\pi:Y\to U$ be a projective surjective morphism with connected fibres +between normal irreducible complex spaces, where $U$ is Stein, and let +$\mathcal L$ be a holomorphic line bundle on $Y$. +If $\pi$ is bimeromorphic, $\mathcal L$ has a holomorphic section +which is not identically zero, and hence an effective Cartier divisor +representative. More generally, after shrinking $U$ around any specified +point, $\mathcal L$ has a nonzero meromorphic section. +\end{lemma} + +\begin{proof} +Suppose first that $\pi$ is bimeromorphic. Grauert's proper direct-image +theorem makes $\mathcal F=\pi_*\mathcal L$ coherent. On the nonempty +open subset $U^\circ$ where $\pi$ is an isomorphism, it is a line bundle. +Choose $u_*\in U^\circ$. Cartan's theorem~A says that global sections +generate $\mathcal F_{u_*}$, so one global section has nonzero image in +$\mathcal F_{u_*}/\mathfrak m_{u_*}\mathcal F_{u_*}$. +Under +\[ + H^0(U,\pi_*\mathcal L)=H^0(Y,\mathcal L) +\] +it gives a section $s$ nonzero at the unique point above $u_*$. +Its zero divisor is effective Cartier and represents $\mathcal L$. +The section may vanish elsewhere; only its meromorphic inverse is +being used to obtain a meromorphic trivialization. + +For the general case, fix $u\in U$ and $y\in\pi^{-1}(u)$, and choose +a $\pi$-ample line bundle $\mathcal H$. Relative Serre generation, +after shrinking to a Stein neighbourhood of $u$, gives an integer +$n$ for which both $\mathcal L\otimes\mathcal H^n$ and +$\mathcal H^n$ are relatively generated. Their proper direct images +are coherent. Cartan's theorem~A and the evaluation maps supply +sections $s$ and $t$ of these two bundles which are both nonzero at +$y$. The quotient $s/t$ is the required nonzero meromorphic section +of $\mathcal L$. At no point is $Y$ assumed Stein. +\end{proof} + +\begin{proposition}[A global algebra for a detected flip] +\label{prog:detected-flip} +Let $(T,B+\mathbf M)$ be a gklt pair on a normal compact K\"ahler +space which is globally strongly $\Q$-factorial, and let $A\in H^{1,1}_{\BC}(T,\R)$ +be its adjoint class. Let $f:T\to Z$ be a projective small contraction +onto a normal compact K\"ahler space. Suppose that the classes of all +$f$-vertical curves lie on one ray $R\subset\overline{\mathrm{NA}}(T)$, that $A\cdot R<0$, +and that a global line bundle $L$ satisfies $L\cdot R<0$. + +Then $\bigoplus_{m\geq0}f_*L^m$ is locally finitely generated. The +relative Proj of a sufficiently divisible Veronese is a projective +small contraction $f^+:T^+\to Z$, where $T^+$ is compact K\"ahler and +globally strongly $\Q$-factorial. The reflexive transform $L^+$ is a +rational line bundle, with a positive power equal to the relatively +ample tautological bundle. The transformed boundary and the same +b-nef datum define a gklt pair on $T^+$, with relatively K\"ahler +adjoint $A^+$. This is the generalized flip, with the usual +strict discrepancy increase at centres in either exceptional locus. + +Moreover, if $t=(A\cdot C)/(L\cdot C)>0$ for an $f$-vertical curve $C$, +then there is a unique $\eta\in H^{1,1}_{\BC}(Z,\R)$ such that +\[ + A=t\,c_1(L)+f^*\eta,\qquad + A^+=t\,c_1(L^+)+(f^+)^*\eta. +\] +Every class on $T$ has the corresponding forward transport with an +actual exceptional-divisor correction. No assertion of surjectivity +on Bott--Chern groups is required. +\end{proposition} + +\begin{proof} +Since $f$ has a global relatively ample bundle and all its vertical +curves lie on $R$, numerical invariance of ampleness on the projective +fibres shows that $-L$ is $f$-ample. + +\smallskip\noindent +\emph{A local rational ordinary adjoint.} +Fix $z\in Z$ and a relatively compact Stein neighbourhood $U$. +Take a projective log resolution $p:V\to T_U$ carrying $\mathbf M$, +and put $\rho=f p$. Write its actual structure boundary as $B_V$, so +\[ + [K_V+B_V]+[\mathbf M_V]=p^*A. +\] +Every coefficient of $B_V$ is less than one. Apply +Lemma~\ref{prog:stein-representatives} to obtain a Cartier +representative $D_L$ of $L$ on $T_U$. The same birational direct-image +argument applies to the coherent rank-one reflexive canonical sheaf: +it supplies a canonical form nonzero at a chosen point of the common +smooth isomorphism locus. Use this meromorphic form on $T_U$ and its +pullback to $V$ to choose compatible canonical representatives, and set +\[ + N=t p^*D_L-K_V-B_V. +\] +For every $\rho$-vertical curve, its degree equals that of $\mathbf M_V$. +Thus $N$ is $\rho$-nef. + +The relative bigness of $N$ uses the additional fact that +$\rho=f\circ p$ is \emph{bimeromorphic}: both the small contraction +$f$ and the resolution $p$ are bimeromorphic. It does not follow from +relative nefness alone. In fact every real Cartier divisor is relatively +big for this projective bimeromorphic map over the Stein neighbourhood. +Here is the required ample-plus-effective decomposition explicitly. +Write $N=\sum_j r_jN_j$, where $r_j>0$, $\sum_jr_j=1$, and the $N_j$ +are rational Cartier divisors in its finite Cartier span. Choose a +$\rho$-ample line bundle and a Cartier representative +$P$ for it using Lemma~\ref{prog:stein-representatives}. +If $q_jN_j$ is integral, apply the birational case of that lemma to +$\mathcal O_V(q_jN_j-P)$. Its nonzero holomorphic section has an +effective Cartier zero divisor $E_j$, giving the actual relation +$q_jN_j\sim P+E_j$. This use depends on $\rho$ being bimeromorphic; +the section is allowed to vanish along exceptional fibres. Therefore +\[ + N\sim_{\R} + \underbrace{\left(\sum_j\frac{r_j}{q_j}\right)P}_{H} + +\underbrace{\sum_j\frac{r_j}{q_j}E_j}_{E}. +\] +Here $H$ is $\rho$-ample and $E\geq0$, proving $\rho$-bigness. +For a map with positive-dimensional general fibres the direct image +used above can be zero; no such bigness claim is made for that setting. +Make the finitely many section choices on a slightly larger Stein +neighbourhood, and now shrink $U$ so its closure is compact in that +neighbourhood. Properness makes the inverse image of this closure +compact. A log resolution of the finitely many resulting divisors +therefore gives a uniform positive bound for the following choice. +Choose $\epsilon>0$ small enough that $(V,B_V+\epsilon E)$ is sub-klt. +The divisor $(1-\epsilon)N+\epsilon H$ is relatively ample. Express it +as a positive combination of rational ample divisors and take general +members of sufficiently high multiples. Relative Bertini +\cite[Theorem~2.20]{DHP}, applied on a log resolution of $B_V+E$, +gives an effective $\Theta\sim_{\R}N$ such that $(V,B_V+\Theta)$ is +sub-klt. The high multiples make the added coefficients arbitrarily +small. All choices are made near the compact fibre; shrink $U$ once +to retain them. + +Put $\Delta=p_*(B_V+\Theta)\geq0$. Write the actual linear equivalence +as $\Theta=N+\sum_j a_j\operatorname{div}_V(g_j)$. +The bimeromorphic map $p$ identifies meromorphic function fields, so +$g_j=p^*h_j$. Pushing down gives +$K_{T_U}+\Delta=tD_L+\sum_j a_j\operatorname{div}_{T_U}(h_j)$; +in particular this divisor is real Cartier. Pulling back this same +identity gives +\[ + K_{T_U}+\Delta\sim_{\R}tD_L, + \qquad p^*(K_{T_U}+\Delta)=K_V+B_V+\Theta. +\] +Thus $(T_U,\Delta)$ is klt. Any finitely many base line bundles in a +relative linear equivalence can first be trivialized by shrinking $U$. +Record the resulting equality as +\begin{equation}\label{prog:detector-rationalization} + K_{T_U}+\sum_i d_iD_i=tD_L+\sum_j b_j\operatorname{div}(g_j), + \qquad d_i>0, +\end{equation} +using the positive support of $\Delta$. After a further relatively +compact shrinking, the displayed divisors have finitely many prime +components: their locally finite supports meet a compact inverse image. +Equality of coefficients in \eqref{prog:detector-rationalization} is a +finite rational affine system in $(d_i,t,b_j)$. The klt condition is +open within this system, as seen on one log resolution; the pullbacks +of its adjoints vary linearly by the displayed identity. A nearby +rational solution therefore gives a rational effective klt boundary +$\Delta_q$ and $t_q\in\Q_{>0}$ with +\begin{equation}\label{prog:detector-local-adjoint} + K_{T_U}+\Delta_q\sim_{\Q}t_qD_L. +\end{equation} +The individual local primes $D_i$ need not be $\Q$-Cartier: the identity +ensures that the total adjoint is $\Q$-Cartier. + +\smallskip\noindent +\emph{One global flip algebra.} +The adjoint in \eqref{prog:detector-local-adjoint} is $f_U$-antiample. +Its ordinary flip and local finite generation follow from +\cite[Theorems~1.14 and~1.18]{FujinoAnalyticBCHM}. Clearing the displayed +linear equivalence identifies a Veronese of its canonical algebra +with a Veronese of $\mathcal R:=\bigoplus_{m\geq0}f_*L^m$ over $U$. +Hence $\mathcal R$ is locally finitely generated by +\cite[Lemma~2.26]{FujinoAnalyticBCHM}. The graded pieces are coherent. Over a connected normal base open +the inverse image is irreducible; products of nonzero sections of line +bundles are nonzero there. Thus its graded section algebra is an +integral domain, without choosing any global meromorphic frame of $L$. +Compactness supplies one sufficiently divisible Veronese +$\mathcal R^{(r)}$ generated in degree one. Its relative Proj $T^+$ +restricts to the ordinary flip over every such $U$. It is therefore +normal, locally klt, and small over $Z$, and has the global relatively +ample bundle +$\mathcal O_{T^+}(1)$. On the common big open this bundle is $L^r$. +Reflexivity gives +\[ + (L^+)^{[r]}\simeq\mathcal O_{T^+}(1). +\] +This proves that $L^+$ is a rational line bundle before any assertion +of factoriality. Relative positivity over the compact K\"ahler base +makes $T^+$ compact K\"ahler. + +For any global rank-one reflexive sheaf $\mathcal F^+$ on $T^+$, +take its coherent reflexive transform $\mathcal F$ on $T$ using a +common resolution. Some $M=\mathcal F^{[m]}$ is a line bundle. +Choose $c\in\Q$ so that $(M+cL)\cdot C=0$, and clear its denominator +to obtain an integral bundle $J=n(M+cL)$ numerically trivial over $f$. +The local rational klt pairs \eqref{prog:detector-local-adjoint} have +antiample adjoint, so Lemma~\ref{prog:integral-descent} yields +one global line bundle $J_Z$ with $J=f^*J_Z$. +On $T^+$, pull back $J_Z$, undo the rational twist $ncL^+$, and compare +on the common big open. After clearing the already established index +of $L^+$, reflexivity gives an invertible positive reflexive power of +$\mathcal F^+$. This proves global strong factoriality. + +\smallskip\noindent +\emph{The original generalized pair.} +The local log-Fano pairs above give $R^if_*\mathcal O_T=0$ for $i>0$ +by \cite[Theorem~5.2]{FujinoAnalyticBCHM}. They make $T$ locally klt, hence rational. +Leray for a projective resolution then shows that $Z$ is rational. +Lemma~\ref{prog:bc-descent}(i) therefore gives +$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$, with injective pullback. Thus $A=t c_1(L)+f^*\eta$ for a unique $\eta$. +Define $A^+=t c_1(L^+)+(f^+)^*\eta$; it is relatively K\"ahler. + +To construct its actual discrepancy divisor, let $\mathcal I$ be the +ideal defined by +\[ + \operatorname{im}(f^*f_*L^r\longrightarrow L^r) + =\mathcal I\otimes L^r. +\] +Principalize $\mathcal I$ on a common resolution $p:W\to T$, +$q:W\to T^+$ carrying $\mathbf M$. If +$\mathcal I\mathcal O_W=\mathcal O_W(-F_L)$, the tautological quotient +gives the actual bundle identity +\[ + q^*\mathcal O_{T^+}(1)=p^*L^r\otimes\mathcal O_W(-F_L). +\] +Hence $E_L=F_L/r$ is effective and exceptional over both sides, and +$[E_L]=p^*c_1(L)-q^*c_1(L^+)$. If $B_W$ is the original structure +boundary, put $B_W^+=B_W-tE_L$. Then +\[ + [K_W+B_W^+]+[\mathbf M_W]=q^*A^+. +\] +Its pushforward is $B^+$, and its discrepancies do not decrease. +The local ordinary comparison in \eqref{prog:detector-local-adjoint} +has discrepancy divisor $t_qE_L$, so its strictness proves the stated +strictness for $tE_L$ as well. Thus the original pair remains gklt. + +Finally, write any $\theta\in H^{1,1}_{\BC}(T,\R)$ uniquely as +$\theta=f^*\eta_\theta+s c_1(L)$ and set +$\theta^+=(f^+)^*\eta_\theta+s c_1(L^+)$. Its correction is the actual +divisor $sE_L$. For a rational line $M$, the number $s$ is rational. Applying +Lemma~\ref{prog:integral-descent} to a Cartier multiple of $M-sL$ +gives $M=sL+f^*M_Z$ as actual rational lines. Reflexive extension +on the common big open gives $M^+=sL^++(f^+)^*M_Z$. +Consequently the cohomological transport agrees with the reflexive +transform on Chern classes. +\end{proof} + + +\subsubsection{Real boundaries and their linear transport} + +The rational ordinary step constructed below also determines the step +for every sufficiently close real boundary. The relevant comparison is +an identity of actual rational lines over the contraction base. + +\begin{proposition}[Ordinary real boundaries] +\label{prog:ordinary-replacement} +Let $T$ be normal compact K\"ahler and globally Weil $\Q$-factorial, +with canonical sheaf a rational line bundle. Let $B\geq0$ be a real +boundary with $(T,B)$ klt. Assume the rational ordinary-step result +of Proposition~\ref{prog:ordinary-step} for $T$ and its subsequent models. Then every +$(K_T+B)$-negative extremal analytic ray has an ordinary step with +projective contractions, compact K\"ahler models, and the expected +opposite relative ample signs. Global Weil $\Q$-factoriality is preserved; +global strong $\Q$-factoriality is preserved when imposed initially. +\end{proposition} +\begin{proof} +Put $D=K_T+B$ and fix a $D$-negative extremal ray $R$. On the finite +positive support of $B$, effectiveness, the klt condition and negativity +on $R$ persist under small coefficient changes. Klt openness is checked +on one log resolution of this support. Choose nearby effective rational +klt boundaries $B_0,\ldots,B_m$ and positive real numbers $u_j$ with +\[ + B=\sum_j u_jB_j,\qquad \sum_j u_j=1, + \qquad D_j\cdot R<0,\quad D_j=K_T+B_j. +\] +For $B=0$ take just $B_0=0$. Every $D_j$ is a global rational line +bundle. The rational-step theorem for $(T,B_0)$ provides a projective +contraction $f:T\to Z$ onto a compact K\"ahler base, contracting +exactly $R$, with $-D_0$ relatively ample. The base is rational by +\cite[Lemma~8.8]{DHP}, so Lemma~\ref{prog:bc-descent} gives +$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$; use case~(ii) when $f$ is of +fiber type. +Let $C$ be a rational curve spanning $R$, and set +$a_j=(D_j\cdot C)/(D_0\cdot C)\in\Q_{>0}$. On every projective +fibre, $-D_j$ is numerically equivalent to the positive multiple +$-a_jD_0$. Numerical invariance of ampleness on that fibre, followed +by the relative-ampleness criterion, makes $-D_j$ relatively ample. +The positive combination $-D$ is relatively ample as a real line +bundle. This also handles a fibre-type contraction. + +Suppose henceforth that $f$ is birational. A Cartier multiple +$M_j=n_j(D_j-a_jD_0)$ is numerically trivial over $Z$ and +$M_j-D_0$ is relatively ample. Lemma~\ref{prog:integral-descent}, +applied to the rational klt pair $(T,B_0)$ gives a rational line bundle +$A_j$ on $Z$ with the actual identity +\begin{equation}\label{prog:real-line-descent} + D_j=a_jD_0+f^*A_j. +\end{equation} +Here and below an identity of rational line bundles means an isomorphism +after a common integral multiple. The descent lemma is used only for +birational $f$. + +For a small $f$, let $f^+:T^+\to Z$ be its rational $D_0$-flip. +The rational theorem supplies a global relatively ample adjoint +$D_0^+$ and preserves the stated factoriality and canonical-sheaf +conditions. The other transformed adjoints $D_j^+$ are therefore +rational line bundles. Transform \eqref{prog:real-line-descent} on the common +big open and extend by reflexivity to obtain +\[ + D_j^+=a_jD_0^++(f^+)^*A_j. +\] +All $D_j^+$ are relatively ample, as is +$D^+=\sum_j u_jD_j^+$. Thus this same small map is the required +ordinary real-boundary flip. The real ordinary discrepancy comparison +proves that $(T^+,B^+)$ is klt and gives the strict increases at +exceptional centres. + +For a divisorial contraction, write +$D_0=f^*D_0'+e_0E$ with $e_0>0$. Transforming +\eqref{prog:real-line-descent} on the target and summing yields +\[ + D-f^*D'=\left(\sum_j u_ja_j\right)e_0E. +\] +This coefficient is positive. The same discrepancy comparison proves +the required assertions. The ambient category was already preserved +by the rational step in both cases. +\end{proof} + + +For later use, let \(D=K_T+B\) drive one of the birational steps just +constructed, and write \(f':T'\to Z\) for its positive morphism +(or the identity for a divisorial contraction). Lemma~\ref{prog:bc-descent} +gives the unique decomposition and forward transform +\begin{equation} + \theta=f^*\eta+s\{D\},\qquad + \theta'=(f')^*\eta+s\{D'\}. + \label{prog:ordinary-forward-transport} +\end{equation} +On a common resolution \(p:W\to T\), \(q:W\to T'\), the ordinary +comparison \(p^*D-q^*D'=F\) is an actual effective real divisor, +exceptional over \(T'\) and over both sides for a flip. Hence +\(p^*\theta-q^*\theta'=s\{F\}\). On Chern classes this is the +reflexive transform, by the line identities in the proof above and +Lemma~\ref{prog:integral-descent}. This construction gives forward +transport without assuming surjectivity onto the next Bott--Chern group. + +\subsubsection{Actual real lines and relative numerical spaces} + +A second polarization argument will be used for maps whose fibers are +Moishezon. It applies to a real combination of global line bundles, +provided that its degrees on the fibers agree with a relatively Kähler +class. + +\begin{lemma}[Polarization by an actual real line bundle] +\label{prog:actual-line-polarization} +Let $f:T\to Z$ be a proper Moishezon map, where $T$ is a compact +K\"ahler space. Suppose that $L\in\operatorname{Pic}(T)\otimes\R$ +and a relatively K\"ahler class $\omega$ satisfy +\[ + L\cdot C=\omega\cdot C + \quad\text{for every compact curve $C$ contracted by $f$.} +\] +Then $L$ is relatively ample and $f$ is projective. +\end{lemma} + +\begin{proof} +Let $V$ be an irreducible positive-dimensional subspace of a reduced +fibre. It is Moishezon. Normalize $V$ and take a smooth projective +modification $\pi:\widetilde V\to V^{\mathrm n}$, chosen so that +an effective exceptional divisor $E$ has $-E$ relatively ample. +The pullback of the K\"ahler class from $V$ to its normalization is +K\"ahler. Consequently +$\pi^*\omega-\delta[E]$ is K\"ahler for sufficiently small +$\delta>0$. On the projective manifold $\widetilde V$ the actual +real line bundle $\pi^*L-\delta E$ has the same curve degrees as +that K\"ahler class, and is therefore ample. For example, subtract +a sufficiently small positive multiple of a fixed ample class from +the K\"ahler class; the corresponding real line bundle is nef by +the projective numerical criterion. Also $\pi^*L$ is nef, and the +decomposition +\[ + \pi^*L=(\pi^*L-\delta E)+\delta E +\] +shows that it is big. Thus +\[ + L^{\dim V}\cdot V=(\pi^*L)^{\dim V}>0. +\] +The real Nakai--Moishezon criterion for proper algebraic spaces +\cite[Theorem~1.6]{FujinoMiyamoto2020}, applied to the algebraizations +of the Moishezon fibres, proves that $L$ is ample on every reduced +fibre. The criterion is insensitive to nilpotents and reducible +components. In the coefficient space of the finitely many global +line bundles occurring in $L$, choose, for each fibre, a rational +simplex containing $L$ in its interior whose vertices restrict to +ample rational lines on that fibre. Openness of fibrewise ampleness +for each of these finitely many rational lines gives a base +neighbourhood on which the entire simplex is relatively ample. +Choose finitely many such neighborhoods covering the base and +intersect their coefficient neighborhoods of $L$. Every rational +point of the resulting neighborhood is relatively ample globally. +A rational simplex in this intersection also expresses $L$ as a +positive combination of relatively ample rational lines. Clearing +the denominator of one rational point proves projectivity. +\end{proof} + +\begin{remark} +The proof uses only positivity of the top self-intersections of +$L$. Equality of curve degrees with $\omega$ does not assert +equality of their higher intersection numbers or of their +Bott--Chern classes. +\end{remark} + + + + +The projective morphism needed for the relative program can be chosen +on a smooth model of the rational quotient. Here the Hodge-theoretic +polarization has an elementary construction, independent of any +birational projectivity-descent assertion. + +\begin{lemma}[Smooth Hodge-theoretic polarization] +\label{prog:smooth-polarization} +Let \(g:W\to S\) be a surjective morphism with connected fibers +between smooth compact Kähler manifolds. If +\(g^*:H^0(S,\Omega_S^2)\to H^0(W,\Omega_W^2)\) is an isomorphism, +then \(g\) is projective. This applies when its general fiber is +rationally connected. +\end{lemma} + +\begin{proof} +We use the smooth argument of \cite[Theorem~3.1, Step~1]{CH}. +Let \(d=\dim W-\dim S\). Choose a rational class \(u\in H^2(W,\Q)\) +close to a Kähler class. Write its Hodge decomposition as +\(u=b+g^*v\), where \(b\) is Kähler and \(v\) has types \((2,0)\) +and \((0,2)\). Hodge type and the projection formula give +\[ + c=g_*(u^d)=g_*(b^d)>0,\qquad + g_*(u^{d+1})=g_*(b^{d+1})+(d+1)c\,v. +\] +Since pushforward is rational, the class +\[ + \ell=u-g^*\frac{g_*(u^{d+1})}{(d+1)c} + =b-g^*\frac{g_*(b^{d+1})}{(d+1)c} +\] +is rational and of type \((1,1)\). Lefschetz's theorem makes it the +Chern class of a rational line. The displayed equality supplies a +metric with positive curvature locally over \(S\), by adding a +potential for the base class. Thus this line is relatively ample. +For rationally connected general fibers, holomorphic two-forms are +pulled back from the smooth base, so the hypothesis holds. +\end{proof} + +\begin{lemma}[A rational Hodge correction over a smooth base] +\label{prog:relative-hodge} +Let $f:T\to S$ be an already projective surjective morphism of normal +compact K\"ahler spaces. Assume that $S$ is smooth, $T$ has globally +strongly $\Q$-factorial klt singularities, and, for a projective resolution +$r:W\to T$, the morphism $g=f\circ r$ satisfies +\[ + g^*:H^0(S,\Omega_S^2)\xrightarrow{\ \simeq\ } + H^0(W,\Omega_W^2). +\] +Then every $\alpha\in H^{1,1}_{\BC}(T,\R)$ can be written +\[ + \alpha=c_1(L)+f^*\gamma, +\] +where $L\in\operatorname{Pic}(T)\otimes_{\mathbb Z}\mathbb R$ is a +finite real combination of global line bundles and +$\gamma\in H^{1,1}(S,\mathbb R)$. +\end{lemma} + +\begin{proof} +The composition $g$ is projective, since all the spaces are compact. +Choose a $g$-ample line bundle on $W$, with integral Chern class $H$. +Set $d=\dim W-\dim S$ and +\[ + c=g_*(H^d)>0. +\] +Thus $c$ is the positive degree of $H^d$ on a general fibre. The +cohomological pushforward here is the usual pushforward between the +smooth compact manifolds $W$ and $S$, defined over $\mathbb Q$ and of +Hodge bidegree $(-d,-d)$. + +Define a rational linear operator on degree-two cohomology by +\[ + \Pi(\eta)=\eta-g^*\!\left(\frac{g_*(\eta\smile H^d)}{c}\right). +\] +Every class of type $(2,0)$ on $W$ is pulled back from $S$, and the +same holds for type $(0,2)$. The projection formula therefore shows +that $\Pi$ kills both types. It preserves type $(1,1)$. Consequently +\[ + \Pi(H^2(W,\mathbb Q))\subset + H^2(W,\mathbb Q)\cap H^{1,1}(W). +\] +By the Lefschetz $(1,1)$ theorem the image consists of rational Chern +classes of global line bundles. Applying $\Pi$ to $r^*\alpha$ gives +\[ + r^*\alpha=c_1(L_W)+g^*\gamma, + \qquad L_W\in\operatorname{Pic}(W)\otimes\mathbb R, + \qquad + \gamma=\frac{g_*(r^*\alpha\smile H^d)}{c}\in H^{1,1}(S,\mathbb R). +\] +For example, express $r^*\alpha$ in a rational basis of $H^2(W,\mathbb Q)$ +and apply $\Pi$ to each basis vector. This gives the required finite +real combination $L_W$. + +Write $L_W=\sum_j a_jL_j$ with actual line bundles $L_j$ on $W$. +For each $j$, let +\[ + \mathcal F_j=(r_*L_j)^{**}. +\] +It is a rank-one reflexive coherent sheaf. Strong +$\mathbb Q$-factoriality gives an integer $m_j>0$ for which +$M_j=\mathcal F_j^{[m_j]}$ is a line bundle on $T$. +The canonical identification over the isomorphism locus of $r$ gives +\[ + L_j^{\otimes m_j}\simeq r^*M_j\otimes\mathcal O_W(E_j) +\] +for an integral $r$-exceptional divisor $E_j$. +One may obtain this comparison directly from the evaluation map for +$r_*L_j$: the two coherent rank-one sheaves agree away from the +exceptional locus, so their invertible transforms differ by an +exceptional divisor. No global meromorphic frame of $L_j$ or of the +canonical sheaf is required. + +Set $L=\sum_j(a_j/m_j)M_j$ and +$E=\sum_j(a_j/m_j)E_j$. The preceding identities yield +\[ + r^*(\alpha-c_1(L)-f^*\gamma)=[E]. +\] +The divisor $E$ is $r$-exceptional and is numerically trivial on every +$r$-contracted curve. The exceptional negativity lemma applied in both +signs gives $E=0$. Injectivity of pullback by a resolution, or pushforward +of the equality of currents, now gives +$\alpha=c_1(L)+f^*\gamma$. +\end{proof} + + +The hypothesis of Lemma~\ref{prog:relative-hodge} persists on each +projective globally strongly \(\Q\)-factorial klt model over this fixed +smooth base: on a common smooth resolution, holomorphic two-forms are +invariant under modifications. Thus the lemma applies separately on +every working model. It supplies real combinations of global lines; +Lemma~\ref{prog:stein-representatives} supplies their Cartier-divisor +representatives on sufficiently small Stein base neighborhoods. + +\begin{lemma}[The projective relative cone in Bott--Chern cohomology] +\label{prog:projective-cone} +Let \(f:T\to S\) be projective between normal compact Kähler spaces +with rational singularities, and assume +\[ + H^{1,1}_{\BC}(T,\R) + =c_1\bigl(\Pic(T)\otimes\R\bigr) + +f^*H^{1,1}_{\BC}(S,\R). +\] +Let \(N_1^{\mathrm{line}}(T/S)\) denote the space of real combinations +of \(f\)-contracted curves modulo degrees of global line bundles, +and let \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) be their +closed effective cone. Their analytic classes give an isomorphism +onto the span of the face +\[ + \mathcal F_f= + \overline{\mathrm{NA}}(T)\cap(f^*\omega_S)^\perp, +\] +where \(\omega_S\) is any Kähler class on \(S\); this isomorphism +identifies \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) with +\(\mathcal F_f\). In particular, relative extremal rays in the line +space are extremal rays of the full analytic cone. +\end{lemma} + +\begin{proof} +The assumed decomposition makes analytic equivalence and line-degree +equivalence identical on combinations of vertical curves. Their map +into the analytic numerical space is therefore well-defined and +injective. We must show that its closed effective cone is the entire +face, including its classes represented by positive currents. + +Fix \(z\in\mathcal F_f\). For every \(\gamma\in +H^{1,1}_{\BC}(S,\R)\), both \(a\omega_S+\gamma\) and +\(a\omega_S-\gamma\) are Kähler for large \(a\). Positivity then +shows that \(z\) annihilates \(f^*\gamma\). Fix a global relatively +ample line \(H\). If a real line \(D\) has nonnegative degree on +every vertical curve, the projective relative numerical criterion +makes \(D+\epsilon H\) relatively ample for every \(\epsilon>0\). +Its Chern class becomes Kähler after adding a sufficiently large +multiple of \(f^*\omega_S\). Therefore +\[ + (c_1(D)+\epsilon c_1(H))\cdot z\geq0, + \qquad c_1(D)\cdot z\geq0. +\] +Applying this to both signs of a relatively numerically trivial line +shows that pairing with \(z\) factors through the finite-dimensional +space of relative line degrees. It is nonnegative on its nef cone. +By duality for this projective relative space, it is represented by +an element of \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\). +The decomposition in the statement makes its analytic image equal +to \(z\). Conversely every vertical effective curve lies in +\(\mathcal F_f\), and finite-dimensional injectivity preserves +closedness. This proves the cone equality and the extremality claim. +\end{proof} diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03b-projective-good-model.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03b-projective-good-model.tex index d4d4e7ed8d2b2f91b1bfc0208240f3e2da755934..3030f6b6689c8aa4a7f87272e7e3e6ed536d2432 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03b-projective-good-model.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03b-projective-good-model.tex @@ -1,439 +1,439 @@ -\subsection{Good models for already projective relative adjoints} - -\begin{proposition}[Projective relative completion] -\label{prog:projective-good-model} -Let \(f:X\to S\) be an already projective surjective morphism of -normal compact Kähler spaces, with \(X\) smooth. Let -\((X,B+\mathbf M)\) be an effective generalized klt pair with adjoint -\(A\), carried by a projective resolution \(p:W\to X\) with smooth -\(W\), so that -\[ - p^*A=c_1(K_W)+[B_W]+\mathbf M_W. -\] -Suppose that there are an actual real line bundle -\(J\in\Pic(X)\otimes\R\) and -\(\beta\in H^{1,1}_{\BC}(S)\) such that -\[ - A=c_1(J)+f^*\beta, - \qquad - N:=p^*J-K_W-B_W - \text{ is nef and big over }fp. -\] -Assume that \(J\) is pseudoeffective over \(S\). -Then there is a finite chosen \(A\)-negative projective program over -\(S\), with globally strongly \(\Q\)-factorial compact Kähler -working spaces, and a nonextracting endpoint \(X_m\) such that -\[ - J_m\sim_{\R}g^*H, - \qquad - A_m=g^*\bigl(c_1(H)+\rho^*\beta\bigr). -\] -Here \(J_m\) is the actual real-line transform of \(J\), -\(g:X_m\to Z\) is a projective connected-fibre morphism, -\(\rho:Z\to S\) is projective, and \(H\) is an actual real line -bundle ample over \(S\). Thus the class on \(Z\) in this formula is -relatively Kähler over \(S\). The generalized pair on \(X_m\) is -gklt. On a common resolution the comparison with \(A\) is effective -and exceptional over \(X_m\), with strictly positive coefficient -at the strict transform of every prime contracted from \(X\). -\end{proposition} - -\begin{proof} -We reduce to fixed ordinary pairs, and construct one global program. -Choose finitely many Stein open sets \(U_\ell\subset S\) and -semianalytic Stein compact subsets \(W_\ell\Subset U_\ell\) whose -interiors cover \(S\). Such compacts are obtained by intersections -with closed polydiscs in local embeddings of \(S\). In particular -they satisfy condition~(P) of \cite{FujinoAnalyticBCHM}: the source -is normal, the base is Stein, the compact is Stein, and its -intersection with any analytic subset defined near it has finitely -many connected components. This remains true on every normal -projective model. Shrinking \(U_\ell\) around \(W_\ell\) preserves -the finite cover. - -\smallskip\noindent -\emph{Actual ordinary replacements.} -On each chart represent the finitely many line bundles in the data -by Cartier divisors. For a projective map to a Stein space this can -be done by twisting a line bundle and its prospective meromorphic -frame by sufficiently positive relative lines and using nonzero -sections. The generic-rank-one direct-image argument is an -alternative only when the map is bimeromorphic. -Relative bigness gives -\(N\sim_{\R}P+E\), with \(P\) relatively ample and \(E\geq0\). -For sufficiently small \(\epsilon>0\), -\((W,B_W+\epsilon E)\) is sub-klt near the compact inverse image. -The line \((1-\epsilon)N+\epsilon P\) is relatively ample. -General divided relative sections give an effective representative -\(T\) such that -\[ - \Theta:=\epsilon E+T\sim_{\R}N, - \qquad (W,B_W+\Theta)\text{ is sub-klt}. -\] -All supports and coefficient margins are fixed near that compact. -With compatible canonical representatives, push the finite -principal-divisor expression for \(\Theta-N\) down and pull it -back again. For \(\Delta_\ell=p_*(B_W+\Theta)\) this gives -\[ - \Delta_\ell\geq0, - \qquad K_X+\Delta_\ell\sim_{\R}J|_{X_{U_\ell}}, - \qquad - p^*(K_X+\Delta_\ell)=K_W+B_W+\Theta. -\] -In the last equality the chosen principal correction is included -in the representative of \(K_X+\Delta_\ell\). -Thus \((X_{U_\ell},\Delta_\ell)\) is ordinary klt and its boundary -is relatively big. Lemma~11.15 of \cite{FujinoAnalyticBCHM}, with -its big part equal to \(\Delta_\ell\) and its remaining part zero, -replaces it, in its actual real linear equivalence class, by a klt -boundary with an effective general relatively ample rational -summand. Its condition on non-klt centres is vacuous. -Choose the reserved ample summand to be \(\Q\)-linearly equivalent -to a small rational multiple of a fixed global \(f\)-ample line. -Its later strict transform is consequently \(\Q\)-Cartier by the -transport of that global line, without a local factoriality claim. - -We make these choices for a finite rational polytope of actual -global line data. Start with a finite rational span containing -\(J\), and add relatively ample rational lines spanning the global -relative degree space. The above construction is valid in a -neighbourhood of \(J\): retain a small part of the ample summand -and use openness of ampleness and the strict klt inequalities. -It is also valid along a segment from \(J\) to a sufficiently -positive adjoint, by adding divided general ample divisors. -Use finitely many such representatives and one log resolution on -each chart. More explicitly, represent a finite simplex inside the -open ample neighbourhood used above; varying its positive -coefficients represents all nearby global line parameters on the -same finite support. Form the joint finite affine system consisting -of the divisor identities on all charts, with common global line -coordinates and separate boundary and principal-divisor -coordinates. This system is rational. Allow all boundary -coefficients to vary, including those originally contributed by -the real boundary \(B\); fix only the reserved rational ample part -and the zero-support constraints. Its projection to the global -line coordinates contains the just constructed neighbourhood, so -it has a rational affine section. Choose that section sufficiently -close to the constructed real affine lift, by rational approximation -in its affine space of sections, so that the strict klt and -effectiveness margins are retained along the compact segment. -In the joint solution space the -strict klt and effectiveness margins permit a rational polytope -containing the scaling segment. This supplies on every chart -a rational polytope of ordinary klt boundaries with a fixed -effective relatively ample rational summand, representing the -restrictions of the same global line parameters. These boundaries, -including the scaling representatives, are fixed before the -program starts. Their divisors need not agree on different charts. - -\smallskip\noindent -\emph{One global scaling construction.} -Apply the global degree-space construction in the proof of -Proposition~\ref{prog:relative-klt} to \(J\) and a general ample -scaling direction in the chosen span. Here are the modifications -that allow real line data and a nonbirational map to \(S\). -The global relatively ample line still gives a compact normalized -slice of the curve cone. The fixed ordinary chart pairs give finite -negative-ray truncations. A general direction avoids equal wall -parameters for independent ray-degree functionals, exactly as in -that proof. A positive wall therefore selects a single ray in the -global degree space. A rational nef support is represented by a -global rational line; a large multiple minus the chartwise ordinary -adjoint is relatively ample. Ordinary relative base point freeness -contracts the ray using the evaluation of this one global line. - -If the contraction is small, choose a global rational line having -negative ray degree and construct its flip by its one global -graded algebra. The local ordinary proof of -Proposition~\ref{prog:detected-flip} applies to this algebra: -all curves of the contraction have proportional degrees for global -lines, so the detector and the driving line are positively -proportional over the contraction; ordinary rational replacement -gives local finite generation. This part of that proof needs only -the already projective contraction and these line degrees. -It does not require a prior assertion about the full Bott--Chern -cone. The relative Proj is the next global model. In a divisorial -step the exceptional prime and -Lemma~\ref{prog:integral-descent} give the same continuation as in -Proposition~\ref{prog:relative-klt}. Killing the degree of any global -line by a rational multiple of the detector, then applying that -lemma, preserves global strong \(\Q\)-factoriality in both cases. -The global lines continue to span the relative degree spaces. -The rational support-simplex argument in that proposition makes -the wall a descended relatively ample real line on its contraction -base and gives a nonempty interval of ampleness immediately after -each step. This argument uses relative line data and projectivity, -not birationality of the map to \(S\). - -We check explicitly the persistence of the ordinary pairs. Expand -the line parameters in finitely many global rational lines. Their -reflexive transforms are actual rational lines by the preceding -strong property. Pushing forward the fixed principal-divisor -identities gives -\[ - K_{X_i}+\Delta_{\ell,i}\sim_{\R}J_i|_{(X_i)_{U_\ell}}. -\] -Hence these ordinary adjoints are \(\R\)-Cartier; no local -factoriality of individual prime divisors is being assumed. -On a common resolution their comparison minus the appropriate -positive multiple of the detector comparison is exceptional and -relatively numerically trivial. Negativity in both signs makes -this difference zero. Every chosen step is consequently negative -for the fixed ordinary pairs. They remain effective and klt, and -their boundaries remain relatively big. The same equality of -comparison divisors, after adding the fixed base pullback, -preserves the generalized pair and its discrepancy inequalities. - -For termination, every already constructed working model is a weak -log canonical model on each chart of a boundary in the fixed -polytope: choose a parameter in its nonempty ample scaling interval. -Indeed, actual descent at a preceding wall \(\lambda_j\) makes -the comparison for \(J+tH\) equal to -\((1-t/\lambda_j)\) times its \(J\)-comparison. For a parameter -in the current interval all preceding \(\lambda_j\) are at least -\(t\), so these comparisons are effective. -Theorem~E of \cite{FujinoAnalyticBCHM} gives finitely many marked -models on each chart. That theorem does not require locally -\(\Q\)-factorial targets. There are therefore finitely many possible -global marked working models. Indeed, isomorphisms between the -restrictions of two existing marked models agree on their common -dense open, and hence on overlaps. A repeated marked working model -would have the same boundary transform and discrepancies, contrary -to strict discrepancy increase at an intervening nontrivial step. -The global program is finite. Relative pseudoeffectivity excludes -a Mori fibre endpoint, so \(J_m\) is nef over \(S\). -This is a finite-cover argument for a single constructed program; -it does not glue independent minimal models. - -\smallskip\noindent -\emph{Semiampleness and the global polarization.} -On the endpoint the fixed ordinary pair is klt, its effective -boundary is relatively big, and its adjoint is nef over -\(W_\ell\). Lemma~11.16 of \cite{FujinoAnalyticBCHM} applies to -this already existing weak model. More explicitly, Lemma~11.15 -replaces the endpoint boundary by \(A_\ell+B_\ell\), where -\(A_\ell\geq0\) is relatively ample, \(B_\ell\geq0\), and the -pair is klt. Theorem~8.3 then gives semiampleness near \(W_\ell\): -its log canonical, klt-existence, real Cartier, ample-summand and -nef-over-compact hypotheses have all been checked. - -The actual equivalences identify these local semiample adjoints -with restrictions of \(J_m\). Their ample models agree on overlaps -by \cite[Lemma~11.3]{FujinoAnalyticBCHM}. To retain a global -polarization, shrink the initial parameter polytope around \(J\) -after the finite program: all its finitely many strict step signs -persist, so the transformed ordinary boundaries remain klt and big. -On each chart the pushforward of its initial common ample rational -summand is effective and relatively big. Apply -\cite[Lemma~11.13]{FujinoAnalyticBCHM} to this common big rational -part and the small endpoint boundary polytope. All pairs are klt, -so the condition on non-klt centres is vacuous. The lemma gives a -common general relatively ample rational summand by one -\(\Q\)-linearly trivial translation. In particular the rational -parametrization is preserved; separate applications of -Lemma~11.15 to individual parameters would not suffice here. -In this finite rational span Theorem~11.17 of \cite{FujinoAnalyticBCHM} -makes the endpoint nef region a rational polytope. Intersect the -finitely many inverse images of these regions and express \(J_m\) -as a positive combination of rational points in its minimal face. -The corresponding global rational lines are semiample on every -chart by the same ordinary base point free argument. After clearing -denominators and taking common sufficiently divisible powers over -the finite cover, each is relatively generated globally: generation on -the charts is generation of its global coherent relative -evaluation map. The product of these finitely many relative -morphisms, followed by Stein factorization, gives a projective -\(g:X_m\to Z\), a projective \(\rho:Z\to S\), and descended -global lines whose positive combination \(H\) is relatively ample. -It realizes the glued ample model and gives -\(J_m\sim_{\R}g^*H\). Adding back \(f_m^*\beta\) proves the -asserted Bott--Chern identity. Composing the effective step -comparisons proves the final discrepancy statement. -\end{proof} - -\begin{corollary}[Semiampleness on an existing projective model] -\label{prog:projective-semiample} -Let \(f:T\to S\) be a projective morphism of normal compact Kähler -spaces with \(T\) globally strongly \(\Q\)-factorial, and let -\(J\in\Pic(T)\otimes\R\) be nef over \(S\). Suppose that on a -finite Stein-compact cover satisfying condition~(P), there are -effective ordinary klt boundaries \(\Delta_\ell\), big over the -base, such that -\(K_T+\Delta_\ell\sim_{\R}J|_{T_{U_\ell}}\). -Then \(J\) has the projective relative semiample contraction and -actual descended ample real line of -Proposition~\ref{prog:projective-good-model}. -In particular this applies to any already existing weak model of -the fixed ordinary chart pairs in that proof; one need not run a -new program for the adjoint in question. -\end{corollary} - -\begin{proof} -Lemma~11.15 and Theorem~8.3 of \cite{FujinoAnalyticBCHM} give -local semiampleness. For an existing weak model this is also -Lemma~11.16. To globalize the polarization, take a finite rational -span of the global lines defining \(J\). On each chart the ample -part of the normalized boundary allows small perturbations in -this span. The joint rational affine construction in the preceding -proof gives a rational family with a common rational ample part; -all boundary coefficients are allowed to vary. Theorem~11.17 then -gives a rational nef region on each chart. The finite-intersection, -rational-vertex and global-evaluation argument in that proof gives -one projective contraction and its actual descended ample real -line. Only its local ample models are identified by uniqueness. -\end{proof} - -\begin{remark}[The two uses of projective completion] -\label{prog:projective-good-model-uses} -For a selective extraction over an already constructed weak model, -the carrier map is projective bimeromorphic. The inherited -exceptional splitting expresses the nef datum, modulo this base, -by actual real lines. Relative nefness follows from the carrier -datum, while relative bigness follows from birationality, using the -ample-plus-effective construction in the ordinary replacement. -The prepared adjoint is a base pullback plus an effective -exceptional divisor supported on exactly the unwanted primes. -Proposition~\ref{prog:projective-good-model} applies. On its endpoint -the remaining error is effective, exceptional over the fixed base, -and relatively nef; negativity makes it zero. Strict discrepancy -improvement prevents contraction of a requested zero-error prime. -The chosen extraction boundary must be effective and klt; in -particular its prescribed primes have the usual admissible -discrepancy range. - -For the graph construction, suppose the already projective graph -resolution \(h:V\to Y_m\) has smooth source and -\[ - A_V=h^*A_{Y_m}+[G],\qquad G\geq0. -\] -Then the global relative real line is \(J=\OO_V(G)\), the base -class is \(A_{Y_m}\), and its carrier line is -\(N=G-K_V-B_V\). Its relative degrees equal those of the prepared -nef datum. That datum is nef and big on the prepared carrier, for -example a pullback of its prepared Kähler class; thus \(N\) is nef -and big over \(Y_m\). This bigness is a separate hypothesis check, -since \(h\) can have positive-dimensional general fibres. -Effectivity of \(G\) gives relative pseudoeffectivity. Apply the -proposition over \(Y_m\). The subsequent negativity argument must -still remove \(G_m\); only then is the endpoint adjoint the pullback -of \(A_{Y_m}\) and good over the original base. - -For transport of all classes, verify separately that all -Bott--Chern classes on the graph carrier are real-line classes -modulo \(Y_m\), using the smooth rationally connected fibration -and the inherited exceptional splitting on the lower-dimensional -base model. Lemma~\ref{prog:projective-cone} then identifies -the relative degree rays with full Bott--Chern rays, so the -detected-step transport applies. The driving identity involving -\(G\) alone does not give this additional property. -\end{remark} - -\begin{corollary}[Ordinary projective Mori output] -\label{prog:projective-mori} -Let \(f:X\to S\) be an already projective surjective morphism of -compact Kähler spaces, with \(X\) smooth. Let \((X,B)\) be an -effective ordinary real klt pair, and let \(D=K_X+B\). If \(D\) -is not pseudoeffective over \(S\), then its projective program over -\(S\), with scaling of a relatively ample actual real line \(N\), -terminates with a Mori fibre contraction. The initial scaling -parameter is chosen so that \(D+TN\) is nef over \(S\). -The working spaces are globally strongly \(\Q\)-factorial and -compact Kähler, and the actual line comparisons and integral -descent of Proposition~\ref{prog:projective-good-model} apply. -\end{corollary} - -\begin{proof} -The line \(N\) is fixed; it need not be a general scaling direction. -In particular several extremal rays may have the same wall -parameter. Choose \(\epsilon>0\) such that \(D+2\epsilon N\) is -still not pseudoeffective over \(S\). On the finite Stein-compact -cover choose divided general effective representatives of \(N\) -so that all the resulting ordinary pairs representing \(D+tN\), -\(0\leq t\leq T\), are klt. For -\(\epsilon\leq t\leq T\) reserve a common effective relatively -ample rational summand. The joint rational-family construction in -Proposition~\ref{prog:projective-good-model} puts these fixed pairs -in the polytopes required by Theorem~E of -\cite{FujinoAnalyticBCHM}. - -Suppose a finite prefix has reached \(X_i\), and put -\(\mu_i=\lambda_{i-1}\), with \(\mu_0=T\). Its transformed -\(D_i+\mu_iN_i\) is nef. Define -\[ - \lambda_i=\min\{t\in[0,\mu_i]:D_i+tN_i - \text{ is nef over }S\}. -\] -The feasible set is a nonempty closed interval. Inductively -\(\lambda_i>2\epsilon\): all previous steps have nonpositive -comparison for \(D+2\epsilon N\), so its transform is still not -pseudoeffective. A first threshold at most \(2\epsilon\) would -make this transform nef by convexity between the current and -preceding thresholds, a contradiction. - -There is a \(D_i\)-negative extremal ray \(R_i\) with -\((D_i+\lambda_iN_i)\cdot R_i=0\). To see attainment without a -generality assumption, let \(t<\lambda_i\) tend to \(\lambda_i\). -A ray negative for \(D_i+tN_i\) has positive \(N_i\)-degree, -because \(D_i+\mu_iN_i\) is nef. When \(t>\lambda_i/2\), it is -also negative for \(D_i+(\lambda_i/2)N_i\). The latter has fixed -ordinary klt big-boundary representatives on the charts, since -\(\lambda_i/2>\epsilon\). Normalize each of these boundaries by -\cite[Lemma~11.15]{FujinoAnalyticBCHM} as \(A_\ell+B'_\ell\), -with \(A_\ell\geq0\) relatively ample and the new pair klt. -The \(A_\ell\)-truncation of the cone theorem for -\(K+B'_\ell\) gives finitely many rays negative for the entire -adjoint in question. The finite cover therefore gives finitely -many global negative rays. One of these rays attains the wall, -as required. - -Choose one such ray, even if the wall vanishes on other rays. -The rational-support argument for an individual negative ray in -Proposition~\ref{prog:relative-klt} gives a global rational nef -support annihilating exactly \(R_i\), and ordinary relative base -point freeness contracts it. For the real ordinary boundary one -may first choose a nearby rational klt boundary that is still -negative on \(R_i\). The contraction therefore has a rational -ordinary antiample adjoint. Its flip, when small, is constructed -from one global detector algebra as above. - -If the contraction is of fibre type, stop. Otherwise the wall -\(w_i=D_i+\lambda_iN_i\) has degree zero on \(R_i\). -In the finite rational span of its actual line factors, this -zero-degree subspace is rational: its defining functional has -rational values on rational lines. Thus \(w_i\) is a real -combination of rational lines of zero ray degree. -Lemma~\ref{prog:integral-descent} descends those lines to the -contraction base. Their combination is nef over \(S\), by lifting -curves through the projective contraction. Pulling it to the -positive side shows that the transformed wall stays nef. The next -threshold therefore satisfies \(\lambda_{i+1}\leq\lambda_i\); -equality is permitted. No interval of ampleness between successive -thresholds is required. - -If \(E_i\geq0\) is the ordinary \(D_i\)-comparison on a common -resolution, this actual wall descent gives, for every real \(t\), -\[ - p_i^*(D_i+tN_i) - \sim_{\R}q_i^*(D_{i+1}+tN_{i+1}) - +(1-t/\lambda_i)E_i. -\] -For \(t\leq\lambda_i\) the error is effective. This proves all -the nonpositivity and fixed-pair persistence assertions used in -the induction, including the lower bound on thresholds. - -At every working model use the \emph{wall parameter} \(t=\lambda_i\), -rather than an interior ample parameter. Since -\(\lambda_i\leq\lambda_j\) for every earlier step, the displayed -comparisons show that \(X_i\) is a weak log canonical model of -the initial fixed chart pairs representing \(D+\lambda_iN\). -Its trace is nef by definition. Theorem~E counts all these marked -weak log canonical models, including the ones at repeated wall -parameters. The finite chart cover gives a finite global marked -list. Repetition would contradict the strict ordinary -\(D\)-discrepancy increase at an intervening step. Thus this -program with the specified \(N\) is finite. Non-pseudoeffectivity -excludes a nef endpoint for \(D\), so the last contraction is a -Mori fibre contraction. The comparison at \(t=1=\lambda_i\) is -crepant: such a step supplies no strict discrepancy improvement -for \(D+N\), while it is still strictly negative for \(D\). -\end{proof} +\subsection{Good models for already projective relative adjoints} + +\begin{proposition}[Projective relative completion] +\label{prog:projective-good-model} +Let \(f:X\to S\) be an already projective surjective morphism of +normal compact Kähler spaces, with \(X\) smooth. Let +\((X,B+\mathbf M)\) be an effective generalized klt pair with adjoint +\(A\), carried by a projective resolution \(p:W\to X\) with smooth +\(W\), so that +\[ + p^*A=c_1(K_W)+[B_W]+\mathbf M_W. +\] +Suppose that there are an actual real line bundle +\(J\in\Pic(X)\otimes\R\) and +\(\beta\in H^{1,1}_{\BC}(S)\) such that +\[ + A=c_1(J)+f^*\beta, + \qquad + N:=p^*J-K_W-B_W + \text{ is nef and big over }fp. +\] +Assume that \(J\) is pseudoeffective over \(S\). +Then there is a finite chosen \(A\)-negative projective program over +\(S\), with globally strongly \(\Q\)-factorial compact Kähler +working spaces, and a nonextracting endpoint \(X_m\) such that +\[ + J_m\sim_{\R}g^*H, + \qquad + A_m=g^*\bigl(c_1(H)+\rho^*\beta\bigr). +\] +Here \(J_m\) is the actual real-line transform of \(J\), +\(g:X_m\to Z\) is a projective connected-fibre morphism, +\(\rho:Z\to S\) is projective, and \(H\) is an actual real line +bundle ample over \(S\). Thus the class on \(Z\) in this formula is +relatively Kähler over \(S\). The generalized pair on \(X_m\) is +gklt. On a common resolution the comparison with \(A\) is effective +and exceptional over \(X_m\), with strictly positive coefficient +at the strict transform of every prime contracted from \(X\). +\end{proposition} + +\begin{proof} +We reduce to fixed ordinary pairs, and construct one global program. +Choose finitely many Stein open sets \(U_\ell\subset S\) and +semianalytic Stein compact subsets \(W_\ell\Subset U_\ell\) whose +interiors cover \(S\). Such compacts are obtained by intersections +with closed polydiscs in local embeddings of \(S\). In particular +they satisfy condition~(P) of \cite{FujinoAnalyticBCHM}: the source +is normal, the base is Stein, the compact is Stein, and its +intersection with any analytic subset defined near it has finitely +many connected components. This remains true on every normal +projective model. Shrinking \(U_\ell\) around \(W_\ell\) preserves +the finite cover. + +\smallskip\noindent +\emph{Actual ordinary replacements.} +On each chart represent the finitely many line bundles in the data +by Cartier divisors. For a projective map to a Stein space this can +be done by twisting a line bundle and its prospective meromorphic +frame by sufficiently positive relative lines and using nonzero +sections. The generic-rank-one direct-image argument is an +alternative only when the map is bimeromorphic. +Relative bigness gives +\(N\sim_{\R}P+E\), with \(P\) relatively ample and \(E\geq0\). +For sufficiently small \(\epsilon>0\), +\((W,B_W+\epsilon E)\) is sub-klt near the compact inverse image. +The line \((1-\epsilon)N+\epsilon P\) is relatively ample. +General divided relative sections give an effective representative +\(T\) such that +\[ + \Theta:=\epsilon E+T\sim_{\R}N, + \qquad (W,B_W+\Theta)\text{ is sub-klt}. +\] +All supports and coefficient margins are fixed near that compact. +With compatible canonical representatives, push the finite +principal-divisor expression for \(\Theta-N\) down and pull it +back again. For \(\Delta_\ell=p_*(B_W+\Theta)\) this gives +\[ + \Delta_\ell\geq0, + \qquad K_X+\Delta_\ell\sim_{\R}J|_{X_{U_\ell}}, + \qquad + p^*(K_X+\Delta_\ell)=K_W+B_W+\Theta. +\] +In the last equality the chosen principal correction is included +in the representative of \(K_X+\Delta_\ell\). +Thus \((X_{U_\ell},\Delta_\ell)\) is ordinary klt and its boundary +is relatively big. Lemma~11.15 of \cite{FujinoAnalyticBCHM}, with +its big part equal to \(\Delta_\ell\) and its remaining part zero, +replaces it, in its actual real linear equivalence class, by a klt +boundary with an effective general relatively ample rational +summand. Its condition on non-klt centres is vacuous. +Choose the reserved ample summand to be \(\Q\)-linearly equivalent +to a small rational multiple of a fixed global \(f\)-ample line. +Its later strict transform is consequently \(\Q\)-Cartier by the +transport of that global line, without a local factoriality claim. + +We make these choices for a finite rational polytope of actual +global line data. Start with a finite rational span containing +\(J\), and add relatively ample rational lines spanning the global +relative degree space. The above construction is valid in a +neighbourhood of \(J\): retain a small part of the ample summand +and use openness of ampleness and the strict klt inequalities. +It is also valid along a segment from \(J\) to a sufficiently +positive adjoint, by adding divided general ample divisors. +Use finitely many such representatives and one log resolution on +each chart. More explicitly, represent a finite simplex inside the +open ample neighbourhood used above; varying its positive +coefficients represents all nearby global line parameters on the +same finite support. Form the joint finite affine system consisting +of the divisor identities on all charts, with common global line +coordinates and separate boundary and principal-divisor +coordinates. This system is rational. Allow all boundary +coefficients to vary, including those originally contributed by +the real boundary \(B\); fix only the reserved rational ample part +and the zero-support constraints. Its projection to the global +line coordinates contains the just constructed neighbourhood, so +it has a rational affine section. Choose that section sufficiently +close to the constructed real affine lift, by rational approximation +in its affine space of sections, so that the strict klt and +effectiveness margins are retained along the compact segment. +In the joint solution space the +strict klt and effectiveness margins permit a rational polytope +containing the scaling segment. This supplies on every chart +a rational polytope of ordinary klt boundaries with a fixed +effective relatively ample rational summand, representing the +restrictions of the same global line parameters. These boundaries, +including the scaling representatives, are fixed before the +program starts. Their divisors need not agree on different charts. + +\smallskip\noindent +\emph{One global scaling construction.} +Apply the global degree-space construction in the proof of +Proposition~\ref{prog:relative-klt} to \(J\) and a general ample +scaling direction in the chosen span. Here are the modifications +that allow real line data and a nonbirational map to \(S\). +The global relatively ample line still gives a compact normalized +slice of the curve cone. The fixed ordinary chart pairs give finite +negative-ray truncations. A general direction avoids equal wall +parameters for independent ray-degree functionals, exactly as in +that proof. A positive wall therefore selects a single ray in the +global degree space. A rational nef support is represented by a +global rational line; a large multiple minus the chartwise ordinary +adjoint is relatively ample. Ordinary relative base point freeness +contracts the ray using the evaluation of this one global line. + +If the contraction is small, choose a global rational line having +negative ray degree and construct its flip by its one global +graded algebra. The local ordinary proof of +Proposition~\ref{prog:detected-flip} applies to this algebra: +all curves of the contraction have proportional degrees for global +lines, so the detector and the driving line are positively +proportional over the contraction; ordinary rational replacement +gives local finite generation. This part of that proof needs only +the already projective contraction and these line degrees. +It does not require a prior assertion about the full Bott--Chern +cone. The relative Proj is the next global model. In a divisorial +step the exceptional prime and +Lemma~\ref{prog:integral-descent} give the same continuation as in +Proposition~\ref{prog:relative-klt}. Killing the degree of any global +line by a rational multiple of the detector, then applying that +lemma, preserves global strong \(\Q\)-factoriality in both cases. +The global lines continue to span the relative degree spaces. +The rational support-simplex argument in that proposition makes +the wall a descended relatively ample real line on its contraction +base and gives a nonempty interval of ampleness immediately after +each step. This argument uses relative line data and projectivity, +not birationality of the map to \(S\). + +We check explicitly the persistence of the ordinary pairs. Expand +the line parameters in finitely many global rational lines. Their +reflexive transforms are actual rational lines by the preceding +strong property. Pushing forward the fixed principal-divisor +identities gives +\[ + K_{X_i}+\Delta_{\ell,i}\sim_{\R}J_i|_{(X_i)_{U_\ell}}. +\] +Hence these ordinary adjoints are \(\R\)-Cartier; no local +factoriality of individual prime divisors is being assumed. +On a common resolution their comparison minus the appropriate +positive multiple of the detector comparison is exceptional and +relatively numerically trivial. Negativity in both signs makes +this difference zero. Every chosen step is consequently negative +for the fixed ordinary pairs. They remain effective and klt, and +their boundaries remain relatively big. The same equality of +comparison divisors, after adding the fixed base pullback, +preserves the generalized pair and its discrepancy inequalities. + +For termination, every already constructed working model is a weak +log canonical model on each chart of a boundary in the fixed +polytope: choose a parameter in its nonempty ample scaling interval. +Indeed, actual descent at a preceding wall \(\lambda_j\) makes +the comparison for \(J+tH\) equal to +\((1-t/\lambda_j)\) times its \(J\)-comparison. For a parameter +in the current interval all preceding \(\lambda_j\) are at least +\(t\), so these comparisons are effective. +Theorem~E of \cite{FujinoAnalyticBCHM} gives finitely many marked +models on each chart. That theorem does not require locally +\(\Q\)-factorial targets. There are therefore finitely many possible +global marked working models. Indeed, isomorphisms between the +restrictions of two existing marked models agree on their common +dense open, and hence on overlaps. A repeated marked working model +would have the same boundary transform and discrepancies, contrary +to strict discrepancy increase at an intervening nontrivial step. +The global program is finite. Relative pseudoeffectivity excludes +a Mori fibre endpoint, so \(J_m\) is nef over \(S\). +This is a finite-cover argument for a single constructed program; +it does not glue independent minimal models. + +\smallskip\noindent +\emph{Semiampleness and the global polarization.} +On the endpoint the fixed ordinary pair is klt, its effective +boundary is relatively big, and its adjoint is nef over +\(W_\ell\). Lemma~11.16 of \cite{FujinoAnalyticBCHM} applies to +this already existing weak model. More explicitly, Lemma~11.15 +replaces the endpoint boundary by \(A_\ell+B_\ell\), where +\(A_\ell\geq0\) is relatively ample, \(B_\ell\geq0\), and the +pair is klt. Theorem~8.3 then gives semiampleness near \(W_\ell\): +its log canonical, klt-existence, real Cartier, ample-summand and +nef-over-compact hypotheses have all been checked. + +The actual equivalences identify these local semiample adjoints +with restrictions of \(J_m\). Their ample models agree on overlaps +by \cite[Lemma~11.3]{FujinoAnalyticBCHM}. To retain a global +polarization, shrink the initial parameter polytope around \(J\) +after the finite program: all its finitely many strict step signs +persist, so the transformed ordinary boundaries remain klt and big. +On each chart the pushforward of its initial common ample rational +summand is effective and relatively big. Apply +\cite[Lemma~11.13]{FujinoAnalyticBCHM} to this common big rational +part and the small endpoint boundary polytope. All pairs are klt, +so the condition on non-klt centres is vacuous. The lemma gives a +common general relatively ample rational summand by one +\(\Q\)-linearly trivial translation. In particular the rational +parametrization is preserved; separate applications of +Lemma~11.15 to individual parameters would not suffice here. +In this finite rational span Theorem~11.17 of \cite{FujinoAnalyticBCHM} +makes the endpoint nef region a rational polytope. Intersect the +finitely many inverse images of these regions and express \(J_m\) +as a positive combination of rational points in its minimal face. +The corresponding global rational lines are semiample on every +chart by the same ordinary base point free argument. After clearing +denominators and taking common sufficiently divisible powers over +the finite cover, each is relatively generated globally: generation on +the charts is generation of its global coherent relative +evaluation map. The product of these finitely many relative +morphisms, followed by Stein factorization, gives a projective +\(g:X_m\to Z\), a projective \(\rho:Z\to S\), and descended +global lines whose positive combination \(H\) is relatively ample. +It realizes the glued ample model and gives +\(J_m\sim_{\R}g^*H\). Adding back \(f_m^*\beta\) proves the +asserted Bott--Chern identity. Composing the effective step +comparisons proves the final discrepancy statement. +\end{proof} + +\begin{corollary}[Semiampleness on an existing projective model] +\label{prog:projective-semiample} +Let \(f:T\to S\) be a projective morphism of normal compact Kähler +spaces with \(T\) globally strongly \(\Q\)-factorial, and let +\(J\in\Pic(T)\otimes\R\) be nef over \(S\). Suppose that on a +finite Stein-compact cover satisfying condition~(P), there are +effective ordinary klt boundaries \(\Delta_\ell\), big over the +base, such that +\(K_T+\Delta_\ell\sim_{\R}J|_{T_{U_\ell}}\). +Then \(J\) has the projective relative semiample contraction and +actual descended ample real line of +Proposition~\ref{prog:projective-good-model}. +In particular this applies to any already existing weak model of +the fixed ordinary chart pairs in that proof; one need not run a +new program for the adjoint in question. +\end{corollary} + +\begin{proof} +Lemma~11.15 and Theorem~8.3 of \cite{FujinoAnalyticBCHM} give +local semiampleness. For an existing weak model this is also +Lemma~11.16. To globalize the polarization, take a finite rational +span of the global lines defining \(J\). On each chart the ample +part of the normalized boundary allows small perturbations in +this span. The joint rational affine construction in the preceding +proof gives a rational family with a common rational ample part; +all boundary coefficients are allowed to vary. Theorem~11.17 then +gives a rational nef region on each chart. The finite-intersection, +rational-vertex and global-evaluation argument in that proof gives +one projective contraction and its actual descended ample real +line. Only its local ample models are identified by uniqueness. +\end{proof} + +\begin{remark}[The two uses of projective completion] +\label{prog:projective-good-model-uses} +For a selective extraction over an already constructed weak model, +the carrier map is projective bimeromorphic. The inherited +exceptional splitting expresses the nef datum, modulo this base, +by actual real lines. Relative nefness follows from the carrier +datum, while relative bigness follows from birationality, using the +ample-plus-effective construction in the ordinary replacement. +The prepared adjoint is a base pullback plus an effective +exceptional divisor supported on exactly the unwanted primes. +Proposition~\ref{prog:projective-good-model} applies. On its endpoint +the remaining error is effective, exceptional over the fixed base, +and relatively nef; negativity makes it zero. Strict discrepancy +improvement prevents contraction of a requested zero-error prime. +The chosen extraction boundary must be effective and klt; in +particular its prescribed primes have the usual admissible +discrepancy range. + +For the graph construction, suppose the already projective graph +resolution \(h:V\to Y_m\) has smooth source and +\[ + A_V=h^*A_{Y_m}+[G],\qquad G\geq0. +\] +Then the global relative real line is \(J=\OO_V(G)\), the base +class is \(A_{Y_m}\), and its carrier line is +\(N=G-K_V-B_V\). Its relative degrees equal those of the prepared +nef datum. That datum is nef and big on the prepared carrier, for +example a pullback of its prepared Kähler class; thus \(N\) is nef +and big over \(Y_m\). This bigness is a separate hypothesis check, +since \(h\) can have positive-dimensional general fibres. +Effectivity of \(G\) gives relative pseudoeffectivity. Apply the +proposition over \(Y_m\). The subsequent negativity argument must +still remove \(G_m\); only then is the endpoint adjoint the pullback +of \(A_{Y_m}\) and good over the original base. + +For transport of all classes, verify separately that all +Bott--Chern classes on the graph carrier are real-line classes +modulo \(Y_m\), using the smooth rationally connected fibration +and the inherited exceptional splitting on the lower-dimensional +base model. Lemma~\ref{prog:projective-cone} then identifies +the relative degree rays with full Bott--Chern rays, so the +detected-step transport applies. The driving identity involving +\(G\) alone does not give this additional property. +\end{remark} + +\begin{corollary}[Ordinary projective Mori output] +\label{prog:projective-mori} +Let \(f:X\to S\) be an already projective surjective morphism of +compact Kähler spaces, with \(X\) smooth. Let \((X,B)\) be an +effective ordinary real klt pair, and let \(D=K_X+B\). If \(D\) +is not pseudoeffective over \(S\), then its projective program over +\(S\), with scaling of a relatively ample actual real line \(N\), +terminates with a Mori fibre contraction. The initial scaling +parameter is chosen so that \(D+TN\) is nef over \(S\). +The working spaces are globally strongly \(\Q\)-factorial and +compact Kähler, and the actual line comparisons and integral +descent of Proposition~\ref{prog:projective-good-model} apply. +\end{corollary} + +\begin{proof} +The line \(N\) is fixed; it need not be a general scaling direction. +In particular several extremal rays may have the same wall +parameter. Choose \(\epsilon>0\) such that \(D+2\epsilon N\) is +still not pseudoeffective over \(S\). On the finite Stein-compact +cover choose divided general effective representatives of \(N\) +so that all the resulting ordinary pairs representing \(D+tN\), +\(0\leq t\leq T\), are klt. For +\(\epsilon\leq t\leq T\) reserve a common effective relatively +ample rational summand. The joint rational-family construction in +Proposition~\ref{prog:projective-good-model} puts these fixed pairs +in the polytopes required by Theorem~E of +\cite{FujinoAnalyticBCHM}. + +Suppose a finite prefix has reached \(X_i\), and put +\(\mu_i=\lambda_{i-1}\), with \(\mu_0=T\). Its transformed +\(D_i+\mu_iN_i\) is nef. Define +\[ + \lambda_i=\min\{t\in[0,\mu_i]:D_i+tN_i + \text{ is nef over }S\}. +\] +The feasible set is a nonempty closed interval. Inductively +\(\lambda_i>2\epsilon\): all previous steps have nonpositive +comparison for \(D+2\epsilon N\), so its transform is still not +pseudoeffective. A first threshold at most \(2\epsilon\) would +make this transform nef by convexity between the current and +preceding thresholds, a contradiction. + +There is a \(D_i\)-negative extremal ray \(R_i\) with +\((D_i+\lambda_iN_i)\cdot R_i=0\). To see attainment without a +generality assumption, let \(t<\lambda_i\) tend to \(\lambda_i\). +A ray negative for \(D_i+tN_i\) has positive \(N_i\)-degree, +because \(D_i+\mu_iN_i\) is nef. When \(t>\lambda_i/2\), it is +also negative for \(D_i+(\lambda_i/2)N_i\). The latter has fixed +ordinary klt big-boundary representatives on the charts, since +\(\lambda_i/2>\epsilon\). Normalize each of these boundaries by +\cite[Lemma~11.15]{FujinoAnalyticBCHM} as \(A_\ell+B'_\ell\), +with \(A_\ell\geq0\) relatively ample and the new pair klt. +The \(A_\ell\)-truncation of the cone theorem for +\(K+B'_\ell\) gives finitely many rays negative for the entire +adjoint in question. The finite cover therefore gives finitely +many global negative rays. One of these rays attains the wall, +as required. + +Choose one such ray, even if the wall vanishes on other rays. +The rational-support argument for an individual negative ray in +Proposition~\ref{prog:relative-klt} gives a global rational nef +support annihilating exactly \(R_i\), and ordinary relative base +point freeness contracts it. For the real ordinary boundary one +may first choose a nearby rational klt boundary that is still +negative on \(R_i\). The contraction therefore has a rational +ordinary antiample adjoint. Its flip, when small, is constructed +from one global detector algebra as above. + +If the contraction is of fibre type, stop. Otherwise the wall +\(w_i=D_i+\lambda_iN_i\) has degree zero on \(R_i\). +In the finite rational span of its actual line factors, this +zero-degree subspace is rational: its defining functional has +rational values on rational lines. Thus \(w_i\) is a real +combination of rational lines of zero ray degree. +Lemma~\ref{prog:integral-descent} descends those lines to the +contraction base. Their combination is nef over \(S\), by lifting +curves through the projective contraction. Pulling it to the +positive side shows that the transformed wall stays nef. The next +threshold therefore satisfies \(\lambda_{i+1}\leq\lambda_i\); +equality is permitted. No interval of ampleness between successive +thresholds is required. + +If \(E_i\geq0\) is the ordinary \(D_i\)-comparison on a common +resolution, this actual wall descent gives, for every real \(t\), +\[ + p_i^*(D_i+tN_i) + \sim_{\R}q_i^*(D_{i+1}+tN_{i+1}) + +(1-t/\lambda_i)E_i. +\] +For \(t\leq\lambda_i\) the error is effective. This proves all +the nonpositivity and fixed-pair persistence assertions used in +the induction, including the lower bound on thresholds. + +At every working model use the \emph{wall parameter} \(t=\lambda_i\), +rather than an interior ample parameter. Since +\(\lambda_i\leq\lambda_j\) for every earlier step, the displayed +comparisons show that \(X_i\) is a weak log canonical model of +the initial fixed chart pairs representing \(D+\lambda_iN\). +Its trace is nef by definition. Theorem~E counts all these marked +weak log canonical models, including the ones at repeated wall +parameters. The finite chart cover gives a finite global marked +list. Repetition would contradict the strict ordinary +\(D\)-discrepancy increase at an intervening step. Thus this +program with the specified \(N\) is finite. Non-pseudoeffectivity +excludes a nef endpoint for \(D\), so the last contraction is a +Mori fibre contraction. The comparison at \(t=1=\lambda_i\) is +crepant: such a step supplies no strict discrepancy improvement +for \(D+N\), while it is still strictly negative for \(D\). +\end{proof} diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-base-presentations.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-base-presentations.tex index fc2dfb15386121b736483e7c6e3c3deee2bc21a2..169d76499ab63bea2f46f21a707490d868f8521a 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-base-presentations.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-base-presentations.tex @@ -1,644 +1,644 @@ -\subsubsection{Base presentations and multiplier ideals} - -The following constructions take place in the category of generalized -data specified above. In particular, the whole adjoint is a -Bott--Chern class, whereas the boundary and nef traces on a singular -space are interpreted as currents. On a smooth carrier the structure -boundary is a real divisor; its negative components may be exceptional -over the space of the pair. The nef part is specified on that carrier -and pulled back to higher models. - -\begin{lemma}[Adjoint-preserving preparation] -\label{prog:base-presentation-preparation} -Let $(X,B+\mathbf M)$ be a generalized pair on a normal compact -K\"ahler space, with globally modified-big total boundary -$B+\mathbf M_X$. Write -\[ - A=[K_X+B+\mathbf M_X],\qquad - \mathcal J_o=\mathcal J(X,B+\mathbf M). -\] -There are generalized data on $X$ with the same adjoint class, a -projective carrier $p:U\to X$ with $U$ smooth and compact K\"ahler, an SNC real -divisor $C$ whose negative components are $p$-exceptional, and a -K\"ahler class $\beta$ such that -\begin{equation}\label{prog:base-presentation-prepared} - p^*A=[K_U+C]+\beta,\qquad - \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor) - \supseteq\mathcal J_o. -\end{equation} -The carrier may be required to dominate any prescribed projective -modification of $X$, with the structure and exceptional divisors -having simultaneous SNC support. The prepared total boundary is -globally modified big. In particular, the preparation preserves the -gklt condition. -\end{lemma} - -\begin{proof} -Choose a common projective log resolution $p_0:U_0\to X$ of the -original nef carrier and the prescribed modification, and write -\[ - p_0^*A=[K_{U_0}+C^0]+\mathbf M_{U_0}. -\] -The class $\mathbf M_{U_0}$ is nef, and the negative part $(C^0)^-$ -is exceptional over $X$. The modified-big preparation -\cite[Lemmas~2.10 and~2.23]{HX} can be made with the prescribed -resolution. We give its divisor and ideal details. Choose an effective divisor $E_0$ with sufficiently large coefficients -and full $p_0$-exceptional support so that the class -\[ - [(C^0)^+]+\mathbf M_{U_0}+[E_0] -\] -is big. Resolve a Kodaira decomposition of this class. On a further -projective modification $a:U\to U_0$ with $U$ smooth one obtains -\[ - a^*\bigl([(C^0)^+]+\mathbf M_{U_0}+[E_0]\bigr) - =\kappa+[D],\qquad D\geq0, -\] -where $\kappa$ is K\"ahler. The support can be made SNC together -with all the other divisors. On any additional blowup, subtract a -sufficiently small effective exceptional class from the pulled-back -K\"ahler class and add it to the divisor; an exceptional divisor -whose negative is relatively ample makes the resulting class -K\"ahler again. - -Put $p=p_0a$ and -\[ - C^{\mathrm{alt}}=D-a^*(C^0)^--a^*E_0-K_{U/U_0}. -\] -Then -\[ - p^*A=[K_U+C^{\mathrm{alt}}]+\kappa, -\] -and every negative component of $C^{\mathrm{alt}}$ is -$p$-exceptional. Let $C^{\mathrm{old}}$ be the crepant structure -boundary of the original data on this same model. For $00),\qquad - f_*\mathcal O_{N_1}=\mathcal O_Z/(f_*\mathcal J_1), - \qquad N_1=V(\mathcal J_1). -\end{equation} -If $N_1$ dominates $Z$, then $f_*\mathcal J_1=0$. -Otherwise there are generalized data $(Z,B_Z+\mathbf N)$, a -projective carrier $\tau:T\to Z$ with $T$ smooth and compact K\"ahler, a real -SNC structure divisor $C_T$, and a K\"ahler class $\beta_T$ such that -\begin{equation}\label{prog:base-presentation-identities} - \begin{gathered} - [K_Z+B_Z+\mathbf N_Z]=\gamma,\qquad - \tau^*\gamma=[K_T+C_T]+\beta_T,\qquad - \mathbf N_T=\beta_T,\\ - B_Z=\tau_*C_T\geq0,\qquad - B_Z+\mathbf N_Z\ \text{is globally modified big}. - \end{gathered} -\end{equation} -The negative components of $C_T$ are $\tau$-exceptional, and -\begin{equation}\label{prog:base-presentation-ideal} - \mathcal J_Z:=\mathcal J(Z,B_Z+\mathbf N) - =\tau_*\mathcal O_T(-\lfloor C_T\rfloor) - \supseteq f_*\mathcal J_1. -\end{equation} -In particular the base data are gklt wherever $f_*\mathcal J_1$ is -the unit ideal. If the source data are gklt, the base data are gklt. -All adjoint identities are preserved by pullback to higher models. -\end{proposition} - -\begin{proof} -Choose a projective resolution $\mu:W\to Z$, with $W$ smooth -compact K\"ahler and reduced exceptional divisor $F$ having SNC -support. Apply Lemma~\ref{prog:base-presentation-preparation} on a -carrier dominating the main component of the base change. We obtain -projective maps $p:U\to X$ and $q:U\to W$ with $fp=\mu q$ and -\begin{equation}\label{prog:base-presentation-source} - [K_U+C]+\beta=p^*A=q^*\mu^*\gamma, - \qquad \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor). -\end{equation} -Here $U$ is smooth compact K\"ahler, $\beta$ is K\"ahler, and all -negative components of $C$ are $p$-exceptional. The maps are -projective by composition and factorization over the separated base -$W$. The map $q$ has connected fibres: this holds over the open set -where $\mu$ is an isomorphism, and its Stein factorization is finite -and bimeromorphic over the normal space $W$. - -The integral divisor $-\lfloor C\rfloor$ has class -\[ - [-\lfloor C\rfloor] - =[K_U+\{C\}]+\beta-(fp)^*\gamma. -\] -The pair $(U,\{C\})$ is klt with SNC boundary. Relative -Kawamata--Viehweg vanishing, first for $p$ and then for $fp$, applies -because the positive class modulo pullbacks is K\"ahler -\cite[Theorem~2.2]{HX}. Equivalently, the real line bundle -$-K_U-C$ is relatively ample, -so the vanishing follows from \cite[Theorem~1.2]{FujinoVanishing2025} -with the identity as its first morphism. Leray gives $R^if_*\mathcal J_1=0$ for -$i>0$. Applying $f_*$ to the sequence defining $N_1$, and using -$f_*\mathcal O_X=\mathcal O_Z$, proves -\eqref{prog:base-presentation-Stein}. If $N_1$ dominates $Z$, the -ideal $f_*\mathcal J_1\subseteq\mathcal O_Z$ vanishes generically -and therefore vanishes. This proves that alternative. - -Suppose from now on that $N_1$ does not dominate $Z$. Every -$q$-horizontal coefficient of $C$ is then less than one. Indeed, -a component with coefficient at least one forces the multiplier -ideal to be proper along its image in $X$, and its image would make -$N_1$ dominate $Z$. If $Z$ is a point, the asserted base presentation -is the dimension-zero presentation. We therefore assume -$b=\dim Z>0$. - -\paragraph{The fibre metric.} -Choose a smooth representative $\rho$ of -$\mu^*\gamma-c_1(K_W)$, a K\"ahler form $\omega_W$, and a -K\"ahler form representing $\beta$, still denoted by $\beta$. -For some $a>0$, -\[ - \beta\geq a q^*\omega_W. -\] -Consider the holomorphic line bundle -\[ - L=\mathcal O_U(-K_{U/W}-\lfloor C\rfloor). -\] -Equation~\eqref{prog:base-presentation-source} gives a singular -Hermitian metric $h$ on $L$ whose curvature current is -\begin{equation}\label{prog:base-presentation-curvature} - \Theta_h(L)=\beta+[\{C\}]-q^*\rho. -\end{equation} -Its weights are smooth functions plus the divisorial logarithms -with coefficients in $\{C\}$. We normalize $dd^c$ so that -$dd^c\log|z|^2=[z=0]$ and write squared-norm weights as $e^{-\varphi}$. -Let $e$ be the canonical meromorphic section of -\[ - K_{U/W}\otimes L=\mathcal O_U(-\lfloor C\rfloor), -\] -so that $\operatorname{div}(e)=-\lfloor C\rfloor$. - -Choose an analytic Zariski open set $W^\circ\subseteq W$ on which -$q$ is a submersion, the horizontal boundary is relatively SNC, -there are no vertical boundary components, $\mu$ is an isomorphism -onto an open subset of $Z$, and coherent base change holds in degree -zero. On $q^{-1}(W^\circ)$, the divisor $-\lfloor C\rfloor$ is -effective and exceptional over $X$. Normality and the contraction -property therefore identify -\[ - q_*(K_{U/W}\otimes L)|_{W^\circ} - =\mathcal O_{W^\circ}e. -\] -Here $K_{U/W}$ is viewed as the relative canonical line bundle. The -fibre norm -\[ - I(w)=\int_{U_w}|e_w|_h^2 -\] -is positive and finite. The fractional SNC weights are integrable; -any zeros of $e_w$ only improve integrability. Relative SNC -coordinates and a partition of unity show that $I$ is smooth: -parameter derivatives act on smooth factors multiplying fixed -integrable monomial weights. - -Set $\psi=-\log I$ on $W^\circ$. Then -\begin{equation}\label{prog:base-presentation-open-positivity} - \rho+dd^c\psi\geq a\omega_W. -\end{equation} -Here are the details for the singular weights. Locally choose -$\ell$ with $dd^c\ell=\rho-a\omega_W$, and put -$h'=he^{-q^*\ell}$. Its weights are psh, and its fibre norm satisfies -$I'=e^{-\ell}I$. Restrict the family to a small coordinate line disk -$\Delta_R$, with centre $0$. The sharp $L^2$ extension theorem gives -a twisted top form extending $e_0$ and satisfying -\begin{equation}\label{prog:base-presentation-sharp-extension} - \int_{q^{-1}(\Delta_R)}|E|_{h'}^2 - \leq\pi R^2 I'(0). -\end{equation} -Top-form norms here are normalized to use Euclidean area in the -base coordinate. We use the Stein hypersurface form of the optimal -extension theorem \cite[Theorem~2.2 and \S3.5, (3.1)]{GuanZhouOptimal2015}, -with defining function the disk coordinate divided by $R$. - -To check applicability, embed the restricted projective family into -projective space over the disk and remove a relative hyperplane -section that does not contain the central fibre. The complement is -Stein. A holomorphic section of $L$ on this Stein space can be chosen -nonzero at a point of the central fibre; deleting its zero divisor -trivializes $L$ and leaves a Stein space. The central fibre is smooth -and connected, hence irreducible, so both deletions leave a dense -open subset of it. For a zero-dimensional fibre choose the deletions -to avoid its point. In the resulting trivialization the weight is -psh, and the sharp extension theorem applies to canonical forms. -The extension extends across the deleted hypersurfaces in reverse -order by local $L^2$ removability: in every original frame the metric -is bounded below by a smooth positive metric. The extension retains -its norm bound and its prescribed restriction to the central fibre. - -After dividing $E$ by the base differential, the rank-one direct -image identifies it with $b(t)e$, where $b$ is holomorphic and -$b(0)=1$. Thus the disk average of $|b(t)|^2I'(t)$ is at most $I'(0)$. -Jensen's inequality and the submean inequality for $\log|b|^2$ give -\[ - \log I'(0) - \geq\operatorname{avg}_{\Delta_R} - \bigl(\log|b(t)|^2+\log I'(t)\bigr) - \geq\operatorname{avg}_{\Delta_R}\log I'(t). -\] -The same argument applies to every sufficiently small disk at every -centre and in every complex line direction. Hence $-\log I'=\psi+\ell$ is psh, which proves -\eqref{prog:base-presentation-open-positivity}. - -\paragraph{Extension over the base.} -Let $D$ be a prime divisor in $W\setminus W^\circ$. Choose a component -$Q$ of $q^*D$ dominating $D$. When $D$ is not $\mu$-exceptional, choose -$Q$ not $p$-exceptional: take a component over the corresponding prime -on $Z$ already on $X$ and then its strict transform. Put -\[ - u=\operatorname{ord}_Q(q^*D),\qquad c=\operatorname{coeff}_Q(C). -\] -At a general point of $Q$, avoiding all other relevant divisors, -there are local coordinates with -\[ - t_1=z_1^u,\qquad t_j=z_j\quad(2\leq j\leq b). -\] -Indeed the map $Q\to D$ is submersive at a general smooth point, -and a local unit in the first equation is absorbed by a holomorphic -$u$th root. Such points can be chosen above every point of a dense -analytic open subset of $D$ by generic smoothness and properness. -Integration over a fixed small polydisk in the remaining fibre -coordinates gives -\begin{equation}\label{prog:base-presentation-divisor-bound} - I(t)\geq c_*|t_1|^{-2(c+u-1)/u} -\end{equation} -for some $c_*>0$ near each such base point. The section and metric -contribute the exponent $c$, while division by the base differential -in the relative canonical form contributes $u-1$. - -For a nonexceptional $D$ the chosen $Q$ has $c\geq0$, so -\eqref{prog:base-presentation-divisor-bound} bounds $\psi$ from above -near its general points. For the finitely many exceptional divisors, -the same estimate supplies a nonnegative integer $m$ such that, if -$k$ is a smooth metric on $\mathcal O_W(F)$ and $s_F$ its divisor -section, -\[ - \varphi=\psi+m\log|s_F|_k^2 -\] -is locally bounded above near general points of every divisor. -Adding local potentials of $\rho+m c_1(k)-a\omega_W$ to -$\varphi$ gives psh weights by -\eqref{prog:base-presentation-open-positivity}. These weights -extend across the divisors and then across the remaining analytic -set of codimension at least two, giving -\begin{equation}\label{prog:base-presentation-positive-current} - S=\rho+m c_1(k)+dd^c\varphi\geq a\omega_W -\end{equation} -on all of $W$. - -\paragraph{A K\"ahler carrier.} -Regularize the current in -\eqref{prog:base-presentation-positive-current} in its class, retaining -a lower bound for the potentials. More precisely, choose an -approximation with analytic singularities such that -\begin{equation}\label{prog:base-presentation-regularization} - S'=\rho+m c_1(k)+dd^c\varphi',\qquad - S'\geq\frac a2\omega_W,\qquad - \varphi'\geq\varphi-O(1). -\end{equation} -These simultaneous properties are the Bergman regularization -properties of \cite{DemaillyRegularization1992}; see also -\cite[pp.~59--60]{PopoviciRegularization2004}. -Resolve the analytic singularities and the support of $F$ by a -sequence of blowups with smooth centres $r:T\to W$, with simultaneous -SNC support, and write -\[ - r^*S'=[H]+\zeta,\qquad H\geq0. -\] -The residual current $\zeta$ has locally bounded potentials. Removing -the divisorial part from the positive current -$r^*S'-(a/2)r^*\omega_W$ shows that -\[ - \zeta\geq\frac a2r^*\omega_W. -\] -The difference has vanishing Lelong numbers, so its class is nef by -regularization. Choose an effective $r$-exceptional divisor $G$ with -$-G$ relatively ample. For $0<\eta\ll1$, the class -$(a/2)r^*\omega_W-\eta[G]$ is K\"ahler. Therefore -\begin{equation}\label{prog:base-presentation-carrier-data} - \begin{split} - \tau&=\mu r,\qquad - \beta_T=[\zeta]-\eta[G],\\ - C_T&=H+\eta G-mr^*F-K_{T/W} - \end{split} -\end{equation} -has K\"ahler $\beta_T$. If $r$ is an isomorphism, omit $G$ and its -correction. The choice of $\eta$ will be decreased once more below. - -The exact class calculation is -\[ - \begin{split} - [K_T+C_T]+\beta_T - &=r^*c_1(K_W)+[H]+[\zeta]-m[r^*F]\\ - &=r^*(c_1(K_W)+[\rho])=\tau^*\gamma. - \end{split} -\] -All possibly negative terms in $C_T$ are $\tau$-exceptional. -Consequently $B_Z=\tau_*C_T$ is effective. Specify the nef b-data by -$\mathbf N_T=\beta_T$; the displayed identity defines generalized -data on $Z$ with adjoint class $\gamma$. At the level of currents, -pushing the identity gives the required trace identity on $Z$: -exceptional divisor currents push to zero, and a $dd^c$ difference -pushes to a $dd^c$ difference. Canonical traces can be computed using -local meromorphic canonical frames; a change of frame changes the -corresponding Chern potentials by the usual logarithmic transition -term. Thus only the whole adjoint, rather -than its individual traces, needs to be a Bott--Chern class. The -class $\beta_T$ plus the strict transform of $B_Z$ is big on $T$, -so its pushforward $B_Z+\mathbf N_Z$ is globally modified big. -These are~\eqref{prog:base-presentation-identities}. - -\paragraph{The multiplier ideal.} -Take a local holomorphic function $v\in f_*\mathcal J_1$. By the -pushforward description of $\mathcal J_1$, -\begin{equation}\label{prog:base-presentation-source-orders} - \operatorname{div}(q^*\mu^*v)\geq\lfloor C\rfloor. -\end{equation} -Thus $q^*\mu^*v\,e$, tensored with a local base canonical frame, -is a holomorphic $L$-valued top form on $U$. Its squared norm is -locally integrable by the fractional SNC weights in -\eqref{prog:base-presentation-curvature}. Over the submersion locus, -fibre integration followed by base integration is precisely this -top-form integral. Properness and a finite coordinate covering -therefore give -\begin{equation}\label{prog:base-presentation-base-integral} - \int |\mu^*v|^2e^{-\psi}<\infty -\end{equation} -locally on $W$; the integral is taken where the original fibre -metric is defined. - -The lower bound on $\varphi'$ in -\eqref{prog:base-presentation-regularization} preserves this -integrability after replacing $\psi$ by -$\varphi'-m\log|s_F|_k^2$. Change variables through $r$. Along a -prime divisor of $T$, let $h,j,k_D,d$ be respectively the coefficients -of $H$, $\operatorname{div}(\tau^*v)$, $K_{T/W}$, and $r^*F$. -The local integrand has exponent -$2(j-h+md+k_D)$, up to a bounded positive factor. Its integrability -implies $j-h+md+k_D>-1$, or, since $j$ is integral, -\[ - j\geq\lfloor h-md-k_D\rfloor. -\] -Therefore -\[ - \operatorname{div}(\tau^*v) - \geq\lfloor H-mr^*F-K_{T/W}\rfloor. -\] -Choose $\eta>0$ sufficiently small that adding $\eta G$ does not -change any of these finitely many round-downs. The resulting -inequality is exactly -$\operatorname{div}(\tau^*v)\geq\lfloor C_T\rfloor$, proving -\eqref{prog:base-presentation-ideal} as an inclusion of ideal sheaves. -In particular, if $f_*\mathcal J_1$ is the unit ideal on an open set, -then so is $\mathcal J_Z$, and the base data are gklt there. - -Finally, the nef data descend to $T$, and the structure boundaries -on higher models are defined by crepant pullback of -\eqref{prog:base-presentation-identities}. On every common refinement -of the source and base carriers, both adjoints are the pullback of -$\gamma$. Thus the identities remain exact on higher models. -\end{proof} - -\begin{lemma}[Contractions of closed subspaces] -\label{prog:base-emc} -Let $D$ be a compact complex space, possibly nonreduced, and let -$\delta\in H^2(D,\R)$ have nonnegative degree on compact curves. -Use the definitions of a Moishezon morphism and of a contraction -defined by a class in \cite[Definitions~2.32 and~2.34]{HX}. -Then the following hold. -\begin{enumerate} -\item If $\delta$ is endowed with a Moishezon contraction, so is -its restriction to every closed analytic subspace of $D$. -\item Let $g:D\to V$ be a projective contraction of compact -complex spaces, so $g_*\OO_D=\OO_V$, and suppose -$\delta=g^*\epsilon$ for a class $\epsilon\in H^2(V,\R)$ -having nonnegative degree on compact curves. Then $\delta$ is -endowed with a Moishezon contraction if and only if $\epsilon$ -is endowed with one. -\end{enumerate} -\end{lemma} - -\begin{proof} -Write $h:D\to S$ for a Moishezon contraction defined by -$\delta$. For a closed subspace $D_0\subset D$, take the Stein -factorization -\[ - D_0\longrightarrow S_0\longrightarrow S -\] -of its restriction. The first map is a contraction, and the second -is finite. Thus a curve in $D_0$ is contracted by the first map -exactly when its $\delta$-degree is zero. Choose a projective -surjection $P\to D$ witnessing that $h$ is Moishezon, with -$P\to S$ projective. The closed subspace -$P_0=P\times_D D_0$ is projective over $S$ and projective -and surjective over $D_0$. It is also projective over $S_0$: -a relatively ample line for $P_0\to S$ remains ample on the -fibres of $P_0\to S_0$. This proves the first assertion, -including its scheme structure. - -For the descent direction of the second assertion, every curve -in a fibre of $g$ has $\delta$-degree zero. A connected -projective complex space is connected by chains of compact curves -after passing to its reduction. Therefore $h$ is constant on -the underlying set of each fibre of $g$. It factors as -$h=k\circ g$ for a holomorphic map $k:V\to S$. -To see that this factorization holds also for nonreduced spaces, -choose a coordinate neighbourhood in $S$ of the image of a fibre. -Properness permits restriction to a neighbourhood of its point -in $V$ whose inverse image maps into that coordinate -neighbourhood. The coordinate functions of $h$ then descend -uniquely through $g_*\OO_D=\OO_V$; their analytic relations -descend through the same equality. These local factorizations -glue. Furthermore, -\[ - k_*\OO_V=h_*\OO_D=\OO_S. -\] -Every irreducible compact curve $C\subset V$ is dominated by -a compact curve $C'\subset D$. Indeed, after base change to -the normalization of $C$, take a component dominating that curve -and a projective multisection. The projection formula gives -\[ - \delta\cdot C' - =\deg(C'\to C)\,\epsilon\cdot C. -\] -Consequently $k$ contracts exactly the curves of -$\epsilon$-degree zero. The composite $P\to D\to V$ -is a projective surjection, and $P\to S$ is projective, so -it witnesses that $k$ is Moishezon. - -Conversely, suppose $k:V\to S$ is a Moishezon contraction -defined by $\epsilon$. Then $k\circ g$ is a contraction, -and the projection formula verifies its curve criterion, treating -separately curves contracted by $g$. If $Q\to V$ is a -projective surjection witnessing that $k$ is Moishezon, then -$D\times_V Q\to D$ is projective and surjective, while -$D\times_V Q\to Q\to S$ is projective. This supplies the -required witness. -\end{proof} - -\begin{lemma}[Crepant transport of the multiplier closed subspace] -\label{prog:multiplier-transport} -Let $g:X\to Y$ be a projective bimeromorphic morphism of -normal compact K\"ahler spaces. Suppose effective generalized -pair data on $X$ and $Y$ have the same globally nef b-part -and are crepant over $Y$. In particular their adjoint classes -$A_X$ and $A_Y$ satisfy $A_X=g^*A_Y$ in Bott--Chern -cohomology. Write $J_X,J_Y$ for their multiplier ideals and -$N_X=V(J_X)$, $N_Y=V(J_Y)$ for their non-klt closed -subspaces. Then -\[ - g_*J_X=J_Y,\qquad R^ig_*J_X=0\quad(i>0), - \qquad g_*\OO_{N_X}=\OO_{N_Y}. -\] -The restriction $N_X\to N_Y$ is a projective contraction. -Hence, whenever $A_Y$ is nef, the restriction of $A_X$ to -$N_X$ is endowed with a Moishezon contraction if and only if -the restriction of $A_Y$ to $N_Y$ is endowed with one. -\end{lemma} - -\begin{proof} -Take a projective common log resolution $r:U\to X$ carrying -the nef datum, and put $s=g\circ r$. By crepancy there is -one real SNC divisor $C$ such that -\[ - K_U+[C]+\beta=s^*A_Y=r^*A_X, -\] -where $\beta$ is globally nef. The multiplier ideals are -\[ - J_X=r_*\OO_U(-\lfloor C\rfloor),\qquad - J_Y=s_*\OO_U(-\lfloor C\rfloor). -\] -Their direct-image equality follows at once. For vanishing, set -$L=\OO_U(-\lfloor C\rfloor)$ and -$\Delta=C-\lfloor C\rfloor$. The pair $(U,\Delta)$ -is klt, and the actual real line -\[ - L-(K_U+\Delta)=-K_U-C -\] -has class $\beta-s^*A_Y$. It is relatively nef over $Y$ -and relatively big because $s$ is bimeromorphic. The same -statements hold over $X$. Relative Kawamata--Viehweg vanishing, -in the form of \cite[Theorem~1.2]{FujinoVanishing2025} with -the first morphism the identity and the real line -$\mathcal H=-K_U-C$, therefore gives -\[ - R^is_*L=R^ir_*L=0\qquad(i>0). -\] -The real-line identity in that theorem is the displayed definition -of $\mathcal H$; its relative positivity is checked by the class -identity above. The klt condition means that there are no proper log -canonical strata to test for relative log bigness. Leray gives -$R^ig_*J_X=0$ for $i>0$. Pushing forward -$0\to J_X\to\OO_X\to\OO_{N_X}\to0$ and using -$g_*\OO_X=\OO_Y$ now proves the asserted structure-sheaf -equality. It also identifies the scheme-theoretic image with -$N_Y$, giving the projective contraction between these closed -subspaces. Apply Lemma~\ref{prog:base-emc} to conclude. -\end{proof} - -\begin{remark} -The preceding lemma also transports the contraction hypothesis -through a crepant projective flip: apply it to the two contractions -to their common base, equipped with the descended adjoint and -the common nef b-data. On that base the boundary is the effective -pushforward of the source boundary, and the nef data remain on -the same carrier; the descended Bott--Chern adjoint and its -identity on a common resolution define the crepant base data -in the whole-adjoint and current-trace category used here. -All assertions concern the fixed adjoint -and its multiplier ideal. No conclusion about the multiplier -ideal of a different driving adjoint is used. -\end{remark} - -\begin{corollary}[The contraction step over a lower-dimensional base] -\label{prog:base-emc-induction} -Use the hypotheses and notation of -Proposition~\ref{prog:base-presentation}. Suppose that -$A=f^*\gamma$ is nef and that its restriction to the original -non-klt closed subspace $N_o=V(J_o)$ is endowed with a -Moishezon contraction. Put -\[ - I=f_*J_1,\qquad N_*=V(I)\subset Z. -\] -Then $\gamma|_{N_*}$ is endowed with a Moishezon contraction. -If the prepared non-klt closed subspace $N_1=V(J_1)$ dominates -$Z$, then $\gamma$ and $A$ are endowed with Moishezon -contractions. Otherwise the generalized pair supplied on $Z$ -has a non-klt closed subspace endowed with the contraction -defined by the restricted class of $\gamma$. In this latter -case, if $\gamma$ is nef and NQC and the assertion -$\mathsf C_{\dim Z}$ is available, then $\gamma$ and $A$ -are endowed with Moishezon contractions. -\end{corollary} - -\begin{proof} -The inclusion $J_o\subset J_1$ gives $N_1\subset N_o$ -as closed analytic subspaces. By -Lemma~\ref{prog:base-emc}, the given contraction restricts to -$N_1$. Proposition~\ref{prog:base-presentation} identifies -\[ - f_*\OO_{N_1}=\OO_Z/I=\OO_{N_*}. -\] -Thus the restriction $N_1\to N_*$ is a projective -contraction. The restricted adjoint is the pullback of -$\gamma|_{N_*}$, so Lemma~\ref{prog:base-emc} descends -the contraction to $N_*$. Nonnegative degrees on curves of -$N_*$ follow from projective multisections and nefness of -$A$. - -If $N_1$ dominates the normal irreducible space $Z$, then -$I=0$: a local function in $I$ pulls back to zero on the -dominating closed subspace and hence vanishes generically on -$Z$. Normality, and in particular reducedness, gives its -vanishing. Therefore $N_*=Z$, which proves the assertion in -this case. - -If $N_1$ does not dominate $Z$, the constructed base ideal -satisfies $I\subset J_Z$. Hence -\[ - V(J_Z)\subset V(I)=N_* -\] -as closed analytic subspaces. Restriction by -Lemma~\ref{prog:base-emc} provides the required contraction -on $V(J_Z)$, with its complete scheme structure. The -generalized pair on $Z$ has globally nef carrier data and -globally modified-big total boundary by -Proposition~\ref{prog:base-presentation}. When the stated -nef and NQC conditions hold, these facts are precisely the -hypotheses of $\mathsf C_{\dim Z}$, which gives the -contraction on $Z$. Finally Lemma~\ref{prog:base-emc} -pulls it back through the projective contraction $f$ to $X$. -The case of an empty non-klt closed subspace has the same -interpretation, with its contraction condition vacuous. -\end{proof} +\subsubsection{Base presentations and multiplier ideals} + +The following constructions take place in the category of generalized +data specified above. In particular, the whole adjoint is a +Bott--Chern class, whereas the boundary and nef traces on a singular +space are interpreted as currents. On a smooth carrier the structure +boundary is a real divisor; its negative components may be exceptional +over the space of the pair. The nef part is specified on that carrier +and pulled back to higher models. + +\begin{lemma}[Adjoint-preserving preparation] +\label{prog:base-presentation-preparation} +Let $(X,B+\mathbf M)$ be a generalized pair on a normal compact +K\"ahler space, with globally modified-big total boundary +$B+\mathbf M_X$. Write +\[ + A=[K_X+B+\mathbf M_X],\qquad + \mathcal J_o=\mathcal J(X,B+\mathbf M). +\] +There are generalized data on $X$ with the same adjoint class, a +projective carrier $p:U\to X$ with $U$ smooth and compact K\"ahler, an SNC real +divisor $C$ whose negative components are $p$-exceptional, and a +K\"ahler class $\beta$ such that +\begin{equation}\label{prog:base-presentation-prepared} + p^*A=[K_U+C]+\beta,\qquad + \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor) + \supseteq\mathcal J_o. +\end{equation} +The carrier may be required to dominate any prescribed projective +modification of $X$, with the structure and exceptional divisors +having simultaneous SNC support. The prepared total boundary is +globally modified big. In particular, the preparation preserves the +gklt condition. +\end{lemma} + +\begin{proof} +Choose a common projective log resolution $p_0:U_0\to X$ of the +original nef carrier and the prescribed modification, and write +\[ + p_0^*A=[K_{U_0}+C^0]+\mathbf M_{U_0}. +\] +The class $\mathbf M_{U_0}$ is nef, and the negative part $(C^0)^-$ +is exceptional over $X$. The modified-big preparation +\cite[Lemmas~2.10 and~2.23]{HX} can be made with the prescribed +resolution. We give its divisor and ideal details. Choose an effective divisor $E_0$ with sufficiently large coefficients +and full $p_0$-exceptional support so that the class +\[ + [(C^0)^+]+\mathbf M_{U_0}+[E_0] +\] +is big. Resolve a Kodaira decomposition of this class. On a further +projective modification $a:U\to U_0$ with $U$ smooth one obtains +\[ + a^*\bigl([(C^0)^+]+\mathbf M_{U_0}+[E_0]\bigr) + =\kappa+[D],\qquad D\geq0, +\] +where $\kappa$ is K\"ahler. The support can be made SNC together +with all the other divisors. On any additional blowup, subtract a +sufficiently small effective exceptional class from the pulled-back +K\"ahler class and add it to the divisor; an exceptional divisor +whose negative is relatively ample makes the resulting class +K\"ahler again. + +Put $p=p_0a$ and +\[ + C^{\mathrm{alt}}=D-a^*(C^0)^--a^*E_0-K_{U/U_0}. +\] +Then +\[ + p^*A=[K_U+C^{\mathrm{alt}}]+\kappa, +\] +and every negative component of $C^{\mathrm{alt}}$ is +$p$-exceptional. Let $C^{\mathrm{old}}$ be the crepant structure +boundary of the original data on this same model. For $00),\qquad + f_*\mathcal O_{N_1}=\mathcal O_Z/(f_*\mathcal J_1), + \qquad N_1=V(\mathcal J_1). +\end{equation} +If $N_1$ dominates $Z$, then $f_*\mathcal J_1=0$. +Otherwise there are generalized data $(Z,B_Z+\mathbf N)$, a +projective carrier $\tau:T\to Z$ with $T$ smooth and compact K\"ahler, a real +SNC structure divisor $C_T$, and a K\"ahler class $\beta_T$ such that +\begin{equation}\label{prog:base-presentation-identities} + \begin{gathered} + [K_Z+B_Z+\mathbf N_Z]=\gamma,\qquad + \tau^*\gamma=[K_T+C_T]+\beta_T,\qquad + \mathbf N_T=\beta_T,\\ + B_Z=\tau_*C_T\geq0,\qquad + B_Z+\mathbf N_Z\ \text{is globally modified big}. + \end{gathered} +\end{equation} +The negative components of $C_T$ are $\tau$-exceptional, and +\begin{equation}\label{prog:base-presentation-ideal} + \mathcal J_Z:=\mathcal J(Z,B_Z+\mathbf N) + =\tau_*\mathcal O_T(-\lfloor C_T\rfloor) + \supseteq f_*\mathcal J_1. +\end{equation} +In particular the base data are gklt wherever $f_*\mathcal J_1$ is +the unit ideal. If the source data are gklt, the base data are gklt. +All adjoint identities are preserved by pullback to higher models. +\end{proposition} + +\begin{proof} +Choose a projective resolution $\mu:W\to Z$, with $W$ smooth +compact K\"ahler and reduced exceptional divisor $F$ having SNC +support. Apply Lemma~\ref{prog:base-presentation-preparation} on a +carrier dominating the main component of the base change. We obtain +projective maps $p:U\to X$ and $q:U\to W$ with $fp=\mu q$ and +\begin{equation}\label{prog:base-presentation-source} + [K_U+C]+\beta=p^*A=q^*\mu^*\gamma, + \qquad \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor). +\end{equation} +Here $U$ is smooth compact K\"ahler, $\beta$ is K\"ahler, and all +negative components of $C$ are $p$-exceptional. The maps are +projective by composition and factorization over the separated base +$W$. The map $q$ has connected fibres: this holds over the open set +where $\mu$ is an isomorphism, and its Stein factorization is finite +and bimeromorphic over the normal space $W$. + +The integral divisor $-\lfloor C\rfloor$ has class +\[ + [-\lfloor C\rfloor] + =[K_U+\{C\}]+\beta-(fp)^*\gamma. +\] +The pair $(U,\{C\})$ is klt with SNC boundary. Relative +Kawamata--Viehweg vanishing, first for $p$ and then for $fp$, applies +because the positive class modulo pullbacks is K\"ahler +\cite[Theorem~2.2]{HX}. Equivalently, the real line bundle +$-K_U-C$ is relatively ample, +so the vanishing follows from \cite[Theorem~1.2]{FujinoVanishing2025} +with the identity as its first morphism. Leray gives $R^if_*\mathcal J_1=0$ for +$i>0$. Applying $f_*$ to the sequence defining $N_1$, and using +$f_*\mathcal O_X=\mathcal O_Z$, proves +\eqref{prog:base-presentation-Stein}. If $N_1$ dominates $Z$, the +ideal $f_*\mathcal J_1\subseteq\mathcal O_Z$ vanishes generically +and therefore vanishes. This proves that alternative. + +Suppose from now on that $N_1$ does not dominate $Z$. Every +$q$-horizontal coefficient of $C$ is then less than one. Indeed, +a component with coefficient at least one forces the multiplier +ideal to be proper along its image in $X$, and its image would make +$N_1$ dominate $Z$. If $Z$ is a point, the asserted base presentation +is the dimension-zero presentation. We therefore assume +$b=\dim Z>0$. + +\paragraph{The fibre metric.} +Choose a smooth representative $\rho$ of +$\mu^*\gamma-c_1(K_W)$, a K\"ahler form $\omega_W$, and a +K\"ahler form representing $\beta$, still denoted by $\beta$. +For some $a>0$, +\[ + \beta\geq a q^*\omega_W. +\] +Consider the holomorphic line bundle +\[ + L=\mathcal O_U(-K_{U/W}-\lfloor C\rfloor). +\] +Equation~\eqref{prog:base-presentation-source} gives a singular +Hermitian metric $h$ on $L$ whose curvature current is +\begin{equation}\label{prog:base-presentation-curvature} + \Theta_h(L)=\beta+[\{C\}]-q^*\rho. +\end{equation} +Its weights are smooth functions plus the divisorial logarithms +with coefficients in $\{C\}$. We normalize $dd^c$ so that +$dd^c\log|z|^2=[z=0]$ and write squared-norm weights as $e^{-\varphi}$. +Let $e$ be the canonical meromorphic section of +\[ + K_{U/W}\otimes L=\mathcal O_U(-\lfloor C\rfloor), +\] +so that $\operatorname{div}(e)=-\lfloor C\rfloor$. + +Choose an analytic Zariski open set $W^\circ\subseteq W$ on which +$q$ is a submersion, the horizontal boundary is relatively SNC, +there are no vertical boundary components, $\mu$ is an isomorphism +onto an open subset of $Z$, and coherent base change holds in degree +zero. On $q^{-1}(W^\circ)$, the divisor $-\lfloor C\rfloor$ is +effective and exceptional over $X$. Normality and the contraction +property therefore identify +\[ + q_*(K_{U/W}\otimes L)|_{W^\circ} + =\mathcal O_{W^\circ}e. +\] +Here $K_{U/W}$ is viewed as the relative canonical line bundle. The +fibre norm +\[ + I(w)=\int_{U_w}|e_w|_h^2 +\] +is positive and finite. The fractional SNC weights are integrable; +any zeros of $e_w$ only improve integrability. Relative SNC +coordinates and a partition of unity show that $I$ is smooth: +parameter derivatives act on smooth factors multiplying fixed +integrable monomial weights. + +Set $\psi=-\log I$ on $W^\circ$. Then +\begin{equation}\label{prog:base-presentation-open-positivity} + \rho+dd^c\psi\geq a\omega_W. +\end{equation} +Here are the details for the singular weights. Locally choose +$\ell$ with $dd^c\ell=\rho-a\omega_W$, and put +$h'=he^{-q^*\ell}$. Its weights are psh, and its fibre norm satisfies +$I'=e^{-\ell}I$. Restrict the family to a small coordinate line disk +$\Delta_R$, with centre $0$. The sharp $L^2$ extension theorem gives +a twisted top form extending $e_0$ and satisfying +\begin{equation}\label{prog:base-presentation-sharp-extension} + \int_{q^{-1}(\Delta_R)}|E|_{h'}^2 + \leq\pi R^2 I'(0). +\end{equation} +Top-form norms here are normalized to use Euclidean area in the +base coordinate. We use the Stein hypersurface form of the optimal +extension theorem \cite[Theorem~2.2 and \S3.5, (3.1)]{GuanZhouOptimal2015}, +with defining function the disk coordinate divided by $R$. + +To check applicability, embed the restricted projective family into +projective space over the disk and remove a relative hyperplane +section that does not contain the central fibre. The complement is +Stein. A holomorphic section of $L$ on this Stein space can be chosen +nonzero at a point of the central fibre; deleting its zero divisor +trivializes $L$ and leaves a Stein space. The central fibre is smooth +and connected, hence irreducible, so both deletions leave a dense +open subset of it. For a zero-dimensional fibre choose the deletions +to avoid its point. In the resulting trivialization the weight is +psh, and the sharp extension theorem applies to canonical forms. +The extension extends across the deleted hypersurfaces in reverse +order by local $L^2$ removability: in every original frame the metric +is bounded below by a smooth positive metric. The extension retains +its norm bound and its prescribed restriction to the central fibre. + +After dividing $E$ by the base differential, the rank-one direct +image identifies it with $b(t)e$, where $b$ is holomorphic and +$b(0)=1$. Thus the disk average of $|b(t)|^2I'(t)$ is at most $I'(0)$. +Jensen's inequality and the submean inequality for $\log|b|^2$ give +\[ + \log I'(0) + \geq\operatorname{avg}_{\Delta_R} + \bigl(\log|b(t)|^2+\log I'(t)\bigr) + \geq\operatorname{avg}_{\Delta_R}\log I'(t). +\] +The same argument applies to every sufficiently small disk at every +centre and in every complex line direction. Hence $-\log I'=\psi+\ell$ is psh, which proves +\eqref{prog:base-presentation-open-positivity}. + +\paragraph{Extension over the base.} +Let $D$ be a prime divisor in $W\setminus W^\circ$. Choose a component +$Q$ of $q^*D$ dominating $D$. When $D$ is not $\mu$-exceptional, choose +$Q$ not $p$-exceptional: take a component over the corresponding prime +on $Z$ already on $X$ and then its strict transform. Put +\[ + u=\operatorname{ord}_Q(q^*D),\qquad c=\operatorname{coeff}_Q(C). +\] +At a general point of $Q$, avoiding all other relevant divisors, +there are local coordinates with +\[ + t_1=z_1^u,\qquad t_j=z_j\quad(2\leq j\leq b). +\] +Indeed the map $Q\to D$ is submersive at a general smooth point, +and a local unit in the first equation is absorbed by a holomorphic +$u$th root. Such points can be chosen above every point of a dense +analytic open subset of $D$ by generic smoothness and properness. +Integration over a fixed small polydisk in the remaining fibre +coordinates gives +\begin{equation}\label{prog:base-presentation-divisor-bound} + I(t)\geq c_*|t_1|^{-2(c+u-1)/u} +\end{equation} +for some $c_*>0$ near each such base point. The section and metric +contribute the exponent $c$, while division by the base differential +in the relative canonical form contributes $u-1$. + +For a nonexceptional $D$ the chosen $Q$ has $c\geq0$, so +\eqref{prog:base-presentation-divisor-bound} bounds $\psi$ from above +near its general points. For the finitely many exceptional divisors, +the same estimate supplies a nonnegative integer $m$ such that, if +$k$ is a smooth metric on $\mathcal O_W(F)$ and $s_F$ its divisor +section, +\[ + \varphi=\psi+m\log|s_F|_k^2 +\] +is locally bounded above near general points of every divisor. +Adding local potentials of $\rho+m c_1(k)-a\omega_W$ to +$\varphi$ gives psh weights by +\eqref{prog:base-presentation-open-positivity}. These weights +extend across the divisors and then across the remaining analytic +set of codimension at least two, giving +\begin{equation}\label{prog:base-presentation-positive-current} + S=\rho+m c_1(k)+dd^c\varphi\geq a\omega_W +\end{equation} +on all of $W$. + +\paragraph{A K\"ahler carrier.} +Regularize the current in +\eqref{prog:base-presentation-positive-current} in its class, retaining +a lower bound for the potentials. More precisely, choose an +approximation with analytic singularities such that +\begin{equation}\label{prog:base-presentation-regularization} + S'=\rho+m c_1(k)+dd^c\varphi',\qquad + S'\geq\frac a2\omega_W,\qquad + \varphi'\geq\varphi-O(1). +\end{equation} +These simultaneous properties are the Bergman regularization +properties of \cite{DemaillyRegularization1992}; see also +\cite[pp.~59--60]{PopoviciRegularization2004}. +Resolve the analytic singularities and the support of $F$ by a +sequence of blowups with smooth centres $r:T\to W$, with simultaneous +SNC support, and write +\[ + r^*S'=[H]+\zeta,\qquad H\geq0. +\] +The residual current $\zeta$ has locally bounded potentials. Removing +the divisorial part from the positive current +$r^*S'-(a/2)r^*\omega_W$ shows that +\[ + \zeta\geq\frac a2r^*\omega_W. +\] +The difference has vanishing Lelong numbers, so its class is nef by +regularization. Choose an effective $r$-exceptional divisor $G$ with +$-G$ relatively ample. For $0<\eta\ll1$, the class +$(a/2)r^*\omega_W-\eta[G]$ is K\"ahler. Therefore +\begin{equation}\label{prog:base-presentation-carrier-data} + \begin{split} + \tau&=\mu r,\qquad + \beta_T=[\zeta]-\eta[G],\\ + C_T&=H+\eta G-mr^*F-K_{T/W} + \end{split} +\end{equation} +has K\"ahler $\beta_T$. If $r$ is an isomorphism, omit $G$ and its +correction. The choice of $\eta$ will be decreased once more below. + +The exact class calculation is +\[ + \begin{split} + [K_T+C_T]+\beta_T + &=r^*c_1(K_W)+[H]+[\zeta]-m[r^*F]\\ + &=r^*(c_1(K_W)+[\rho])=\tau^*\gamma. + \end{split} +\] +All possibly negative terms in $C_T$ are $\tau$-exceptional. +Consequently $B_Z=\tau_*C_T$ is effective. Specify the nef b-data by +$\mathbf N_T=\beta_T$; the displayed identity defines generalized +data on $Z$ with adjoint class $\gamma$. At the level of currents, +pushing the identity gives the required trace identity on $Z$: +exceptional divisor currents push to zero, and a $dd^c$ difference +pushes to a $dd^c$ difference. Canonical traces can be computed using +local meromorphic canonical frames; a change of frame changes the +corresponding Chern potentials by the usual logarithmic transition +term. Thus only the whole adjoint, rather +than its individual traces, needs to be a Bott--Chern class. The +class $\beta_T$ plus the strict transform of $B_Z$ is big on $T$, +so its pushforward $B_Z+\mathbf N_Z$ is globally modified big. +These are~\eqref{prog:base-presentation-identities}. + +\paragraph{The multiplier ideal.} +Take a local holomorphic function $v\in f_*\mathcal J_1$. By the +pushforward description of $\mathcal J_1$, +\begin{equation}\label{prog:base-presentation-source-orders} + \operatorname{div}(q^*\mu^*v)\geq\lfloor C\rfloor. +\end{equation} +Thus $q^*\mu^*v\,e$, tensored with a local base canonical frame, +is a holomorphic $L$-valued top form on $U$. Its squared norm is +locally integrable by the fractional SNC weights in +\eqref{prog:base-presentation-curvature}. Over the submersion locus, +fibre integration followed by base integration is precisely this +top-form integral. Properness and a finite coordinate covering +therefore give +\begin{equation}\label{prog:base-presentation-base-integral} + \int |\mu^*v|^2e^{-\psi}<\infty +\end{equation} +locally on $W$; the integral is taken where the original fibre +metric is defined. + +The lower bound on $\varphi'$ in +\eqref{prog:base-presentation-regularization} preserves this +integrability after replacing $\psi$ by +$\varphi'-m\log|s_F|_k^2$. Change variables through $r$. Along a +prime divisor of $T$, let $h,j,k_D,d$ be respectively the coefficients +of $H$, $\operatorname{div}(\tau^*v)$, $K_{T/W}$, and $r^*F$. +The local integrand has exponent +$2(j-h+md+k_D)$, up to a bounded positive factor. Its integrability +implies $j-h+md+k_D>-1$, or, since $j$ is integral, +\[ + j\geq\lfloor h-md-k_D\rfloor. +\] +Therefore +\[ + \operatorname{div}(\tau^*v) + \geq\lfloor H-mr^*F-K_{T/W}\rfloor. +\] +Choose $\eta>0$ sufficiently small that adding $\eta G$ does not +change any of these finitely many round-downs. The resulting +inequality is exactly +$\operatorname{div}(\tau^*v)\geq\lfloor C_T\rfloor$, proving +\eqref{prog:base-presentation-ideal} as an inclusion of ideal sheaves. +In particular, if $f_*\mathcal J_1$ is the unit ideal on an open set, +then so is $\mathcal J_Z$, and the base data are gklt there. + +Finally, the nef data descend to $T$, and the structure boundaries +on higher models are defined by crepant pullback of +\eqref{prog:base-presentation-identities}. On every common refinement +of the source and base carriers, both adjoints are the pullback of +$\gamma$. Thus the identities remain exact on higher models. +\end{proof} + +\begin{lemma}[Contractions of closed subspaces] +\label{prog:base-emc} +Let $D$ be a compact complex space, possibly nonreduced, and let +$\delta\in H^2(D,\R)$ have nonnegative degree on compact curves. +Use the definitions of a Moishezon morphism and of a contraction +defined by a class in \cite[Definitions~2.32 and~2.34]{HX}. +Then the following hold. +\begin{enumerate} +\item If $\delta$ is endowed with a Moishezon contraction, so is +its restriction to every closed analytic subspace of $D$. +\item Let $g:D\to V$ be a projective contraction of compact +complex spaces, so $g_*\OO_D=\OO_V$, and suppose +$\delta=g^*\epsilon$ for a class $\epsilon\in H^2(V,\R)$ +having nonnegative degree on compact curves. Then $\delta$ is +endowed with a Moishezon contraction if and only if $\epsilon$ +is endowed with one. +\end{enumerate} +\end{lemma} + +\begin{proof} +Write $h:D\to S$ for a Moishezon contraction defined by +$\delta$. For a closed subspace $D_0\subset D$, take the Stein +factorization +\[ + D_0\longrightarrow S_0\longrightarrow S +\] +of its restriction. The first map is a contraction, and the second +is finite. Thus a curve in $D_0$ is contracted by the first map +exactly when its $\delta$-degree is zero. Choose a projective +surjection $P\to D$ witnessing that $h$ is Moishezon, with +$P\to S$ projective. The closed subspace +$P_0=P\times_D D_0$ is projective over $S$ and projective +and surjective over $D_0$. It is also projective over $S_0$: +a relatively ample line for $P_0\to S$ remains ample on the +fibres of $P_0\to S_0$. This proves the first assertion, +including its scheme structure. + +For the descent direction of the second assertion, every curve +in a fibre of $g$ has $\delta$-degree zero. A connected +projective complex space is connected by chains of compact curves +after passing to its reduction. Therefore $h$ is constant on +the underlying set of each fibre of $g$. It factors as +$h=k\circ g$ for a holomorphic map $k:V\to S$. +To see that this factorization holds also for nonreduced spaces, +choose a coordinate neighbourhood in $S$ of the image of a fibre. +Properness permits restriction to a neighbourhood of its point +in $V$ whose inverse image maps into that coordinate +neighbourhood. The coordinate functions of $h$ then descend +uniquely through $g_*\OO_D=\OO_V$; their analytic relations +descend through the same equality. These local factorizations +glue. Furthermore, +\[ + k_*\OO_V=h_*\OO_D=\OO_S. +\] +Every irreducible compact curve $C\subset V$ is dominated by +a compact curve $C'\subset D$. Indeed, after base change to +the normalization of $C$, take a component dominating that curve +and a projective multisection. The projection formula gives +\[ + \delta\cdot C' + =\deg(C'\to C)\,\epsilon\cdot C. +\] +Consequently $k$ contracts exactly the curves of +$\epsilon$-degree zero. The composite $P\to D\to V$ +is a projective surjection, and $P\to S$ is projective, so +it witnesses that $k$ is Moishezon. + +Conversely, suppose $k:V\to S$ is a Moishezon contraction +defined by $\epsilon$. Then $k\circ g$ is a contraction, +and the projection formula verifies its curve criterion, treating +separately curves contracted by $g$. If $Q\to V$ is a +projective surjection witnessing that $k$ is Moishezon, then +$D\times_V Q\to D$ is projective and surjective, while +$D\times_V Q\to Q\to S$ is projective. This supplies the +required witness. +\end{proof} + +\begin{lemma}[Crepant transport of the multiplier closed subspace] +\label{prog:multiplier-transport} +Let $g:X\to Y$ be a projective bimeromorphic morphism of +normal compact K\"ahler spaces. Suppose effective generalized +pair data on $X$ and $Y$ have the same globally nef b-part +and are crepant over $Y$. In particular their adjoint classes +$A_X$ and $A_Y$ satisfy $A_X=g^*A_Y$ in Bott--Chern +cohomology. Write $J_X,J_Y$ for their multiplier ideals and +$N_X=V(J_X)$, $N_Y=V(J_Y)$ for their non-klt closed +subspaces. Then +\[ + g_*J_X=J_Y,\qquad R^ig_*J_X=0\quad(i>0), + \qquad g_*\OO_{N_X}=\OO_{N_Y}. +\] +The restriction $N_X\to N_Y$ is a projective contraction. +Hence, whenever $A_Y$ is nef, the restriction of $A_X$ to +$N_X$ is endowed with a Moishezon contraction if and only if +the restriction of $A_Y$ to $N_Y$ is endowed with one. +\end{lemma} + +\begin{proof} +Take a projective common log resolution $r:U\to X$ carrying +the nef datum, and put $s=g\circ r$. By crepancy there is +one real SNC divisor $C$ such that +\[ + K_U+[C]+\beta=s^*A_Y=r^*A_X, +\] +where $\beta$ is globally nef. The multiplier ideals are +\[ + J_X=r_*\OO_U(-\lfloor C\rfloor),\qquad + J_Y=s_*\OO_U(-\lfloor C\rfloor). +\] +Their direct-image equality follows at once. For vanishing, set +$L=\OO_U(-\lfloor C\rfloor)$ and +$\Delta=C-\lfloor C\rfloor$. The pair $(U,\Delta)$ +is klt, and the actual real line +\[ + L-(K_U+\Delta)=-K_U-C +\] +has class $\beta-s^*A_Y$. It is relatively nef over $Y$ +and relatively big because $s$ is bimeromorphic. The same +statements hold over $X$. Relative Kawamata--Viehweg vanishing, +in the form of \cite[Theorem~1.2]{FujinoVanishing2025} with +the first morphism the identity and the real line +$\mathcal H=-K_U-C$, therefore gives +\[ + R^is_*L=R^ir_*L=0\qquad(i>0). +\] +The real-line identity in that theorem is the displayed definition +of $\mathcal H$; its relative positivity is checked by the class +identity above. The klt condition means that there are no proper log +canonical strata to test for relative log bigness. Leray gives +$R^ig_*J_X=0$ for $i>0$. Pushing forward +$0\to J_X\to\OO_X\to\OO_{N_X}\to0$ and using +$g_*\OO_X=\OO_Y$ now proves the asserted structure-sheaf +equality. It also identifies the scheme-theoretic image with +$N_Y$, giving the projective contraction between these closed +subspaces. Apply Lemma~\ref{prog:base-emc} to conclude. +\end{proof} + +\begin{remark} +The preceding lemma also transports the contraction hypothesis +through a crepant projective flip: apply it to the two contractions +to their common base, equipped with the descended adjoint and +the common nef b-data. On that base the boundary is the effective +pushforward of the source boundary, and the nef data remain on +the same carrier; the descended Bott--Chern adjoint and its +identity on a common resolution define the crepant base data +in the whole-adjoint and current-trace category used here. +All assertions concern the fixed adjoint +and its multiplier ideal. No conclusion about the multiplier +ideal of a different driving adjoint is used. +\end{remark} + +\begin{corollary}[The contraction step over a lower-dimensional base] +\label{prog:base-emc-induction} +Use the hypotheses and notation of +Proposition~\ref{prog:base-presentation}. Suppose that +$A=f^*\gamma$ is nef and that its restriction to the original +non-klt closed subspace $N_o=V(J_o)$ is endowed with a +Moishezon contraction. Put +\[ + I=f_*J_1,\qquad N_*=V(I)\subset Z. +\] +Then $\gamma|_{N_*}$ is endowed with a Moishezon contraction. +If the prepared non-klt closed subspace $N_1=V(J_1)$ dominates +$Z$, then $\gamma$ and $A$ are endowed with Moishezon +contractions. Otherwise the generalized pair supplied on $Z$ +has a non-klt closed subspace endowed with the contraction +defined by the restricted class of $\gamma$. In this latter +case, if $\gamma$ is nef and NQC and the assertion +$\mathsf C_{\dim Z}$ is available, then $\gamma$ and $A$ +are endowed with Moishezon contractions. +\end{corollary} + +\begin{proof} +The inclusion $J_o\subset J_1$ gives $N_1\subset N_o$ +as closed analytic subspaces. By +Lemma~\ref{prog:base-emc}, the given contraction restricts to +$N_1$. Proposition~\ref{prog:base-presentation} identifies +\[ + f_*\OO_{N_1}=\OO_Z/I=\OO_{N_*}. +\] +Thus the restriction $N_1\to N_*$ is a projective +contraction. The restricted adjoint is the pullback of +$\gamma|_{N_*}$, so Lemma~\ref{prog:base-emc} descends +the contraction to $N_*$. Nonnegative degrees on curves of +$N_*$ follow from projective multisections and nefness of +$A$. + +If $N_1$ dominates the normal irreducible space $Z$, then +$I=0$: a local function in $I$ pulls back to zero on the +dominating closed subspace and hence vanishes generically on +$Z$. Normality, and in particular reducedness, gives its +vanishing. Therefore $N_*=Z$, which proves the assertion in +this case. + +If $N_1$ does not dominate $Z$, the constructed base ideal +satisfies $I\subset J_Z$. Hence +\[ + V(J_Z)\subset V(I)=N_* +\] +as closed analytic subspaces. Restriction by +Lemma~\ref{prog:base-emc} provides the required contraction +on $V(J_Z)$, with its complete scheme structure. The +generalized pair on $Z$ has globally nef carrier data and +globally modified-big total boundary by +Proposition~\ref{prog:base-presentation}. When the stated +nef and NQC conditions hold, these facts are precisely the +hypotheses of $\mathsf C_{\dim Z}$, which gives the +contraction on $Z$. Finally Lemma~\ref{prog:base-emc} +pulls it back through the projective contraction $f$ to $X$. +The case of an empty non-klt closed subspace has the same +interpretation, with its contraction condition vacuous. +\end{proof} diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-restricted-induction.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-restricted-induction.tex index 222494796066246c4a079d2c82db5b5c7b58e419..759c17634532afca877563c94a5d94ffd231373c 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-restricted-induction.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-restricted-induction.tex @@ -1,1925 +1,1925 @@ -\subsection{A restricted dimension induction} -\label{prog:dimension-induction} - -All generalized nef data in this subsection are globally nef on a fixed -compact K\"ahler carrier. For the degree arguments we use weak NQC: -a positive combination of rational degree-two classes having -nonnegative degree on compact curves. Strong NQC implies this -condition, and the contraction assertion admits this weak form -\cite[Definition~2.43]{HX}. The nef adjoint itself remains nef in -Bott--Chern cohomology throughout. A modified-big boundary means the total trace -$B+\mathbf M_X$ is globally modified big. Relative assertions retain -these absolute hypotheses. A marked model records its bimeromorphic map -from the specified source; two markings agree only under an isomorphism -commuting with those maps. Weak models below may be arbitrary normal compact K\"ahler weak log -canonical models. Their generalized nef traces are understood as -closed currents pushed forward from the fixed nef carrier; only the -whole adjoint is required to be a Bott--Chern class. In particular no -factoriality, and no separate Bott--Chern class for either $K_Y+B_Y$ -or the nef trace, is required of an auxiliary weak target. This -category contains both the working models and the adjunction strata -used in special termination. Chosen program outputs remain globally -strongly $\mathbb Q$-factorial. - -For dimension at most $d$, write $\mathsf B_d$ for semiampleness of a -nef gklt adjoint with modified-big boundary, including a Moishezon -connected contraction onto a normal compact K\"ahler target. It makes -no assertion that every such contraction is projective. -Write $\mathsf C_d$ for the NQC non-klt contraction assertion of -\cite[Theorem~1.6]{HX}, and $\mathsf C_{d,\mathrm{big}}$ for its -big-adjoint case. The relative assertions are: -\begin{itemize} -\item $\mathsf M_d$: a relatively pseudo-effective gklt adjoint over -any proper map to a normal compact K\"ahler base $S$, with the globally -nef and modified-big data just specified, has a chosen nonextracting -good log terminal model over $S$. Its working space is globally -strongly $\mathbb Q$-factorial and compact K\"ahler; its connected -relative canonical morphism has normal compact K\"ahler target and -is Moishezon. From a smooth common carrier, -all Bott--Chern classes have forward traces, and the chosen model has -the rational degree-two and Bott--Chern exceptional splittings proved -below. -\item $\mathsf F_d$: a compact polytope of these data on one fixed -carrier has a polyhedral relative effective locus, finitely many -relative canonical chambers with chosen good terminal models, and -finitely many marked weak models, accounting for every weak model -in the category specified above. -\end{itemize} -Passing to the proper image and then to its normal Stein factor allows -all base maps to be taken surjective with connected fibres. These -operations preserve the compact K\"ahler base category. - -The independent inputs are the analytic cone theorem, the projective -results of Proposition~\ref{prog:projective-good-model}, relative -vanishing, adjunction, relative-canonical positivity, and the -nef-and-big K\"ahler criterion. The base presentations and -multiplier-ideal descent used in the induction are proved below. -The projective constructions use the actual-line lemmas of the preceding -subsection. No assertion of termination of an arbitrary sequence of -flips is used. - -\input{sections/03c-base-presentations} - -\subsubsection{A fixed integral grid and cohomology transport} - -\begin{lemma}[Integral degree-two descent for a locally log-Fano map] -\label{prog:integral-h2} -Let $f:X\to Z$ be a projective surjective morphism with connected fibres -between normal complex analytic spaces. Assume that $f_*\mathcal O_X= -\mathcal O_Z$, that $R^if_*\mathcal O_X=0$ for $i>0$, and that every point -of $Z$ has a neighbourhood $U$ on which there is a klt pair -$(X_U,\Delta_U)$ with $-(K_{X_U}+\Delta_U)$ ample over $U$. -Then $f^*:H^2(Z,\mathbb Z)\to H^2(X,\mathbb Z)$ is injective, and its -image consists exactly of the integral classes having degree zero on -every compact curve contracted by $f$. -\end{lemma} - -\begin{proof} -Properness, normality and connected fibres identify both -$f_*\mathcal O_X=\mathcal O_Z$ and -$f_*\mathcal O_X^*=\mathcal O_Z^*$. Apply $Rf_*$ to the analytic -exponential sequence. The local surjectivity of -$\mathcal O_Z\to\mathcal O_Z^*$ and the stated coherent vanishing give -\[ - R^1f_*\mathbb Z_X=0, - \qquad - R^1f_*\mathcal O_X^*\simeq R^2f_*\mathbb Z_X. -\] -Let $\xi\in H^2(X,\mathbb Z)$ have zero degrees on contracted curves. -A germ of its edge image in $R^2f_*\mathbb Z_X$ is, under this -isomorphism, represented after shrinking $U$ by an actual line bundle -$L$ on $X_U$. Equality of these germs means that, after further -shrinking, $c_1(L)$ and $\xi|_{X_U}$ have the same class. In particular -$L$ has degree zero on every curve in every fibre above this smaller -neighbourhood. - -On a sufficiently small Stein base neighbourhood, twisting by a -relatively ample line and taking a quotient of two nonzero relative -sections gives a meromorphic Cartier representative for $L$. Thus -the Cartier-divisor form of base point freeness applies. -The line bundle $L$ is relatively nef, and -$aL-(K_{X_U}+\Delta_U)$ is relatively ample for every real $a$. -The projective analytic base-point-free theorem applies to the Cartier -line $L$. Its conclusion is generation for \emph{every} sufficiently -large integer power, not only for a divisible subsequence -\cite[Theorems~6.2 and~6.5]{FujinoAnalyticBCHM}. Thus, over a further relatively compact -neighbourhood, both $L^m$ and $L^{m+1}$ are relatively generated. -A generated line of degree zero on every curve of a projective connected -fibre defines a constant projective map on that fibre: the restriction -of the generated line is the pullback of $\mathcal O(1)$, and a -positive-dimensional projective image contains a curve of positive -degree. The corresponding relative image is therefore finite and -bimeromorphic over the normal base, and is the base itself. Consequently -\[ - L^m\simeq f^*M_m,\qquad L^{m+1}\simeq f^*M_{m+1} -\] -for line bundles on the neighbourhood in $Z$. Taking their quotient -proves $L\simeq f^*(M_{m+1}\otimes M_m^{-1})$. Its germ in -$R^1f_*\mathcal O_X^*$ is zero, so the edge image of $\xi$ is zero. - -The integral Leray spectral sequence, using -$R^1f_*\mathbb Z_X=0$, gives -\[ - 0\longrightarrow H^2(Z,\mathbb Z) - \xrightarrow{f^*}H^2(X,\mathbb Z) - \longrightarrow H^0(Z,R^2f_*\mathbb Z_X). -\] -Exactness at the middle term proves the assertion. The argument is -integral throughout and does not discard torsion. The converse follows -by restriction of a pulled-back class to a fibre. -\end{proof} - -\begin{lemma}[A fixed integral grid along existing crepant steps] -\label{prog:fixed-integral-grid} -Let $X_0$ be a compact K\"ahler space, and suppose that -\[ - \alpha_0=\sum_{j=1}^r a_j\gamma_{0j},\qquad a_j>0, - \qquad \gamma_{0j}\in H^2(X_0,\mathbb Q), -\] -where each $\gamma_{0j}$ has nonnegative degree on compact curves. -Fix integers $m_j>0$ such that $m_j\gamma_{0j}$ is the image of an -integral cohomology class. Consider an existing sequence of projective -birational steps -\[ - X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1}, -\] -where $f_i^+$ is the identity for a divisorial step, $f_i$ satisfies -Lemma~\ref{prog:integral-h2}, and $\alpha_i$ is trivial on its -contracted ray. Assume every $f_i$-contracted curve has degree -proportional to that ray for the classes in the display. Then the -classes $\gamma_{ij}$ descend through $f_i$ and pull back through -$f_i^+$, the same integers $m_j$ remain integral multiples at every -stage, and the transported classes remain nonnegative on curves. -In particular, for every integral compact curve $C\subset X_i$, -\[ - \alpha_i\cdot C>0\quad\Longrightarrow\quad - \alpha_i\cdot C\geq\delta, - \qquad \delta:=\min_j\frac{a_j}{m_j}>0. -\] -\end{lemma} - -\begin{proof} -Positivity of the coefficients and nonnegativity of the summands imply -that every $\gamma_{ij}$ annihilates the contracted ray. Apply the -integral descent lemma to the chosen integral lift of -$m_j\gamma_{ij}$ and pull its descended class back to $X_{i+1}$. -This keeps $m_j$ fixed. - -To verify nonnegativity, take a curve on $X_{i+1}$. If it is contracted -by $f_i^+$, every descended summand has degree zero. Otherwise its -image is a curve $\Gamma\subset Z_i$. The projective map $f_i$ has a -compact curve mapping onto $\Gamma$ with positive degree: take a -component of its inverse image dominating $\Gamma$ and intersect with -sufficiently many relatively ample hyperplanes. Nonnegativity of the -old summand on this curve implies nonnegativity on $\Gamma$, hence on -the curve on $X_{i+1}$. This argument uses a positive-degree lift, not -a degree-one section. Finally $m_j\gamma_{ij}\cdot C$ is a -nonnegative integer, giving the stated lower bound. -\end{proof} - -\begin{corollary}[Uniform crepancy parameter; no termination assertion] -\label{prog:uniform-canonical-parameter} -Suppose additionally that the working spaces are $d$-dimensional klt -spaces, that $\alpha_i$ are nef, and that the ordinary canonical -negative-ray length bound $0<-K_{X_i}\cdot C\leq2d$ is available. -For a fixed real number $t>2d/\delta$, every negative extremal ray -chosen in an existing $(K_{X_i}+t\alpha_i)$-program is -$\alpha_i$-trivial. If the steps are projective ordinary canonical -steps, Lemma~\ref{prog:fixed-integral-grid} therefore applies -inductively with this same value of $t$. - -If $\alpha_0$ is not big and $\alpha_0-c_1(K_{X_0})$ is big, then -$K_{X_i}+t\alpha_i$ remains non-pseudo-effective along these crepant -steps. Consequently a terminating such program ends with a Mori -fibre contraction. This corollary does not assert that the requisite -contractions exist or that the program terminates. -\end{corollary} - -\begin{proof} -A negative ray is $K_{X_i}$-negative because $\alpha_i$ is nef. Choose -its rational curve generator with the ordinary length bound. If its -$\alpha_i$-degree were positive, then -\[ - (K_{X_i}+t\alpha_i)\cdot C\geq-2d+t\delta>0, -\] -a contradiction. The step is thus an ordinary canonical step with -$\alpha_i$ pulled back from its base, and the preceding lemma applies. -On a common resolution the pullback of -$\alpha_{i+1}-c_1(K_{X_{i+1}})$ is the pullback of -$\alpha_i-c_1(K_{X_i})$ plus the effective canonical comparison. -Thus it stays big. Equality of the crepant pullbacks of $\alpha_i$ -preserves its nonbigness. The -identity -\[ - (t+1)\alpha_i=(c_1(K_{X_i})+t\alpha_i) - +(\alpha_i-c_1(K_{X_i})) -\] -excludes pseudo-effectivity of its first summand. A nef endpoint is -therefore impossible. -\end{proof} - - -\begin{lemma}[Cohomology splitting along detected steps] -\label{prog:detected-cohomology} -Let $X_0$ be a smooth compact K\"ahler manifold. Consider a finite -sequence of divisorial contractions and small flips -\[ - X_0\dashrightarrow X_1\dashrightarrow\cdots - \dashrightarrow X_r -\] -between normal globally strongly $\mathbb Q$-factorial compact -K\"ahler spaces with rational singularities. -Assume that every negative contraction $f_i:X_i\to Z_i$ is a -projective contraction of one ray of the analytic cone, that $Z_i$ -has rational singularities, and that each small flip is the detected -flip of a global rational line $L_i$ with $L_i\cdot R_i<0$. -For a divisorial contraction assume the usual single exceptional -prime. Then, for every projective smooth resolution $p:U\to X_i$, -\begin{align*} - H^2(U,\mathbb R) - &=p^*H^2(X_i,\mathbb R) - \oplus\bigoplus_{E\text{ exceptional for }p}\mathbb R[E],\\ - H^{1,1}(U,\mathbb R) - &=p^*H^{1,1}_{\rm BC}(X_i,\mathbb R) - \oplus\bigoplus_{E\text{ exceptional for }p}\mathbb R[E]. -\end{align*} -The first equality also holds over $\mathbb Q$. There are compatible -forward linear transforms on $H^2(-,\mathbb R)$ and on Bott--Chern -cohomology. They are surjective, are isomorphisms for a small flip, -and preserve rational degree-two classes. On a common resolution the -pullback difference is an actual exceptional divisor class; for a -flip it is exceptional over both spaces. - -The exceptional quotient also takes the Chern class of a holomorphic -line bundle on $U$ to a rational holomorphic line class on $X_i$. -\end{lemma} - -\begin{proof} -For a modification of a smooth compact K\"ahler manifold the two -splittings are the degree-two modification formula. Suppose that they -hold for $X=X_i$. We use only the following birational descent fact: -for a proper bimeromorphic map between normal compact spaces in -Fujiki's class with rational singularities, pullback is injective on -$H^2(-,\mathbb R)$ and on Bott--Chern cohomology, and its image in -either group consists of the classes having degree zero on every -contracted curve. This is the bimeromorphic case of -\cite[Lemma~2.6(1)]{DH2024}. - -First note that a degree-two class $\xi\in H^2(X,\mathbb R)$ has -the same degrees on curves as a Bott--Chern class. Indeed, pull it to -a smooth projective resolution $a:W\to X$ and take the $(1,1)$-part -of its Hodge decomposition. The induction hypothesis writes that -part as $a^*\beta+[F]$, with $F$ exceptional over $X$. On every -$a$-contracted curve, $a^*\xi$ and its $(2,0)$ and $(0,2)$ parts -have degree zero. Hence $F$ has degree zero on every such curve. -Exceptional negativity applied to both signs gives $F=0$. -Projective multisections lifting a curve of $X$ then show that -$\xi$ and $\beta$ have the same degrees on every curve of $X$. - -Suppose first that $X\dashrightarrow X^+$ is a small detected flip. -Write $f:X\to Z$ and $f^+:X^+\to Z$, and fix an integral curve -$C$ on its negative ray. For $\xi\in H^2(X,\mathbb R)$ set -\[ - s=\frac{\xi\cdot C}{c_1(L)\cdot C}. -\] -By the preceding paragraph, $\xi-s c_1(L)$ vanishes on every -$f$-contracted curve. Thus it is $f^*\eta$ for a unique -$\eta\in H^2(Z,\mathbb R)$. Define -\[ - T\xi=(f^+)^*\eta+s c_1(L^+). -\] -For a Bott--Chern class, the same descent takes place in -Bott--Chern cohomology, so the two transforms agree. The detected -flip construction gives an effective rational divisor $E_L$ on a -common projective resolution $a:W\to X$, $b:W\to X^+$ such that -\[ - a^*c_1(L)-b^*c_1(L^+)=[E_L]. -\] -Consequently -\[ - a^*\xi=b^*T\xi+s[E_L]. -\] -If $\xi$ is rational, then $s$ is rational. Since the image of -$f^*$ on real degree-two cohomology is the realification of a -rational subspace, its unique preimage $\eta$ is rational as well. -This proves rationality of $T\xi$. - -The sets of prime divisors exceptional for $a$ and $b$ are identical: -the two working spaces are small bimeromorphic. The old splitting -and the displayed pullback identity therefore span -\[ - H^2(W,\mathbb R) - =b^*H^2(X^+,\mathbb R) - +\operatorname{span}_{\mathbb R}\{[E]:E\text{ is }b - \text{-exceptional}\}, -\] -and similarly in bidegree $(1,1)$. The sum is direct. If a pullback -from $X^+$ equals an exceptional divisor class, that divisor has -degree zero on all $b$-contracted curves, so both signs of negativity -make it zero; pullback injectivity then makes the original class -zero. In particular the dimensions on the two sides agree, $T$ is -injective by the old direct sum, and $T$ is surjective. The rational -version follows either from the same argument or from realification. - -For a divisorial contraction $f:X\to X^+$, let $D$ be its -exceptional prime. Its degree on the contracted ray is negative: -otherwise $D$ would be $f$-nef, contrary to exceptional negativity. -For any $\xi$, subtract -\[ - \frac{\xi\cdot C}{D\cdot C}[D] -\] -and descend the result by the same birational cohomology lemma. -This defines $T\xi$. On a common resolution its pullback difference -is a multiple of the total transform of $D$. The target's exceptional -prime set is the old one together with the strict transform of $D$. -The same spanning and directness argument gives the target splitting -and surjectivity; its kernel is $\mathbb R[D]$. Rationality is again -immediate from the quotient of rational intersection numbers. - -These arguments initially prove the splitting on a chosen common -resolution. To obtain it on any projective smooth resolution -$U\to X^+$, dominate that resolution and the chosen one by a smooth -common projective resolution. Apply the smooth degree-two -modification formula upstairs and push down to $U$. An exceptional -divisor pushes either to zero or to a divisor exceptional over $X^+$. -This proves spanning on $U$; negativity gives directness there. - -Finally let $\mathcal H$ be a holomorphic line on $U$ and put -$\mathcal Q=(p_*\mathcal H)^{**}$. Global strong -$\mathbb Q$-factoriality gives an $m>0$ for which -$\mathcal M=\mathcal Q^{[m]}$ is invertible. The coherent evaluation -comparison gives -\[ - \mathcal H^{\otimes m} - \simeq p^*\mathcal M\otimes\mathcal O_U(F) -\] -for an integral $p$-exceptional divisor $F$. This comparison is -obtained from the coherent direct image and evaluation map; it does -not assume that $\mathcal H$ has a global meromorphic section. -Dividing first Chern classes by $m$ proves the line assertion. -\end{proof} - - -\begin{corollary}[Stability under projective extraction] -\label{prog:extraction-cohomology} -Let $r:Y\to X$ be a projective bimeromorphic morphism of normal compact -K\"ahler spaces with rational singularities. Suppose that $Y$ is globally -strongly $\mathbb Q$-factorial and that $X$ has the splitting property in -the preceding lemma. Then $Y$ has that property as well. More precisely, -if $E_1,\ldots,E_s$ are the prime divisors exceptional for $r$, then -\[ - H^2(Y,\mathbb R) - =r^*H^2(X,\mathbb R)\oplus - \bigoplus_{j=1}^{s}\mathbb R[E_j], -\] -and the analogous statements hold over $\mathbb Q$ and in -Bott--Chern cohomology. -\end{corollary} - -\begin{proof} -Choose any projective smooth resolution $q:W\to Y$. The composite -$p=rq$ is a projective resolution of $X$. Its exceptional primes are -exactly the $q$-exceptional primes together with the strict transforms -$\widetilde E_j$ of the $E_j$. Since every $E_j$ is $\mathbb Q$-Cartier, -\[ - [\widetilde E_j]=q^*[E_j]+[F_j] -\] -for a rational $q$-exceptional divisor $F_j$. Substitute these identities -in the splitting for $p$. The result spans $H^2(W,\mathbb R)$ by -$q^*H^2(Y,\mathbb R)$ and the $q$-exceptional divisor classes. Their sum -is direct: an exceptional divisor whose class is a pullback has degree -zero on every $q$-contracted curve, and applying exceptional negativity -to both signs makes the divisor zero. Pullback injectivity then applies. -The same argument in bidegree $(1,1)$ and over $\mathbb Q$ proves the -splitting property for $Y$. - -Now expand $q^*\xi$, for $\xi\in H^2(Y,\mathbb R)$, using the original -$p$-splitting and replace the $[\widetilde E_j]$ as above. The difference -between $q^*\xi$ and a class in -$q^*(r^*H^2(X,\mathbb R)+\sum_j\mathbb R[E_j])$ is $q$-exceptional, -and is therefore zero by the direct sum just proved. Thus the displayed -formula for $H^2(Y,\mathbb R)$ spans. Its directness follows from -exceptional negativity for $r$ and pullback injectivity. The rational -and Bott--Chern statements follow by the identical argument. The line -bundle assertion of the preceding lemma uses only global strong -$\mathbb Q$-factoriality and hence applies to $Y$ as well. -\end{proof} - - - -\paragraph{Support in the special-termination comparison.} -\label{prog:special-support} -The following support comparison applies to the detected steps in -the dimension induction. Let $\psi:X\dashrightarrow X^+$ be a detected -small negative step with maps $f:X\to Z$ and $f^+:X^+\to Z$, -and let $L$ be its negative detector. For a sufficiently -divisible $r>0$, put -\[ - \operatorname{im}(f^*f_*L^r\longrightarrow L^r) - =\mathcal I\otimes L^r. -\] -The zero set of $\mathcal I$ is exactly the exceptional locus of $f$. -Outside that locus $f$ is an isomorphism, so the evaluation map is -surjective. On a positive-dimensional projective fibre, every section -of $L^r$ vanishes: its restriction to each integral curve has negative -degree, and those curves cover every positive-dimensional fibre -component. Connectedness supplies the whole nontrivial fibre. - -On a resolution principalizing $\mathcal I$, the detector comparison -is a positive multiple of the effective divisor defined by -$\mathcal I\mathcal O_W$. Consequently its support is the full -inverse image of the exceptional locus. The driving adjoint differs -from a positive real multiple of $c_1(L)$ by a class pulled back from -the contraction base. Its comparison divisor is therefore the same -positive multiple of the detector comparison: their class difference -is exceptional and zero, so both-sign negativity makes it zero as a -divisor. The comparison for a longer -negative program is at least this first-step divisor, by discrepancy -monotonicity on a common resolution. - -Suppose the beginning and end restrictions to a surviving normalized -lc stratum $S$ are isomorphic and their strict adjunction data agree. -The general point of this stratum avoids every intervening surgery: -a zero-discrepancy place remains such a place, whereas strict -comparison would increase its discrepancy if its center were contained -in the exceptional locus. Thus its strict transform $S_W$ on a common -resolution is not contained in the comparison support. Iterated strict -adjunction identifies the restriction of the effective comparison with -the difference of the identified adjoints, so its class is zero. -A nonzero effective divisor on a positive-dimensional compact K\"ahler -stratum has positive K\"ahler mass. The restricted divisor is therefore -zero. If the first step meeting the stratum were nontrivial there, -the full inverse-image support and surjectivity $S_W\to S$ would make -this restriction nonzero, a contradiction. For a zero-dimensional -stratum, generic-point avoidance already excludes an intersection. -The whole program is consequently an isomorphism near the stratum. -This is the same strict-adjunction support argument used in the proof -of Theorem~\ref{prog:special-termination}; it does not use its -lower-dimensional abundance assumption. - -\begin{proposition}[A uniform relative NQC bound] -\label{prog:relative-nqc-grid} -Let $f_0:U\to Z$ be an already projective surjective morphism with -connected fibres between smooth compact -K\"ahler manifolds, and suppose that -\[ - f_0^*:H^0(Z,\Omega_Z^2)\longrightarrow H^0(U,\Omega_U^2) -\] -is an isomorphism. Let $\alpha_0$ be a nef class with an expression -\[ - \alpha_0=\sum_{j=1}^k r_j\eta_{j,0},\qquad - r_j>0,\quad \eta_{j,0}\in H^2(U,\mathbb Q), -\] -where $\eta_{j,0}$ has nonnegative degree on every curve vertical -over $Z$. Let $D_0=K_U+\Delta_U+\mathbf M_U$ be a generalized klt -adjoint. Consider its projective relative program for -$D_0+a\alpha_0$, under the projective local ordinary reduction described in the -proof of Lemma~\ref{prog:base-point-free-induction}. - -If $\alpha_0=0$, triviality is immediate. Otherwise there are -positive integers $m_j$, chosen once on $U$, and a number -\[ - \delta=\min_j\frac{r_j}{m_j}>0 -\] -such that, for $a\delta>2\dim U$, the entire relative program is -$\alpha_0$-trivial. The same number $a$ works at every step. -\end{proposition} - -\begin{proof} -\emph{Rational classes modulo the fixed base.} -Choose an $f_0$-ample line bundle with integral Chern class $H$, and -put $e=\dim U-\dim Z$ and $c=f_{0*}(H^e)>0$. The rational operator -\[ - \Pi(\eta)=\eta- - f_0^*\!\left(\frac{f_{0*}(\eta\smile H^e)}{c}\right) -\] -preserves type $(1,1)$ and kills types $(2,0)$ and $(0,2)$. -Indeed these latter classes are pulled back from $Z$, and the -projection formula applies. Thus -$\Pi(H^2(U,\mathbb Q))\subset H^{1,1}(U)\cap H^2(U,\mathbb Q)$. -By the Lefschetz $(1,1)$ theorem, choose an actual line bundle -$L_{j,0}$ and an integer $m_j>0$ with -\[ - \eta_{j,0}=\frac1{m_j}c_1(L_{j,0})+f_0^*\gamma_j, - \qquad \gamma_j\in H^2(Z,\mathbb Q). -\] -The original summands $\eta_{j,0}$ need not have type $(1,1)$. -Their base components account for this. Their weighted sum gives -\begin{equation}\label{prog:eq:relative-nqc} - \alpha_0=\sum_j\frac{r_j}{m_j}c_1(L_{j,0})+f_0^*\Gamma, - \qquad \Gamma=\sum_jr_j\gamma_j\in H^{1,1}(Z,\mathbb R). -\end{equation} -The final assertion follows because $f_0^*$ is an injective Hodge -map and the other terms have type $(1,1)$. The identity is therefore -also one of Bott--Chern classes. -Each $L_{j,0}$ has nonnegative integral degree on every -$Z$-vertical curve. - -\smallskip\noindent -\emph{Induction across an actual contraction.} -Suppose at a finite stage $f_i:U_i\to Z$ we have actual line bundles -$L_{j,i}$, nef over $Z$, and -\[ - \alpha_i=\sum_j\frac{r_j}{m_j}c_1(L_{j,i})+f_i^*\Gamma. -\] -Assume also that $\alpha_i$ is nef and that the transformed -$D_i$ is generalized klt. These assertions hold initially. -Let $R$ be a $(D_i+a\alpha_i)$-negative extremal ray over $Z$. -It is $D_i$-negative because $\alpha_i$ is nef. Lemma~\ref{prog:relative-hodge} and Lemma~\ref{prog:projective-cone} -identify the relative curve cone with the corresponding face of the -full analytic cone, at the initial model and after every step. The -analytic cone theorem therefore supplies a rational generator $C$ with -\[ - 0<-D_i\cdot C\leq2\dim U. -\] -All $L_{j,i}\cdot C$ are nonnegative integers. If -$\alpha_i\cdot C>0$, one of them is at least one, and hence -$\alpha_i\cdot C\geq\delta$. Therefore -\[ - (D_i+a\alpha_i)\cdot C - \geq-2\dim U+a\delta>0, -\] -a contradiction. We have $\alpha_i\cdot C=0$ and -$L_{j,i}\cdot C=0$ for every $j$. Every curve contracted by the -projective extremal contraction $h_i:U_i\to Y_i$ has class on -$R$ when tested by global line bundles, so the same vanishing -holds for all of them. - -On each fixed Stein chart of $Z$, choose the effective real -ordinary representative of the initial relatively ample nef -datum once, before running the program. Its actual real linear -equivalence to the fixed real line-bundle representative persists -under pushforward. The Bott--Chern comparison with the generalized -adjoint modulo the fixed base class persists separately by -exceptional negativity, as detailed below. -Thus every $h_i$ is, locally on $Y_i$, a contraction with an -ordinary real klt log-Fano boundary. On a relatively compact base -chart, rational approximation in the finite affine system of Cartier -adjoint identities gives a rational effective klt boundary with the -same antiample sign, as in Proposition~\ref{prog:detected-flip}. -Lemma~\ref{prog:integral-descent} therefore applies. -Consequently -\[ - M_{j,i}:=(h_i)_*L_{j,i}\ \text{is a line bundle},\qquad - h_i^*M_{j,i}=L_{j,i}. -\] -For a divisorial contraction set $L_{j,i+1}=M_{j,i}$. -For a flip $h_i^+:U_{i+1}\to Y_i$, set -$L_{j,i+1}=(h_i^+)^*M_{j,i}$. -These are actual line bundles with the \emph{same} integers $m_j$. -The descended Bott--Chern class -\[ - \alpha_{Y_i} - =\sum_j\frac{r_j}{m_j}c_1(M_{j,i})+f_{Y_i}^*\Gamma -\] -satisfies $h_i^*\alpha_{Y_i}=\alpha_i$. -It is nef by descent of nefness through the projective surjection -$h_i$; its pullback to the next model is $\alpha_{i+1}$. -This proves that the step is crepant for $\alpha_i$. Hence it is -also a $D_i$-negative step, and the original generalized pair -remains gklt. - -Finally each $M_{j,i}$ is nef over $Z$. For a $Z$-vertical curve -$B\subset Y_i$, projectivity supplies a curve $B'\subset U_i$ -mapping onto it with some positive degree $d_B$. Then -\[ - d_B(M_{j,i}\cdot B)=L_{j,i}\cdot B'\geq0. -\] -No assertion $d_B=1$ is required. Pullback preserves this relative -nefness, so the induction applies on $U_{i+1}$. Its curve degrees -are again integers because the transported objects are line -bundles, not because any curves were lifted with degree one. -This proves the fixed bound through the whole program. -\end{proof} - -\begin{remark}[Scope of the degree grid] -The transported rational classes -\[ - \eta_{j,i}=\frac1{m_j}c_1(L_{j,i})+f_i^*\gamma_j -\] -have degrees in $m_j^{-1}\mathbb Z_{\geq0}$ on curves vertical over -the fixed smooth base $Z$. This relative assertion is exactly what -the initial projective program requires. It does not assert -nonnegativity of these individual classes on all curves of every -birational model, and uses neither rational connectedness of -resolution fibres nor Graber--Harris--Starr. -\end{remark} - -\subsubsection{The negative-part comparison for a nef adjoint} -We write $N_\sigma(\xi)$ for the divisorial negative part of a -pseudo-effective class. We use its homogeneity, subadditivity, -continuity after adding a vanishing K\"ahler perturbation, and the -identities -\[ - N_\sigma(r^*\xi+[E])=N_\sigma(r^*\xi)+E, - \qquad r_*N_\sigma(r^*\xi)=N_\sigma(\xi) -\] -for an effective $r$-exceptional divisor $E$. A nef class has zero -negative part. These are the divisorial Zariski-decomposition -properties of \cite[Section~3]{Boucksom2004} and -\cite[Appendix~A]{DHY}. No identity -$N_\sigma(r^*\xi)=r^*N_\sigma(\xi)$ for arbitrary $\xi$ is assumed. - -\begin{lemma}[The absolute negative part after a relative program] -\label{prog:negative-part-program} -Let $\mu:U\to X$ be a projective resolution, let $\alpha$ be nef on $X$, -let $c>0$, and let $F_U\geq0$ be $\mu$-exceptional. Put -\[ - D_U=c\mu^*\alpha+[F_U]. -\] -Suppose that $\phi:U\dasharrow V$ is a finite $D_U$-negative program, -possibly relative to another base, and denote its transformed class and -divisor by $D_V$ and $F_V=\phi_*F_U$. Then -\[ - N_\sigma(D_V)=F_V. -\] -\end{lemma} - -\begin{proof} -Take a common projective resolution $p:W\to U$, $q:W\to V$. -The negativity comparison for the finite program is -\[ - p^*D_U=q^*D_V+[E], - \qquad E\geq0,\quad E\text{ is }q\text{-exceptional}. -\] -This comparison is valid for a relative negative program: its proof -uses the negativity of the contracted rays and the positive adjoint on -each flipped side, not absolute nefness of the final adjoint. -Since $p^*F_U$ is effective and exceptional over $X$, the exceptional -identity and nefness of $(\mu p)^*\alpha$ give -\[ - N_\sigma(p^*D_U)=p^*F_U - =N_\sigma(q^*D_V)+E. -\] -Pushing forward by $q$ proves the assertion. In particular, the -negative part in this statement is absolute, even when the program -that produced $V$ was relative. -\end{proof} - -\subsubsection{Descent of the effective vertical divisor} - -\begin{lemma}[Vertical numerical triviality] -\label{prog:vertical-divisor} -Let $g:V\to S$ be a projective surjective morphism with connected fibres. -Assume that $S$ is globally Weil $\mathbb{Q}$-factorial. Let $F\geq0$ be -an effective real Cartier divisor on $V$, vertical over $S$, such that -\[ - F\cdot C=0 - \quad\text{for every curve }C\text{ contracted by }g. -\] -Then there is an effective real Cartier divisor $F_S$ on $S$ such that, -as actual divisors, -\[ - F=g^*F_S. -\] -\end{lemma} - -\begin{proof} -For a prime divisor $P\subset S$, write $m_Q$ for the multiplicity of -$Q$ in $g^*P$, where $Q$ ranges over the prime divisors dominating $P$. -These multiplicities may be computed over the smooth generic locus of -$P$, where $P$ is Cartier. Put -\[ - b_P=\min_{g(Q)=P}\frac{\operatorname{coeff}_Q F}{m_Q}, - \qquad F_S=\sum_P b_PP. -\] -Only finitely many $b_P$ are nonzero, since any such $P$ is the image of -a component of $F$. Thus $F_S$ is a genuine effective Weil real divisor. -Global Weil $\mathbb{Q}$-factoriality makes this finite divisor real -Cartier. - -We first check equality over codimension one in $S$. Work near a -general point of $P$ and restrict to a transverse disk. Resolving the -source, and cutting by general relative ample hypersurfaces if the -relative dimension exceeds one, reduces to a projective surface over -that disk with connected fibres. These operations can be performed -over a relatively compact Stein neighbourhood; they do not require -global sections on $S$. The intersection matrix of the components of -the special fibre is negative semidefinite, with kernel generated by -the full fibre with its multiplicities. The restriction of $F$ has -zero intersection with every fibre component. Its coefficients are -therefore proportional to those multiplicities. Equivalently, all the -ratios in the definition of $b_P$ are equal. When $g$ is birational this -assertion over the generic point of $P$ is immediate. - -Consequently -\[ - G=F-g^*F_S -\] -is a signed real Cartier divisor supported over a subset of codimension -at least two in $S$. Moreover, $G$ is numerically trivial over $S$. -We check that such a signed exceptional divisor is zero. The assertion -is local on $S$. Over a relatively compact Stein neighbourhood, take -$d=\dim V-\dim S$ general relative ample hypersurfaces and resolve their -intersection. They give a projective generically finite morphism -$H\to S$. The cuts may be chosen to meet any prescribed component of -$G$ in a nonzero divisorial trace. After normalization and Stein -factorization, $H\to S$ factors as a projective birational morphism -$H\to S'$ followed by a finite morphism $S'\to S$. The restricted -divisor $G|_H$ is exceptional for $H\to S'$ and numerically trivial -over $S'$. Applying the ordinary exceptional negativity lemma to both -$G|_H$ and $-G|_H$ gives $G|_H=0$. The choice of the cuts therefore -excludes every nonzero component of $G$. Hence $G=0$. - -The surface argument above is also the usual proof of the -degenerate-divisor negativity statement: after subtracting the minimum -multiple of the full fibre, an effective residual fibre divisor omits -a component and cannot be numerically trivial. Notice that projectivity -of $g$ was a hypothesis; it was not deduced from factoriality. -\end{proof} - -\subsubsection{The negative part on the lower-dimensional base} - -\begin{lemma}[Detecting the negative part by pullback currents] -\label{prog:negative-part-base} -Let $g:V\to S$ be a surjective morphism. Suppose that $\alpha_S$ is nef, -$F_S\geq0$ is real Cartier, $c>0$, and -\[ - D_S=c\alpha_S+[F_S],\qquad D_V=g^*D_S,\qquad - N_\sigma(D_V)=g^*F_S. -\] -Then $N_\sigma(D_S)=F_S$. -\end{lemma} - -\begin{proof} -Nefness gives $N_\sigma(D_S)\leq F_S$. Fix a prime divisor $P\subset S$ -and a prime divisor $Q\subset V$ dominating it. Write -$m=\operatorname{mult}_Q(g^*P)>0$ and $b=\operatorname{coeff}_P F_S$. -At their generic smooth points, pullback of positive currents satisfies -\[ - \nu(g^*T,Q)=m\nu(T,P). -\] -Indeed, the divisorial term $\nu(T,P)[P]$ pulls back with multiplicity -$m$, and the residual current has zero generic Lelong number there. - -Choose K\"ahler forms $\omega_S,\omega_V$ and a constant $C>0$ for which -$C\omega_V-g^*\omega_S$ is K\"ahler. For $\varepsilon>0$, let $T_\varepsilon$ -be any positive current in the big class $D_S+\varepsilon[\omega_S]$. -Minimality of the multiplicity and monotonicity under adding a nef -class give -\begin{align*} - m\nu(T_\varepsilon,P) - &=\nu(g^*T_\varepsilon,Q)\\ - &\geq\nu(D_V+\varepsilon g^*[\omega_S],Q)\\ - &\geq\nu(D_V+C\varepsilon[\omega_V],Q). -\end{align*} -Taking the infimum over $T_\varepsilon$, and then letting -$\varepsilon\downarrow0$, gives -\[ - m\nu(D_S,P)\geq\nu(D_V,Q) - =\operatorname{coeff}_Q(g^*F_S)=mb. -\] -This proves the reverse inequality for every $P$. -All multiplicities are unchanged by resolving away from the generic -points in question, so the same argument applies to the normal spaces -under consideration. - -The $\varepsilon$ perturbation is necessary: at a pseudoeffective -boundary class, the infimum over its exact positive currents need not -equal its minimal multiplicity. The argument uses that equality only -for the big perturbed classes. -\end{proof} - -\begin{lemma}[Pullback when the positive part is nef] -\label{prog:negative-part-pullback} -Suppose -\[ - \xi=P+[F],\qquad P\text{ nef},\quad F\geq0\text{ real Cartier}, - \qquad N_\sigma(\xi)=F. -\] -For every projective resolution $p:W\to S$, -\[ - N_\sigma(p^*\xi)=p^*F. -\] -\end{lemma} - -\begin{proof} -Write $N'=N_\sigma(p^*\xi)$. Since $p^*P$ is nef, $N'\leq p^*F$. -Birational invariance gives $p_*N'=F$, so -$E=p^*F-N'\geq0$ is $p$-exceptional. The positive part of $p^*\xi$ is -\[ - p^*P+[E], -\] -and is modified nef. If $E\neq0$, exceptional negativity in its -covering-curve form provides a component of $E$ covered by curves $C_t$ -contracted by $p$, with $E\cdot C_t<0$ -\cite[Appendix~A, Lemma~A.3]{DHY}. A modified nef class has nonnegative -intersection with a general member of a family of curves covering a -prime divisor: use currents with arbitrarily small negative part and -zero generic divisorial Lelong number, restrict to a general member, -and let the negative bound tend to zero. But here -\[ - (p^*P+E)\cdot C_t=E\cdot C_t<0, -\] -a contradiction. Thus $E=0$. -\end{proof} - -\begin{proposition}[Any lower-dimensional good model suffices] -\label{prog:nef-good-comparison} -Suppose the initial relative program has the data in -Lemma~\ref{prog:negative-part-program}. Assume in addition that -its class $\alpha_V$ is nef and agrees with $\mu^*\alpha$ on a common -resolution, and that there is an already projective connected-fibre -morphism $g:V\to S$ with -\[ -\alpha_V=g^*\alpha_S,\qquad D_V=g^*D_S. -\] -Suppose there is a projective small modification -$s:S^{\mathrm q}\to S$ such that $S^{\mathrm q}$ is globally Weil -$\mathbb Q$-factorial; the identity is allowed. Assume that the -lower-dimensional adjoint $s^*D_S$ has a good model: -there are a normal compact K\"ahler space $S_m$, a common projective -resolution $p:W\to S^{\mathrm q}$, $q:W\to S_m$, and a nef semiample -class $D_m$ such that -\[ - p^*s^*D_S=q^*D_m+[E], - \qquad E\geq0,\quad E\text{ is }q\text{-exceptional}. -\] -Then $\alpha$ is semiample on $X$. If the contraction defining -semiampleness of $D_m$ is Moishezon, the resulting contraction of -$\alpha$ is Moishezon as well. In particular, it is unnecessary to -lift a program on $S$ to $V$, or to require that a chosen program -producing $S_m$ preserve $\alpha_S$ at every intermediate step. -\end{proposition} - -\begin{proof} -Nefness descends under the surjective morphism $g$, so $\alpha_S$ is nef. -The class of $F_V$ is pulled back from $S$, so $F_V$ is numerically -trivial over $S$. It is therefore vertical: an effective horizontal -component would restrict to a nonzero effective divisor on a general -projective fibre, with positive intersection against a suitable power -of an ample class, contradicting numerical triviality. For a -birational $g$, verticality is automatic by dimension. -First take a projective resolution $\pi:\widetilde V\to V$ of the main -component of $V\times_S S^{\mathrm q}$. The induced map -$\widetilde g:\widetilde V\to S^{\mathrm q}$ is projective and has -connected fibres. Pull back $\alpha_V,D_V,F_V$ along $\pi$, and pull -back $\alpha_S,D_S$ along $s$. All displayed class identities are -preserved, and $\pi^*F_V$ remains an effective vertical divisor. -Lemma~\ref{prog:negative-part-program} gives -$N_\sigma(D_V)=F_V$ before this modification. Because -$D_V=c\alpha_V+[F_V]$ with $\alpha_V$ nef, -Lemma~\ref{prog:negative-part-pullback} then gives -\[ - N_\sigma(\pi^*D_V)=\pi^*F_V. -\] -We may therefore replace $(V,S,g)$ by -$(\widetilde V,S^{\mathrm q},\widetilde g)$ and suppress the new -superscripts in the rest of the proof. The good-model comparison now -reads $p^*D_S=q^*D_m+[E]$. No assertion that the crepant subboundary on -$\widetilde V$ is effective is needed: this space is used only for -classes, currents and the effective divisor $\pi^*F_V$. The -lower-dimensional generalized pair is the crepant pullback to the -small model $S^{\mathrm q}$. - -The class identity $[F_V]=g^*(D_S-c\alpha_S)$ shows that $F_V$ is -numerically trivial over $S$. Lemma~\ref{prog:vertical-divisor} -gives an actual effective real Cartier divisor $F_S$ with -$F_V=g^*F_S$. Injectivity of pullback yields -\[ - D_S=c\alpha_S+[F_S]. -\] -The established identity $N_\sigma(D_V)=F_V$ and -Lemma~\ref{prog:negative-part-base} therefore give -$N_\sigma(D_S)=F_S$, and -Lemma~\ref{prog:negative-part-pullback} gives -\[ - N_\sigma(p^*D_S)=p^*F_S. -\] -On the other hand $q^*D_m$ is nef, so the exceptional-divisor identity -applied to the good-model comparison gives -\[ - N_\sigma(p^*D_S)=E. -\] -Thus $E=p^*F_S$ as actual divisors. Subtracting them from the comparison -proves the exact class equality -\[ - q^*D_m=c\,p^*\alpha_S. -\] -This proves the needed preservation of $\alpha_S$ from the final good -model, without an assumption about its intermediate models. - -Choose a contraction $h:S_m\to T$ to a normal compact K\"ahler space -and a K\"ahler class $\kappa$ on $T$ with $D_m=h^*\kappa$. Taking a -resolution of the main component of $V\times_S W$, and then a common -projective resolution with $U$, produces a projective modification -$r:R\to X$ and a holomorphic map $\ell:R\to T$ satisfying -\[ - r^*\alpha=\frac1c\ell^*\kappa. -\] -For every curve $C$ in a fibre of $r$ this equality implies -$\ell(C)$ is a point, since a K\"ahler class has positive degree on -every nonconstant image curve. The fibres of the projective modification -$r$ are connected projective complex spaces, hence are connected by -chains of curves. Therefore $\ell$ is -constant on each fibre of $r$. The factorization theorem for a proper -map onto a normal space gives a holomorphic map $f:X\to T$ with -$\ell=f\circ r$. Pushing forward the class equality by $r$ gives -\[ - \alpha=f^*(\kappa/c). -\] -The maps from the main fibre-product component to $S_m$ have connected -general fibres; their Stein factorizations are finite birational over -the normal space $S_m$, hence have connected fibres everywhere. -Together with the connected fibres of $h$, this shows that $\ell$, -and therefore $f$, has connected fibres. This is the required -semiampleness of $\alpha$. - -Finally, $R\to S_m$ is projective: it is obtained from the projective -map $g$ by base change, followed by projective resolutions and the -projective map $q$. If $h$ is Moishezon, choose a projective surjection -$H\to S_m$ such that $H\to T$ is projective. A component of -$R\times_{S_m}H$ dominating $R$ is projective and surjective over $X$, -and its map to $T$ is projective. This is a Moishezon witness for $f$. -\end{proof} - - -\begin{lemma}[The contraction step in the dimension induction] -\label{prog:base-point-free-induction} -Assume $\mathsf C_{d-1}$, $\mathsf B_{d-1}$ and $\mathsf M_{d-1}$. -Then $\mathsf C_{d,\mathrm{big}}$ and $\mathsf B_d$ hold. -\end{lemma} -\begin{proof} -\smallskip\noindent -\emph{The big non-klt contraction.} -Apply the construction of \cite[Section~3]{HX}. On its smooth -modification $\nu:U\to X$ it writes -\[ - \nu^*\alpha=[D_U]+\eta_U, - \qquad D_U\geq0,\quad \eta_U\text{ K\"ahler}. -\] -The induction over the jumping coefficients of the multiplier -ideal reduces the new reduced stratum to dimension at most $d-1$. -The given contraction on the old non-klt subspace and -$\mathsf C_{d-1}$ provide the Moishezon maps of those strata. -Every map that is promoted to a projective map in that construction -contracts exactly the curves on which $\nu^*\alpha$ vanishes. -On each such curve the actual real line bundle $-D_U$ has degree -$\eta_U\cdot C$. On a further projective common resolution -$q:U''\to U$, choose an effective exceptional divisor $E$ with -$-E$ relatively ample and choose $\varepsilon>0$ sufficiently small. -Use the decomposition -\[ - q^*\nu^*\alpha=[D'']+\eta'',\qquad - D''=q^*D_U+\varepsilon E,\qquad - \eta''=q^*\eta_U-\varepsilon[E]. -\] -Here $\eta''$ is K\"ahler, and the actual real line bundle $-D''$ -has degree $\eta''\cdot C$ on every contracted curve. Restrict this -decomposition to the smooth reduced components in the gluing -construction. Lemma~\ref{prog:actual-line-polarization} then supplies -precisely the relative ampleness needed in place of -\cite[Lemma~2.42]{HX}, including the common-resolution step in its -Claim~3.5. The exceptional term $-\varepsilon[E]$ makes the -positive part $\eta''$ K\"ahler on the common resolution. - -The exact sequences of multiplier ideals, relative vanishing, -finite pushouts, and extension from sufficiently thickened -$\operatorname{Supp}D_U$ in that proof then apply unchanged. -They give the birational Moishezon contraction in -$\mathsf C_{d,\mathrm{big}}$. In particular the gluing step uses -neither generalized termination nor a projectivity criterion for -an undetected ray. - -\smallskip\noindent -\emph{Preparation for both cases of $\mathsf B_d$.} -The modified-big perturbation -\cite[Lemma~2.23]{HX} supplies a nef datum that dominates a -K\"ahler class on its carrier. Subtracting a sufficiently small -multiple of the pullback of a K\"ahler class $\omega_X$ still leaves -a nef datum on that carrier. Thus -$\alpha$ is a gklt adjoint plus $\varepsilon\omega_X$, and the cone -argument of \cite[Lemma~2.45]{HX} makes $\alpha$ NQC. -This preparation applies whether or not $\alpha$ is big. - -\smallskip\noindent -\emph{The nef-and-big case of $\mathsf B_d$.} -For a gklt pair the multiplier ideal is the unit ideal, so the -preceding big non-klt assertion now applies and gives a birational -contraction $h:X\to Y$. -On a projective resolution $p:W\to X$ chosen projective over $Y$, -relative Kawamata--Viehweg vanishing for the effective exceptional -divisor $-\lfloor B_W\rfloor$ gives -$R^ih_*\mathcal O_X=0$ for $i>0$. Thus $Y$ has rational -singularities. Proper birational Bott--Chern descent -\cite[Lemma~8.7]{DHP} gives $\alpha=h^*\gamma$. -The descended adjoint is gklt, nef and big, and has no trivial -curve. The criterion \cite[Theorem~4.3]{HLX} makes $\gamma$ -K\"ahler. - -For completeness, the projectivity input in that criterion is -restricted to a map $S\to Z'$ between smooth compact K\"ahler -manifolds with rationally connected general fibre: $S$ is the -chosen smooth divisor on a resolution and $Z'$ is the resolution -of the null-locus component. The smooth-source, smooth-base -argument in \cite[Theorem~3.1, Step~1]{CH} applies. Its subsequent -relative program starts with this projective map. Thus this application of the criterion uses only the -smooth-source, smooth-base projectivity argument and an -already-projective relative program. - -\smallskip\noindent -\emph{The non-big case of $\mathsf B_d$.} -Use a projective small factorialization and the modified-big -perturbation of the pair. The non-pseudo-effectivity of $K_X$ -gives an MRC fibration by \cite{Ou2025}. Choose a smooth K\"ahler -MRC base $Z$ and a smooth modification $\mu:U\to X$ such that -$U\to Z$ is projective. Only the smooth case of -\cite[Theorem~3.1, Step~1]{CH} is needed: pullback identifies -the holomorphic two-forms since the general fibre is rationally -connected. Lemma~\ref{prog:relative-hodge} identifies any $(1,1)$-class on $U$, -modulo a class pulled back from $Z$, with an actual real line bundle. -Lemma~\ref{prog:projective-cone} identifies its relative rays -with rays of the analytic cone; both statements persist along the -projective relative construction. - -Write, as in \cite[Theorem~4.1, Claim~4.1]{HX}, -\[ - D_U(a)=K_U+\Delta_U+\mathbf M_U+a\alpha_U - =(a+1)\alpha_U+F_U, - \quad \alpha_U=\mu^*\alpha,\quad - \Delta_U,F_U\geq0, -\] -where $\Delta_U$ is klt and $F_U$ is $\mu$-exceptional. -The modified-big replacement is chosen, as in the cited Claim~4.1, -so that the nef datum $\mathbf M_U$ on this smooth carrier is -K\"ahler. If a further projective resolution is needed, subtract a -sufficiently small effective exceptional divisor from its nef part -and add that divisor to the subboundary; this preserves the -adjoint and the klt inequalities. -Choose $a_0$ using the uniform bound of -Proposition~\ref{prog:relative-nqc-grid}. It makes the relative -$D_U(a_0)$-program $\alpha_U$-trivial and preserves that choice -through every step. This is a -\emph{projective} relative program from its first step: the nef -data are represented by real line bundles modulo $Z$. -Lemma~\ref{prog:relative-hodge} applies on every -intermediate model over this fixed smooth base. -More precisely, put $r:U\to Z$ and choose one actual global real -line bundle $L$ with -\[ - \mathbf M_U+a_0\alpha_U=c_1(L)+r^*\gamma. -\] -The bundle $L$ is relatively ample, because $\mathbf M_U$ is -K\"ahler and $\alpha_U$ is nef. On each of a fixed finite collection -of relatively compact Stein charts of $Z$, choose an effective -real divisor $\Theta$ representing $L$, with $(U,\Delta_U+\Theta)$ -klt. The representatives can retain a positive relatively ample -part in their boundaries. Here the equality -$K_U+\Delta_U+\Theta\sim_{\R}K_U+\Delta_U+L$ is an actual -real linear equivalence of line bundles; the equality with -$D_U(a_0)$ modulo $r^*\gamma$ is separately an equality of -Bott--Chern classes. - -Fix these ordinary pairs on the initial charts. Their actual -linear equivalences and their Bott--Chern comparisons with $D_U(a_0)$ -modulo the fixed base class persist separately throughout a global -relative program. Proposition~\ref{prog:projective-good-model} applies: -the source is smooth, its adjoint is represented by an actual global -real line modulo $Z$, its carrier nef part is relatively ample, and -$D_U(a_0)=(a_0+1)\mu^*\alpha+[F_U]$ is pseudo-effective. -It gives a finite program $U\dasharrow V$ and a good relative endpoint. -This is one global program controlled by fixed ordinary chart pairs. -Its construction uses global line algebras, not a choice of unrelated -local minimal models. Proposition~\ref{prog:relative-nqc-grid} makes -every step $\alpha_U$-trivial with the same value $a_0$. - -Here is an explicit ample-model comparison that gives the required -descent of $\alpha_V$. Both $D_V(a_0)$ and $\alpha_V$ are nef over -$Z$. Put $a_1=a_0+1$ and $a_2=a_0+2$. Before the program starts, -choose fixed ordinary representatives on the finite chart cover for -all three relatively ample nef parts $\mathbf M_U+a_j\alpha_U$, -$j=0,1,2$, on one simultaneous log resolution. The -$\alpha_U$-crepancy of the constructed program makes its endpoint -$V$ a weak model for each of these three fixed ordinary adjoints. -Their traces $D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V$ are nef. -Corollary~\ref{prog:projective-semiample} therefore makes -\[ - D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V -\] -semiample over $Z$. Their zero curves are exactly the common zero -curves of $D_V(a_0)$ and $\alpha_V$. Their projective ample-model -contractions therefore have the same fibres: these fibres are -connected by curves, and each contraction is constant on the -fibres of the other. Identify the two normal targets, writing the -common contraction as $g:V\to S$. If -$D_V(a_j)=g^*\lambda_j$, with $\lambda_j$ relatively K\"ahler over -$Z$, subtraction gives -\[ - \alpha_V=g^*(\lambda_2-\lambda_1)=g^*\alpha_S. -\] -This is an equality of Bott--Chern classes, not merely of curve -degrees. Since the initial program is $\alpha_U$-trivial, it is -also $D_U(a_1)$-negative. From now on put $a=a_1$ and suppress -the argument in $D_U(a)$ and $D_V(a)$. - -The maps $V\to Z$ and $S\to Z$ are projective. Moreover $g$ is -projective: a relatively ample bundle for $V\to Z$ restricts to -an ample bundle on every fibre of $g$, and hence is $g$-ample. -Each $D_U(a_j)$ is not big globally: pushing a big such class to -$X$ would make $(a_j+1)\alpha$ big. If it were big over the smooth -MRC base $Z$, relative-canonical positivity -\cite[Theorem~2.48]{HX}, together with the pseudo-effectivity of -$K_Z$, would make it big globally. It is therefore not big over -$Z$. This property persists to its relative semiample model, so -its relative canonical morphism has $\dim S0$, -with the map to $S$ retained throughout. -\end{lemma} - -\begin{proof} -After the smooth preparation in \cite[Proposition~5.12]{HX}, write the -adjoint as -\[ - A=D+\omega,\qquad D\geq0,\quad \omega\text{ K\"ahler}, - \qquad \operatorname{Supp}D=N_+(A). -\] -Here $N_+(A)$ denotes the reduced divisor with support -$\bigcap_{0<\epsilon\ll1}\operatorname{Supp}N_\sigma(A-\epsilon\omega)$; -this support is constant for sufficiently small positive $\epsilon$ -and is independent of the chosen Kähler class $\omega$ -\cite[Lemma~5.3]{HX}. At a nonterminal scaling step, -\[ - A_i\cdot R_i<0,\qquad - (A_i+t_i\omega_i)\cdot R_i=0,\qquad t_i>0. -\] -Consequently $\omega_i\cdot R_i>0$ and $D_i\cdot R_i<0$. -A component of the actual effective divisor $D_i$ supplies a global -rational Cartier detector. The supporting contraction follows from -$\mathsf B_d$, and its projectivity and flip follow from -Proposition~\ref{prog:detected-flip}. This argument does not require -$\omega_i$ to remain nef. - -To apply $\mathsf B_d$ to a gdlt presentation, lower its finitely many -floor coefficients on the initial smooth carrier and add the same small -class to the K\"ahler nef datum. The adjoint does not change, and the -resulting gklt presentation persists through the same negative steps. -The original gdlt presentation is retained for adjunction to the floor. -Thus the special-termination proof of \cite[Lemma~5.8]{HX} applies with -$\mathsf F_{d-1}$ on its smooth lower-dimensional carrier. Its other -inputs are adjunction, discrepancy monotonicity, and the local -Cartier-index calculation. The last neighbourhood-isomorphism step -uses the evaluation-ideal support argument in -the \hyperref[prog:special-support]{support comparison above}: the comparison of -the first detected flip has support equal to the full inverse image -of its exceptional locus, and later comparisons dominate it. -One does not infer disjointness of images merely from a vanishing -restriction of an arbitrary exceptional divisor. The theta induction of -\cite[Proposition~5.11]{HX} now gives a weak model. Applying -$\mathsf B_d$ to its nef adjoint makes it good, and -Lemma~\ref{prog:weak-to-terminal} gives the log terminal model. -All its steps are detected or belong to the already-projective -relative extraction in that lemma. - -For completeness, the negative-part identity needed in this argument -is only -\[ - N(p^*\xi+E)=N(p^*\xi)+E\quad(E\geq0\text{ $p$-exceptional}), -\] -not the stronger formula $N(p^*\xi)=p^*N(\xi)$ for an arbitrary -pseudo-effective class. The comparison with a nef endpoint identifies -its exceptional error with the negative part. The formula for -$N_+$ follows by writing a K\"ahler form upstairs as -$p^*\omega-F$, with $F$ positive on every exceptional prime, and applying -the displayed identity to -$p^*(\xi-\epsilon\omega)+E+\epsilon F$. Pushforward identifies the -nonexceptional coefficients; $\epsilon F$ supplies all exceptional -primes. These are exactly the uses in the theta argument. - -Now put $P=f^*\omega_S$ and suppose $A+cP$ is big for some $c$. -Fix $\delta>0$ smaller than the $\omega_S$-degree of every compact -integral curve on $S$ and enlarge $c$ so $c>4d/\delta$. -Use the following direct preparation, which keeps the original -unshifted globally nef datum. On a smooth projective carrier -$\nu:W\to X$ write -\[ - \nu^*(A+cP)=[G]+\omega,\qquad G\geq0,\quad\omega\text{ K\"ahler}, - \qquad [K_W+B_W]+M_W=\nu^*A, -\] -with the non-K\"ahler support property used in the big-class -preparation. Here $\operatorname{Ex}(\nu)$ below denotes the reduced -divisor formed by all $\nu$-exceptional primes. Write -$B_W=B_W^+-B_W^-$. For small $\epsilon>0$ put -\[ - \begin{split} - B'&=B_W^++\epsilon B_W^-+ - \epsilon(G+\operatorname{Ex}(\nu)),\\ - M'_W&=M_W+c\nu^*P+\epsilon\omega,\\ - b&=B_W^++\frac{\epsilon}{1+\epsilon}\operatorname{Ex}(\nu). - \end{split} -\] -The new boundary is klt and $M'_W$ is K\"ahler. Direct addition gives -\[ - A'=[K_W+B']+M'_W - =(1+\epsilon)\bigl([K_W+b]+M_W+c\nu^*P\bigr). -\] -Moreover $B'\geq b$. In every theta branch the cumulative added -boundary satisfies $0\leq C\leq1-B'$, so -\[ - b+\frac{C}{1+\epsilon} - \leq b+\frac{1-b}{1+\epsilon}<1. -\] -Thus the unshifted presentation -$[K_W+b+C/(1+\epsilon)]+M_W$ is genuinely gklt with the -\emph{original} globally nef datum. This is the presentation used -for the horizontal length bound. No base form is subtracted from -an arbitrary newly prepared nef datum. - -A horizontal negative ray would have $P$-degree at least $\delta$; -the bound $0<-A_{\rm unshifted}\cdot C\leq4d$ would contradict -$c\delta>4d$. Every constructed step is therefore over $S$. -Its projective connected fibres are contracted by $f$, so $f$ factors -through its base and through the flip. The final finite extraction is -also over $S$. - -The shifted nef endpoint is semiample by $\mathsf B_d$. Its zero face -contains no horizontal curve: decompose such a curve by the unshifted -adjoint's cone theorem; nefness of the shifted class forces all -summands into the zero face, and a summand of positive $P$-degree again -contradicts the length bound. The shifted class is big, so the -connected semiampleness map is bimeromorphic. Its connected fibres are Moishezon and are connected -by chains of compact curves. The preceding zero-face argument makes -the map to $S$ constant on each such curve, hence on every fibre. -The analytic factorization lemma therefore factors the map to $S$ -through the semiampleness map. Subtracting $c\omega_S$ from the -normalized class on its target gives the required relative K\"ahler class for the unshifted adjoint. The effective -resolution error from the preparation is removed by the usual -negativity comparison, giving a model of the original data. - -The forward traces and exceptional splittings persist by -Lemma~\ref{prog:detected-cohomology}. In the last extraction, -the newly extracted prime classes move from the old exceptional span -into the pullback span; they are actual rational Cartier classes. -Both-sign negativity proves directness as before. -\end{proof} - -\begin{lemma}[Changing the base of a relatively trivial adjoint] -\label{prog:graph-good-model} -Suppose $g:X\to Y$ is projective, its general fibre is rationally -connected, and a gklt adjoint satisfies $A_X=g^*A_Y$. Assume the -prepared nef datum is K\"ahler on a smooth carrier. Let -$Y\dashrightarrow Y_m$ be the chosen nonextracting good model -supplied by $\mathsf M_{d-1}$ over $S$, with its exceptional -cohomology splitting, and let $\dim Y2d/\delta$, where every -integral curve on $S$ has $\omega_S$-degree greater than -$\delta>0$. Set -\[ - A_c=A_0+cP=D+N,\qquad D=K_U+B_U,\quad N=M_U+cP. -\] -Since $A_c$ is not big globally, relative-canonical positivity -\cite[Theorem~2.48(2)]{HX}, together with pseudo-effectivity of $K_Z$, -shows that $A_c$ is not big over $Z$, whether or not it is globally -pseudo-effective. The class $N$ is big over $Z$. Consequently $D$ -cannot be pseudo-effective over $Z$: otherwise $D+N=A_c$ would be -big over $Z$. The smooth projective RC Hodge projection represents -every class modulo $Z$ by an actual real line. The ordinary global klt adjoint $D$ is therefore relatively -non-pseudo-effective. The projective ordinary Mori construction of -Corollary~\ref{prog:projective-mori}, with scaling of the actual -relatively ample real line represented by $N$, terminates in a Mori -fibre space. Lemma~\ref{prog:relative-hodge} and -Lemma~\ref{prog:projective-cone} identify every relative ray -with a full analytic ray. Its nef b-data retain the fixed globally -nef carrier $U$. - -First suppose $A_c$ is not pseudo-effective globally. The retained -relative-canonical positivity theorem on $h$, together with -pseudo-effectivity of $K_Z$, then says that $A_c$ is not -pseudo-effective over $Z$. The terminating relative program ends -in a Mori contraction at a threshold $\tau>1$. At every step its -scaled zero ray has $A_c$-degree negative. Inductively the map to -$S$ is retained: if such a ray were horizontal, its $P$-degree -would exceed $\delta$, while the gklt cone theorem for $A_0$ gives -a rational generator of $-A_0$-degree at most $2d$, contradicting -$A_c\cdot R\leq0$. Once $P\cdot R=0$, the step is negative or -crepant for $A_0$, so its gklt presentation persists. The same -argument applies to the final Mori ray. This would be an -$A_c$-negative Mori contraction over $S$, contradicting the -preserved pseudo-effectivity over $S$. Hence $A_c$ is -pseudo-effective globally. - -It is not big globally. Relative-canonical positivity now implies -that it is not big over $Z$. Its pseudo-effective threshold for -$D+tN$ over $Z$ is exactly one: for $t>1$ the class is big, while -pseudo-effectivity for $t<1$ would make $A_c$ big because $N$ is -big over $Z$. The same finite relative program therefore ends in -a Mori contraction $g:U'\to Y$ at threshold one. -All birational steps, including any steps at threshold one, and -the final zero ray are over $S$ by the preceding length argument -with $A_c\cdot R\leq0$. For a zero ray with positive $P$-degree, -$A_0$ would again have degree less than $-c\delta$. -Thus $Y$ carries a map to $S$ and $\dim Y0$ below the -$\omega_c$-degree of every compact integral curve in $X_c$. Fix $\bar\lambda<\delta/(\delta+2d)$ and for -$0<\lambda\leq\bar\lambda$ put -\[ - A_{\lambda,i}=(1-\lambda)H_i+\lambda A_{u,i}. -\] -The same cone estimate shows more than nefness: for a sufficiently -small $\epsilon>0$, uniform on the finite list and the chosen radial -interval, $A_{\lambda,i}-\epsilon H_i$ is nef. Indeed on a -horizontal $A_{u,i}$-negative rational ray its degree is greater than -$(1-\lambda-\epsilon)\delta-2d\lambda>0$; on the nonnegative -part of the cone and on the vertical cone it is nonnegative. -It follows for every class $\xi\in\overline{\mathrm{NA}}(Y_i)$ that -\[ - A_{\lambda,i}\cdot\xi=0 - \quad\Longleftrightarrow\quad - H_i\cdot\xi=A_{u,i}\cdot\xi=0. -\] -In particular its null curves are exactly those contracted by -$\psi_i$. The central data are crepant through these maps over $X_c$; -thus the convex data defining $A_{\lambda,i}$ remain gklt with -globally modified-big boundary. Apply $\mathsf B_d$ to $A_{\lambda,i}$. Its connected -semiampleness map and $\psi_i$ both have Moishezon fibres, and each -connected such fibre is connected by chains of compact curves. -Equality of their contracted curves makes each map constant on the -fibres of the other. The analytic factorization lemma identifies -the two maps. Thus the selected relative canonical model is also -the canonical model over $S$ on this radial interval. -This proves local polyhedrality and the finite local list; compactness -gives the global finite list of canonical models and chosen terminal -models. - -\paragraph{Uniform preparation preserving every weak model.} -\label{prog:uniform-preparation} -Write $A_u=[K_X+B_u+\mathbf M_{u,X}]$ for the adjoints in the -parameter polytope. We first record why a weak model of these gklt data -extracts no divisor. On a common resolution $p:U\to X$, $q:U\to Y$ -put $D=p^*A_u-q^*A_{u,Y}$, using the actual structure-boundary -comparison. The weak-model discrepancy inequalities give $p_*D\geq0$, -and $-D$ is $p$-nef because $A_{u,Y}$ is nef over the fixed base. -Negativity therefore gives $D\geq0$. An extracted prime on $Y$ has -coefficient one in the weak-model boundary, whereas its coefficient in -the structure boundary of the gklt source is less than one; it would -give a negative coefficient of $D$. Hence there is no such prime. -Consequently $D$ is $q$-exceptional and the target is itself gklt. - -Fix a parameter $u_0$. The boundary plus nef part is globally -modified big by hypothesis. The modified-big preparation -\cite[Lemma~2.23]{HX} gives, on a fixed projective smooth carrier -$a:W\to X$, an effective klt boundary $\Gamma$, a K\"ahler class -$\gamma_0$, and an effective $a$-exceptional real divisor $F$ such that -\begin{equation}\label{prog:eq:uniform-preparation} - [K_W+\Gamma]+\gamma_0=a^*A_{u_0}+[F]. -\end{equation} -Here is the effectivity detail in this use of the preparation. Its new -nef datum dominates a K\"ahler form on a carrier. After resolving the -new structure boundary, choose an effective exceptional divisor whose -negative is relatively ample and subtract a sufficiently small multiple -from the pulled-back datum to make it K\"ahler. Add the same multiple -to the structure boundary, -then replace its negative coefficients by zero and, if needed, add -small positive coefficients on the remaining exceptional primes. -All coefficients stay below one. Only exceptional coefficients have -been increased; their increase is the divisor $F\geq0$ in -\eqref{prog:eq:uniform-preparation}. - -On a sufficiently small closed polyhedral neighbourhood of $u_0$ put -\[ - \gamma_u=\gamma_0+a^*(A_u-A_{u_0}). -\] -These classes remain K\"ahler by openness, and the same $\Gamma$ and -the same $F$ satisfy -\[ - [K_W+\Gamma]+\gamma_u=a^*A_u+[F]. -\] -Thus this is one affine family of gklt pairs on one smooth carrier, -with a fixed effective boundary and K\"ahler nef data. - -Let $\psi:X\dashrightarrow Y$ be \emph{any} weak model at a parameter -$u$ in this neighbourhood, and resolve the induced map $W\dashrightarrow -Y$ by $r:T\to W$, $q:T\to Y$. Its original comparison is -\[ - (ar)^*A_u=q^*A_{u,Y}+E_u, - \qquad E_u\geq0\text{ is }q\text{-exceptional}. -\] -Since $\psi$ is nonextracting, every $a$-exceptional prime is exceptional -over $Y$. Hence $r^*F$ is also $q$-exceptional, and -\begin{equation}\label{prog:eq:prepared-weak} - r^*\bigl([K_W+\Gamma]+\gamma_u\bigr) - =q^*A_{u,Y}+E_u+r^*F. -\end{equation} -This proves that $W\dashrightarrow Y$ is a weak model of the prepared -pair. In detail, subtract the effective exceptional error on the right -from its structure boundary on $T$ and push down to $Y$. The resulting -boundary is precisely the transform of $\Gamma$, it is effective, the -adjoint is the already existing Bott--Chern class $A_{u,Y}$, and all -discrepancies are at least those of the prepared gklt pair. This uses generalized data at the level of currents. On $T$ keep -the nef datum $r^*\gamma_u$, and subtract the displayed effective -$q$-exceptional error from the prepared structure boundary. Pushing -forward by $q$ gives the boundary $\Gamma_Y$ and the trace of this -fixed nef b-class. Their sum with the canonical current represents -the already existing locally exact adjoint $A_{u,Y}$; the resolution -identity defines its structure boundary. The generalized discrepancies -have not decreased. This argument neither requires $K_Y+\Gamma_Y$ to -be real Cartier nor asserts that the nef trace is separately a -Bott--Chern class. It therefore applies to every K\"ahler weak target, -including the adjunction strata used in special termination. - -\paragraph{Every weak model occurs in the finite canonical list.} -Choose a small box in $H^{1,1}(W,\mathbb R)$ which spans that vector -space and whose addition to the compact family $\{\gamma_u\}$ stays -inside the K\"ahler cone. For any $Y$ above choose a K\"ahler class -$\eta_Y$ and define $\eta_W=r_*q^*\eta_Y$. Smooth-source cohomology -puts $\eta_W$ in $H^{1,1}(W,\mathbb R)$ and gives an actual -$r$-exceptional real divisor $D_\eta$ with -\[ - r^*\eta_W-q^*\eta_Y=[D_\eta]. -\] -Its negative is $r$-nef, so $D_\eta\geq0$ by exceptional negativity. -As $W\dashrightarrow Y$ extracts no divisors, $D_\eta$ is also -$q$-exceptional. Scale $\eta_Y$ by a sufficiently small positive number -so that $\eta_W$ lies in the fixed box. Adding this identity to -\eqref{prog:eq:prepared-weak} yields an effective -$q$-exceptional comparison with target class $A_{u,Y}+\eta_Y$, which -is K\"ahler over the original base. Thus $W\dashrightarrow Y$ is the -canonical model of a member of this one enlarged prepared polytope. -Its canonical-model list is finite by the first part of the geography -proof. The induced list of maps from $X$ is therefore finite as well. -A finite cover of the original compact parameter polytope completes -the proof for all weak models. - -\end{proof} - - -\begin{theorem}[The restricted package in every dimension] -\label{prog:restricted-package} -The assertions $\mathsf B_d$, $\mathsf C_d$, $\mathsf M_d$ and -$\mathsf F_d$ hold in every finite dimension, for the globally nef -carrier data and globally modified-big boundary traces specified above. -All birational steps used to prove them are detected steps or belong -to an already-projective relative construction with the stated -relative-line algebraicity. A general semiample morphism is asserted -to be Moishezon; projectivity is asserted only when the construction -provides an actual relatively ample line. -\end{theorem} -\begin{proof} -The assertions in dimensions zero and one follow from degrees and -factorization of maps of compact curves. Suppose the package holds -in dimensions less than $d$. Lemma~\ref{prog:base-point-free-induction} -first proves $\mathsf C_{d,\mathrm{big}}$ and then $\mathsf B_d$. -The former uses $\mathsf C_{d-1}$; the nonbig part of the latter -uses only a chosen $\mathsf M_{d-1}$ model and the explicit -negative-part comparison. In particular, no lower program is lifted. - -With $\mathsf B_d$ available, Proposition~\ref{prog:ordinary-step} -constructs detected negative steps in dimension $d$. -Lemma~\ref{prog:big-good-model}, using the already known -$\mathsf F_{d-1}$, gives the big-shift case of $\mathsf M_d$. -Lemma~\ref{prog:relative-model} gives its remaining case, using -$\mathsf M_{d-1}$ and the projective graph construction. This proves -$\mathsf M_d$ over every proper compact K\"ahler base, with its -chosen-model invariant. Lemma~\ref{prog:finite-geography}, whose -hypotheses are \emph{both} $\mathsf B_d$ and $\mathsf M_d$, now -proves $\mathsf F_d$. - -It remains to prove $\mathsf C_d$ when its nef NQC adjoint $\alpha$ -is not big. First use -Lemma~\ref{prog:base-presentation-preparation}, preserving the -adjoint $\alpha$ and making the nef datum K\"ahler on a smooth -carrier. Its multiplier-ideal inclusion allows the given contraction -to restrict to the prepared non-klt closed subspace by -Lemma~\ref{prog:base-emc}. Next take the projective dlt preparation -in \cite[Section~6]{HX}, keeping its crepant structure data. The -working space is klt and globally strongly $\mathbb Q$-factorial. -The retained crepant structure boundary need not be lc; it is -effective, including the extracted primes of coefficient at least one. On a common carrier the nef datum is -nef and big. Consequently the total boundary trace is modified big, -and $\alpha-c_1(K_X)$ is a big Bott--Chern class. The contraction on -the non-klt closed subspace transports through this preparation by -Lemma~\ref{prog:multiplier-transport}. Choose an NQC expression with -positive coefficients and integral multiples of its rational -degree-two summands. Corollary~\ref{prog:uniform-canonical-parameter} -provides a single number $t$ such that each negative ray of -$K_X+t\alpha$ is $\alpha$-trivial. Choose a K\"ahler scaling class -$\omega$ with $K_X+t\alpha+\omega$ nef. Every selected negative -ray is ordinary $K_X$-negative; the actual rational canonical line -is its detector. Proposition~\ref{prog:ordinary-step} supplies its -projective contraction or flip, and the fixed integral grid persists. -The descended $\alpha$ remains nef by projective descent of nefness. - -The driving class is non-pseudo-effective throughout, by the -corollary. Its initial pseudo-effective scaling threshold is positive; -choose a cutoff $\varepsilon>0$ below that threshold. On the fixed -initial carrier the compact segment of globally nef data -\[ - t\alpha+[\varepsilon,1]\omega -\] -has modified-big total boundary. At its scaling parameter each working -model is a marked weak model of a member of this segment. Each selected -step is negative for the fixed driving adjoint, so its discrepancies -do not decrease and increase at a valuation affected by that step. -An infinite sequence would repeat a marked model by $\mathsf F_d$; -the cumulative nonzero effective comparison for the fixed adjoint -would then contradict equality of that marking. Hence this program -terminates. Non-pseudo-effectivity excludes a nef endpoint, leaving -an $\alpha$-trivial ordinary Mori contraction $f:X'\to Z$. - -The structure data of the prepared contraction problem remain -crepant throughout this $\alpha$-trivial program. Their effective -traces and globally nef and big carrier datum therefore persist. -Lemma~\ref{prog:multiplier-transport} transports the contraction on -the non-klt closed subspace through each birational step. -Lemma~\ref{prog:bc-descent} gives -\[ - \alpha_{X'}=f^*\gamma -\] -in Bott--Chern cohomology. The class $\gamma$ is nef by projective -descent; Lemma~\ref{prog:integral-h2} descends the fixed rational -NQC summands, and projective multisections preserve their -nonnegative degrees. - -Apply Corollary~\ref{prog:base-emc-induction} to this projective -Mori contraction. In its notation the prepared non-klt subspace -$N_1$ has Stein space $V(f_*J_1)$ over $Z$. If $N_1$ dominates -$Z$, this Stein space is $Z$ itself and its inherited contraction -already supplies the contraction of $\gamma$. Otherwise the base -presentation has multiplier ideal $J_Z\supset f_*J_1$, so the -contraction restricts to $V(J_Z)$ as a closed subspace of -$V(f_*J_1)$. Since $\dim Z0$, is finite. If the driving adjoint is not -pseudo-effective, the scaling terminates with a Mori fibre space. -\end{corollary} -\begin{proof} -Proposition~\ref{prog:ordinary-step} constructs each step and -Lemma~\ref{prog:detected-cohomology} supplies the forward traces from -the fixed carrier. Each reached model at a scaling parameter is a -marked weak model of that parameter's adjoint. The compact parameter -segment satisfies $\mathsf F_d$, so there are finitely many such -markings. Strict increase for the fixed driving adjoint rules out -repetition. If that adjoint is not pseudo-effective, the positive -pseudo-effective scaling threshold gives a fixed positive cutoff -containing every parameter of a continuing program. The finite -program cannot end nef, hence ends with the claimed Mori map. -\end{proof} +\subsection{A restricted dimension induction} +\label{prog:dimension-induction} + +All generalized nef data in this subsection are globally nef on a fixed +compact K\"ahler carrier. For the degree arguments we use weak NQC: +a positive combination of rational degree-two classes having +nonnegative degree on compact curves. Strong NQC implies this +condition, and the contraction assertion admits this weak form +\cite[Definition~2.43]{HX}. The nef adjoint itself remains nef in +Bott--Chern cohomology throughout. A modified-big boundary means the total trace +$B+\mathbf M_X$ is globally modified big. Relative assertions retain +these absolute hypotheses. A marked model records its bimeromorphic map +from the specified source; two markings agree only under an isomorphism +commuting with those maps. Weak models below may be arbitrary normal compact K\"ahler weak log +canonical models. Their generalized nef traces are understood as +closed currents pushed forward from the fixed nef carrier; only the +whole adjoint is required to be a Bott--Chern class. In particular no +factoriality, and no separate Bott--Chern class for either $K_Y+B_Y$ +or the nef trace, is required of an auxiliary weak target. This +category contains both the working models and the adjunction strata +used in special termination. Chosen program outputs remain globally +strongly $\mathbb Q$-factorial. + +For dimension at most $d$, write $\mathsf B_d$ for semiampleness of a +nef gklt adjoint with modified-big boundary, including a Moishezon +connected contraction onto a normal compact K\"ahler target. It makes +no assertion that every such contraction is projective. +Write $\mathsf C_d$ for the NQC non-klt contraction assertion of +\cite[Theorem~1.6]{HX}, and $\mathsf C_{d,\mathrm{big}}$ for its +big-adjoint case. The relative assertions are: +\begin{itemize} +\item $\mathsf M_d$: a relatively pseudo-effective gklt adjoint over +any proper map to a normal compact K\"ahler base $S$, with the globally +nef and modified-big data just specified, has a chosen nonextracting +good log terminal model over $S$. Its working space is globally +strongly $\mathbb Q$-factorial and compact K\"ahler; its connected +relative canonical morphism has normal compact K\"ahler target and +is Moishezon. From a smooth common carrier, +all Bott--Chern classes have forward traces, and the chosen model has +the rational degree-two and Bott--Chern exceptional splittings proved +below. +\item $\mathsf F_d$: a compact polytope of these data on one fixed +carrier has a polyhedral relative effective locus, finitely many +relative canonical chambers with chosen good terminal models, and +finitely many marked weak models, accounting for every weak model +in the category specified above. +\end{itemize} +Passing to the proper image and then to its normal Stein factor allows +all base maps to be taken surjective with connected fibres. These +operations preserve the compact K\"ahler base category. + +The independent inputs are the analytic cone theorem, the projective +results of Proposition~\ref{prog:projective-good-model}, relative +vanishing, adjunction, relative-canonical positivity, and the +nef-and-big K\"ahler criterion. The base presentations and +multiplier-ideal descent used in the induction are proved below. +The projective constructions use the actual-line lemmas of the preceding +subsection. No assertion of termination of an arbitrary sequence of +flips is used. + +\input{sections/03c-base-presentations} + +\subsubsection{A fixed integral grid and cohomology transport} + +\begin{lemma}[Integral degree-two descent for a locally log-Fano map] +\label{prog:integral-h2} +Let $f:X\to Z$ be a projective surjective morphism with connected fibres +between normal complex analytic spaces. Assume that $f_*\mathcal O_X= +\mathcal O_Z$, that $R^if_*\mathcal O_X=0$ for $i>0$, and that every point +of $Z$ has a neighbourhood $U$ on which there is a klt pair +$(X_U,\Delta_U)$ with $-(K_{X_U}+\Delta_U)$ ample over $U$. +Then $f^*:H^2(Z,\mathbb Z)\to H^2(X,\mathbb Z)$ is injective, and its +image consists exactly of the integral classes having degree zero on +every compact curve contracted by $f$. +\end{lemma} + +\begin{proof} +Properness, normality and connected fibres identify both +$f_*\mathcal O_X=\mathcal O_Z$ and +$f_*\mathcal O_X^*=\mathcal O_Z^*$. Apply $Rf_*$ to the analytic +exponential sequence. The local surjectivity of +$\mathcal O_Z\to\mathcal O_Z^*$ and the stated coherent vanishing give +\[ + R^1f_*\mathbb Z_X=0, + \qquad + R^1f_*\mathcal O_X^*\simeq R^2f_*\mathbb Z_X. +\] +Let $\xi\in H^2(X,\mathbb Z)$ have zero degrees on contracted curves. +A germ of its edge image in $R^2f_*\mathbb Z_X$ is, under this +isomorphism, represented after shrinking $U$ by an actual line bundle +$L$ on $X_U$. Equality of these germs means that, after further +shrinking, $c_1(L)$ and $\xi|_{X_U}$ have the same class. In particular +$L$ has degree zero on every curve in every fibre above this smaller +neighbourhood. + +On a sufficiently small Stein base neighbourhood, twisting by a +relatively ample line and taking a quotient of two nonzero relative +sections gives a meromorphic Cartier representative for $L$. Thus +the Cartier-divisor form of base point freeness applies. +The line bundle $L$ is relatively nef, and +$aL-(K_{X_U}+\Delta_U)$ is relatively ample for every real $a$. +The projective analytic base-point-free theorem applies to the Cartier +line $L$. Its conclusion is generation for \emph{every} sufficiently +large integer power, not only for a divisible subsequence +\cite[Theorems~6.2 and~6.5]{FujinoAnalyticBCHM}. Thus, over a further relatively compact +neighbourhood, both $L^m$ and $L^{m+1}$ are relatively generated. +A generated line of degree zero on every curve of a projective connected +fibre defines a constant projective map on that fibre: the restriction +of the generated line is the pullback of $\mathcal O(1)$, and a +positive-dimensional projective image contains a curve of positive +degree. The corresponding relative image is therefore finite and +bimeromorphic over the normal base, and is the base itself. Consequently +\[ + L^m\simeq f^*M_m,\qquad L^{m+1}\simeq f^*M_{m+1} +\] +for line bundles on the neighbourhood in $Z$. Taking their quotient +proves $L\simeq f^*(M_{m+1}\otimes M_m^{-1})$. Its germ in +$R^1f_*\mathcal O_X^*$ is zero, so the edge image of $\xi$ is zero. + +The integral Leray spectral sequence, using +$R^1f_*\mathbb Z_X=0$, gives +\[ + 0\longrightarrow H^2(Z,\mathbb Z) + \xrightarrow{f^*}H^2(X,\mathbb Z) + \longrightarrow H^0(Z,R^2f_*\mathbb Z_X). +\] +Exactness at the middle term proves the assertion. The argument is +integral throughout and does not discard torsion. The converse follows +by restriction of a pulled-back class to a fibre. +\end{proof} + +\begin{lemma}[A fixed integral grid along existing crepant steps] +\label{prog:fixed-integral-grid} +Let $X_0$ be a compact K\"ahler space, and suppose that +\[ + \alpha_0=\sum_{j=1}^r a_j\gamma_{0j},\qquad a_j>0, + \qquad \gamma_{0j}\in H^2(X_0,\mathbb Q), +\] +where each $\gamma_{0j}$ has nonnegative degree on compact curves. +Fix integers $m_j>0$ such that $m_j\gamma_{0j}$ is the image of an +integral cohomology class. Consider an existing sequence of projective +birational steps +\[ + X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1}, +\] +where $f_i^+$ is the identity for a divisorial step, $f_i$ satisfies +Lemma~\ref{prog:integral-h2}, and $\alpha_i$ is trivial on its +contracted ray. Assume every $f_i$-contracted curve has degree +proportional to that ray for the classes in the display. Then the +classes $\gamma_{ij}$ descend through $f_i$ and pull back through +$f_i^+$, the same integers $m_j$ remain integral multiples at every +stage, and the transported classes remain nonnegative on curves. +In particular, for every integral compact curve $C\subset X_i$, +\[ + \alpha_i\cdot C>0\quad\Longrightarrow\quad + \alpha_i\cdot C\geq\delta, + \qquad \delta:=\min_j\frac{a_j}{m_j}>0. +\] +\end{lemma} + +\begin{proof} +Positivity of the coefficients and nonnegativity of the summands imply +that every $\gamma_{ij}$ annihilates the contracted ray. Apply the +integral descent lemma to the chosen integral lift of +$m_j\gamma_{ij}$ and pull its descended class back to $X_{i+1}$. +This keeps $m_j$ fixed. + +To verify nonnegativity, take a curve on $X_{i+1}$. If it is contracted +by $f_i^+$, every descended summand has degree zero. Otherwise its +image is a curve $\Gamma\subset Z_i$. The projective map $f_i$ has a +compact curve mapping onto $\Gamma$ with positive degree: take a +component of its inverse image dominating $\Gamma$ and intersect with +sufficiently many relatively ample hyperplanes. Nonnegativity of the +old summand on this curve implies nonnegativity on $\Gamma$, hence on +the curve on $X_{i+1}$. This argument uses a positive-degree lift, not +a degree-one section. Finally $m_j\gamma_{ij}\cdot C$ is a +nonnegative integer, giving the stated lower bound. +\end{proof} + +\begin{corollary}[Uniform crepancy parameter; no termination assertion] +\label{prog:uniform-canonical-parameter} +Suppose additionally that the working spaces are $d$-dimensional klt +spaces, that $\alpha_i$ are nef, and that the ordinary canonical +negative-ray length bound $0<-K_{X_i}\cdot C\leq2d$ is available. +For a fixed real number $t>2d/\delta$, every negative extremal ray +chosen in an existing $(K_{X_i}+t\alpha_i)$-program is +$\alpha_i$-trivial. If the steps are projective ordinary canonical +steps, Lemma~\ref{prog:fixed-integral-grid} therefore applies +inductively with this same value of $t$. + +If $\alpha_0$ is not big and $\alpha_0-c_1(K_{X_0})$ is big, then +$K_{X_i}+t\alpha_i$ remains non-pseudo-effective along these crepant +steps. Consequently a terminating such program ends with a Mori +fibre contraction. This corollary does not assert that the requisite +contractions exist or that the program terminates. +\end{corollary} + +\begin{proof} +A negative ray is $K_{X_i}$-negative because $\alpha_i$ is nef. Choose +its rational curve generator with the ordinary length bound. If its +$\alpha_i$-degree were positive, then +\[ + (K_{X_i}+t\alpha_i)\cdot C\geq-2d+t\delta>0, +\] +a contradiction. The step is thus an ordinary canonical step with +$\alpha_i$ pulled back from its base, and the preceding lemma applies. +On a common resolution the pullback of +$\alpha_{i+1}-c_1(K_{X_{i+1}})$ is the pullback of +$\alpha_i-c_1(K_{X_i})$ plus the effective canonical comparison. +Thus it stays big. Equality of the crepant pullbacks of $\alpha_i$ +preserves its nonbigness. The +identity +\[ + (t+1)\alpha_i=(c_1(K_{X_i})+t\alpha_i) + +(\alpha_i-c_1(K_{X_i})) +\] +excludes pseudo-effectivity of its first summand. A nef endpoint is +therefore impossible. +\end{proof} + + +\begin{lemma}[Cohomology splitting along detected steps] +\label{prog:detected-cohomology} +Let $X_0$ be a smooth compact K\"ahler manifold. Consider a finite +sequence of divisorial contractions and small flips +\[ + X_0\dashrightarrow X_1\dashrightarrow\cdots + \dashrightarrow X_r +\] +between normal globally strongly $\mathbb Q$-factorial compact +K\"ahler spaces with rational singularities. +Assume that every negative contraction $f_i:X_i\to Z_i$ is a +projective contraction of one ray of the analytic cone, that $Z_i$ +has rational singularities, and that each small flip is the detected +flip of a global rational line $L_i$ with $L_i\cdot R_i<0$. +For a divisorial contraction assume the usual single exceptional +prime. Then, for every projective smooth resolution $p:U\to X_i$, +\begin{align*} + H^2(U,\mathbb R) + &=p^*H^2(X_i,\mathbb R) + \oplus\bigoplus_{E\text{ exceptional for }p}\mathbb R[E],\\ + H^{1,1}(U,\mathbb R) + &=p^*H^{1,1}_{\rm BC}(X_i,\mathbb R) + \oplus\bigoplus_{E\text{ exceptional for }p}\mathbb R[E]. +\end{align*} +The first equality also holds over $\mathbb Q$. There are compatible +forward linear transforms on $H^2(-,\mathbb R)$ and on Bott--Chern +cohomology. They are surjective, are isomorphisms for a small flip, +and preserve rational degree-two classes. On a common resolution the +pullback difference is an actual exceptional divisor class; for a +flip it is exceptional over both spaces. + +The exceptional quotient also takes the Chern class of a holomorphic +line bundle on $U$ to a rational holomorphic line class on $X_i$. +\end{lemma} + +\begin{proof} +For a modification of a smooth compact K\"ahler manifold the two +splittings are the degree-two modification formula. Suppose that they +hold for $X=X_i$. We use only the following birational descent fact: +for a proper bimeromorphic map between normal compact spaces in +Fujiki's class with rational singularities, pullback is injective on +$H^2(-,\mathbb R)$ and on Bott--Chern cohomology, and its image in +either group consists of the classes having degree zero on every +contracted curve. This is the bimeromorphic case of +\cite[Lemma~2.6(1)]{DH2024}. + +First note that a degree-two class $\xi\in H^2(X,\mathbb R)$ has +the same degrees on curves as a Bott--Chern class. Indeed, pull it to +a smooth projective resolution $a:W\to X$ and take the $(1,1)$-part +of its Hodge decomposition. The induction hypothesis writes that +part as $a^*\beta+[F]$, with $F$ exceptional over $X$. On every +$a$-contracted curve, $a^*\xi$ and its $(2,0)$ and $(0,2)$ parts +have degree zero. Hence $F$ has degree zero on every such curve. +Exceptional negativity applied to both signs gives $F=0$. +Projective multisections lifting a curve of $X$ then show that +$\xi$ and $\beta$ have the same degrees on every curve of $X$. + +Suppose first that $X\dashrightarrow X^+$ is a small detected flip. +Write $f:X\to Z$ and $f^+:X^+\to Z$, and fix an integral curve +$C$ on its negative ray. For $\xi\in H^2(X,\mathbb R)$ set +\[ + s=\frac{\xi\cdot C}{c_1(L)\cdot C}. +\] +By the preceding paragraph, $\xi-s c_1(L)$ vanishes on every +$f$-contracted curve. Thus it is $f^*\eta$ for a unique +$\eta\in H^2(Z,\mathbb R)$. Define +\[ + T\xi=(f^+)^*\eta+s c_1(L^+). +\] +For a Bott--Chern class, the same descent takes place in +Bott--Chern cohomology, so the two transforms agree. The detected +flip construction gives an effective rational divisor $E_L$ on a +common projective resolution $a:W\to X$, $b:W\to X^+$ such that +\[ + a^*c_1(L)-b^*c_1(L^+)=[E_L]. +\] +Consequently +\[ + a^*\xi=b^*T\xi+s[E_L]. +\] +If $\xi$ is rational, then $s$ is rational. Since the image of +$f^*$ on real degree-two cohomology is the realification of a +rational subspace, its unique preimage $\eta$ is rational as well. +This proves rationality of $T\xi$. + +The sets of prime divisors exceptional for $a$ and $b$ are identical: +the two working spaces are small bimeromorphic. The old splitting +and the displayed pullback identity therefore span +\[ + H^2(W,\mathbb R) + =b^*H^2(X^+,\mathbb R) + +\operatorname{span}_{\mathbb R}\{[E]:E\text{ is }b + \text{-exceptional}\}, +\] +and similarly in bidegree $(1,1)$. The sum is direct. If a pullback +from $X^+$ equals an exceptional divisor class, that divisor has +degree zero on all $b$-contracted curves, so both signs of negativity +make it zero; pullback injectivity then makes the original class +zero. In particular the dimensions on the two sides agree, $T$ is +injective by the old direct sum, and $T$ is surjective. The rational +version follows either from the same argument or from realification. + +For a divisorial contraction $f:X\to X^+$, let $D$ be its +exceptional prime. Its degree on the contracted ray is negative: +otherwise $D$ would be $f$-nef, contrary to exceptional negativity. +For any $\xi$, subtract +\[ + \frac{\xi\cdot C}{D\cdot C}[D] +\] +and descend the result by the same birational cohomology lemma. +This defines $T\xi$. On a common resolution its pullback difference +is a multiple of the total transform of $D$. The target's exceptional +prime set is the old one together with the strict transform of $D$. +The same spanning and directness argument gives the target splitting +and surjectivity; its kernel is $\mathbb R[D]$. Rationality is again +immediate from the quotient of rational intersection numbers. + +These arguments initially prove the splitting on a chosen common +resolution. To obtain it on any projective smooth resolution +$U\to X^+$, dominate that resolution and the chosen one by a smooth +common projective resolution. Apply the smooth degree-two +modification formula upstairs and push down to $U$. An exceptional +divisor pushes either to zero or to a divisor exceptional over $X^+$. +This proves spanning on $U$; negativity gives directness there. + +Finally let $\mathcal H$ be a holomorphic line on $U$ and put +$\mathcal Q=(p_*\mathcal H)^{**}$. Global strong +$\mathbb Q$-factoriality gives an $m>0$ for which +$\mathcal M=\mathcal Q^{[m]}$ is invertible. The coherent evaluation +comparison gives +\[ + \mathcal H^{\otimes m} + \simeq p^*\mathcal M\otimes\mathcal O_U(F) +\] +for an integral $p$-exceptional divisor $F$. This comparison is +obtained from the coherent direct image and evaluation map; it does +not assume that $\mathcal H$ has a global meromorphic section. +Dividing first Chern classes by $m$ proves the line assertion. +\end{proof} + + +\begin{corollary}[Stability under projective extraction] +\label{prog:extraction-cohomology} +Let $r:Y\to X$ be a projective bimeromorphic morphism of normal compact +K\"ahler spaces with rational singularities. Suppose that $Y$ is globally +strongly $\mathbb Q$-factorial and that $X$ has the splitting property in +the preceding lemma. Then $Y$ has that property as well. More precisely, +if $E_1,\ldots,E_s$ are the prime divisors exceptional for $r$, then +\[ + H^2(Y,\mathbb R) + =r^*H^2(X,\mathbb R)\oplus + \bigoplus_{j=1}^{s}\mathbb R[E_j], +\] +and the analogous statements hold over $\mathbb Q$ and in +Bott--Chern cohomology. +\end{corollary} + +\begin{proof} +Choose any projective smooth resolution $q:W\to Y$. The composite +$p=rq$ is a projective resolution of $X$. Its exceptional primes are +exactly the $q$-exceptional primes together with the strict transforms +$\widetilde E_j$ of the $E_j$. Since every $E_j$ is $\mathbb Q$-Cartier, +\[ + [\widetilde E_j]=q^*[E_j]+[F_j] +\] +for a rational $q$-exceptional divisor $F_j$. Substitute these identities +in the splitting for $p$. The result spans $H^2(W,\mathbb R)$ by +$q^*H^2(Y,\mathbb R)$ and the $q$-exceptional divisor classes. Their sum +is direct: an exceptional divisor whose class is a pullback has degree +zero on every $q$-contracted curve, and applying exceptional negativity +to both signs makes the divisor zero. Pullback injectivity then applies. +The same argument in bidegree $(1,1)$ and over $\mathbb Q$ proves the +splitting property for $Y$. + +Now expand $q^*\xi$, for $\xi\in H^2(Y,\mathbb R)$, using the original +$p$-splitting and replace the $[\widetilde E_j]$ as above. The difference +between $q^*\xi$ and a class in +$q^*(r^*H^2(X,\mathbb R)+\sum_j\mathbb R[E_j])$ is $q$-exceptional, +and is therefore zero by the direct sum just proved. Thus the displayed +formula for $H^2(Y,\mathbb R)$ spans. Its directness follows from +exceptional negativity for $r$ and pullback injectivity. The rational +and Bott--Chern statements follow by the identical argument. The line +bundle assertion of the preceding lemma uses only global strong +$\mathbb Q$-factoriality and hence applies to $Y$ as well. +\end{proof} + + + +\paragraph{Support in the special-termination comparison.} +\label{prog:special-support} +The following support comparison applies to the detected steps in +the dimension induction. Let $\psi:X\dashrightarrow X^+$ be a detected +small negative step with maps $f:X\to Z$ and $f^+:X^+\to Z$, +and let $L$ be its negative detector. For a sufficiently +divisible $r>0$, put +\[ + \operatorname{im}(f^*f_*L^r\longrightarrow L^r) + =\mathcal I\otimes L^r. +\] +The zero set of $\mathcal I$ is exactly the exceptional locus of $f$. +Outside that locus $f$ is an isomorphism, so the evaluation map is +surjective. On a positive-dimensional projective fibre, every section +of $L^r$ vanishes: its restriction to each integral curve has negative +degree, and those curves cover every positive-dimensional fibre +component. Connectedness supplies the whole nontrivial fibre. + +On a resolution principalizing $\mathcal I$, the detector comparison +is a positive multiple of the effective divisor defined by +$\mathcal I\mathcal O_W$. Consequently its support is the full +inverse image of the exceptional locus. The driving adjoint differs +from a positive real multiple of $c_1(L)$ by a class pulled back from +the contraction base. Its comparison divisor is therefore the same +positive multiple of the detector comparison: their class difference +is exceptional and zero, so both-sign negativity makes it zero as a +divisor. The comparison for a longer +negative program is at least this first-step divisor, by discrepancy +monotonicity on a common resolution. + +Suppose the beginning and end restrictions to a surviving normalized +lc stratum $S$ are isomorphic and their strict adjunction data agree. +The general point of this stratum avoids every intervening surgery: +a zero-discrepancy place remains such a place, whereas strict +comparison would increase its discrepancy if its center were contained +in the exceptional locus. Thus its strict transform $S_W$ on a common +resolution is not contained in the comparison support. Iterated strict +adjunction identifies the restriction of the effective comparison with +the difference of the identified adjoints, so its class is zero. +A nonzero effective divisor on a positive-dimensional compact K\"ahler +stratum has positive K\"ahler mass. The restricted divisor is therefore +zero. If the first step meeting the stratum were nontrivial there, +the full inverse-image support and surjectivity $S_W\to S$ would make +this restriction nonzero, a contradiction. For a zero-dimensional +stratum, generic-point avoidance already excludes an intersection. +The whole program is consequently an isomorphism near the stratum. +This is the same strict-adjunction support argument used in the proof +of Theorem~\ref{prog:special-termination}; it does not use its +lower-dimensional abundance assumption. + +\begin{proposition}[A uniform relative NQC bound] +\label{prog:relative-nqc-grid} +Let $f_0:U\to Z$ be an already projective surjective morphism with +connected fibres between smooth compact +K\"ahler manifolds, and suppose that +\[ + f_0^*:H^0(Z,\Omega_Z^2)\longrightarrow H^0(U,\Omega_U^2) +\] +is an isomorphism. Let $\alpha_0$ be a nef class with an expression +\[ + \alpha_0=\sum_{j=1}^k r_j\eta_{j,0},\qquad + r_j>0,\quad \eta_{j,0}\in H^2(U,\mathbb Q), +\] +where $\eta_{j,0}$ has nonnegative degree on every curve vertical +over $Z$. Let $D_0=K_U+\Delta_U+\mathbf M_U$ be a generalized klt +adjoint. Consider its projective relative program for +$D_0+a\alpha_0$, under the projective local ordinary reduction described in the +proof of Lemma~\ref{prog:base-point-free-induction}. + +If $\alpha_0=0$, triviality is immediate. Otherwise there are +positive integers $m_j$, chosen once on $U$, and a number +\[ + \delta=\min_j\frac{r_j}{m_j}>0 +\] +such that, for $a\delta>2\dim U$, the entire relative program is +$\alpha_0$-trivial. The same number $a$ works at every step. +\end{proposition} + +\begin{proof} +\emph{Rational classes modulo the fixed base.} +Choose an $f_0$-ample line bundle with integral Chern class $H$, and +put $e=\dim U-\dim Z$ and $c=f_{0*}(H^e)>0$. The rational operator +\[ + \Pi(\eta)=\eta- + f_0^*\!\left(\frac{f_{0*}(\eta\smile H^e)}{c}\right) +\] +preserves type $(1,1)$ and kills types $(2,0)$ and $(0,2)$. +Indeed these latter classes are pulled back from $Z$, and the +projection formula applies. Thus +$\Pi(H^2(U,\mathbb Q))\subset H^{1,1}(U)\cap H^2(U,\mathbb Q)$. +By the Lefschetz $(1,1)$ theorem, choose an actual line bundle +$L_{j,0}$ and an integer $m_j>0$ with +\[ + \eta_{j,0}=\frac1{m_j}c_1(L_{j,0})+f_0^*\gamma_j, + \qquad \gamma_j\in H^2(Z,\mathbb Q). +\] +The original summands $\eta_{j,0}$ need not have type $(1,1)$. +Their base components account for this. Their weighted sum gives +\begin{equation}\label{prog:eq:relative-nqc} + \alpha_0=\sum_j\frac{r_j}{m_j}c_1(L_{j,0})+f_0^*\Gamma, + \qquad \Gamma=\sum_jr_j\gamma_j\in H^{1,1}(Z,\mathbb R). +\end{equation} +The final assertion follows because $f_0^*$ is an injective Hodge +map and the other terms have type $(1,1)$. The identity is therefore +also one of Bott--Chern classes. +Each $L_{j,0}$ has nonnegative integral degree on every +$Z$-vertical curve. + +\smallskip\noindent +\emph{Induction across an actual contraction.} +Suppose at a finite stage $f_i:U_i\to Z$ we have actual line bundles +$L_{j,i}$, nef over $Z$, and +\[ + \alpha_i=\sum_j\frac{r_j}{m_j}c_1(L_{j,i})+f_i^*\Gamma. +\] +Assume also that $\alpha_i$ is nef and that the transformed +$D_i$ is generalized klt. These assertions hold initially. +Let $R$ be a $(D_i+a\alpha_i)$-negative extremal ray over $Z$. +It is $D_i$-negative because $\alpha_i$ is nef. Lemma~\ref{prog:relative-hodge} and Lemma~\ref{prog:projective-cone} +identify the relative curve cone with the corresponding face of the +full analytic cone, at the initial model and after every step. The +analytic cone theorem therefore supplies a rational generator $C$ with +\[ + 0<-D_i\cdot C\leq2\dim U. +\] +All $L_{j,i}\cdot C$ are nonnegative integers. If +$\alpha_i\cdot C>0$, one of them is at least one, and hence +$\alpha_i\cdot C\geq\delta$. Therefore +\[ + (D_i+a\alpha_i)\cdot C + \geq-2\dim U+a\delta>0, +\] +a contradiction. We have $\alpha_i\cdot C=0$ and +$L_{j,i}\cdot C=0$ for every $j$. Every curve contracted by the +projective extremal contraction $h_i:U_i\to Y_i$ has class on +$R$ when tested by global line bundles, so the same vanishing +holds for all of them. + +On each fixed Stein chart of $Z$, choose the effective real +ordinary representative of the initial relatively ample nef +datum once, before running the program. Its actual real linear +equivalence to the fixed real line-bundle representative persists +under pushforward. The Bott--Chern comparison with the generalized +adjoint modulo the fixed base class persists separately by +exceptional negativity, as detailed below. +Thus every $h_i$ is, locally on $Y_i$, a contraction with an +ordinary real klt log-Fano boundary. On a relatively compact base +chart, rational approximation in the finite affine system of Cartier +adjoint identities gives a rational effective klt boundary with the +same antiample sign, as in Proposition~\ref{prog:detected-flip}. +Lemma~\ref{prog:integral-descent} therefore applies. +Consequently +\[ + M_{j,i}:=(h_i)_*L_{j,i}\ \text{is a line bundle},\qquad + h_i^*M_{j,i}=L_{j,i}. +\] +For a divisorial contraction set $L_{j,i+1}=M_{j,i}$. +For a flip $h_i^+:U_{i+1}\to Y_i$, set +$L_{j,i+1}=(h_i^+)^*M_{j,i}$. +These are actual line bundles with the \emph{same} integers $m_j$. +The descended Bott--Chern class +\[ + \alpha_{Y_i} + =\sum_j\frac{r_j}{m_j}c_1(M_{j,i})+f_{Y_i}^*\Gamma +\] +satisfies $h_i^*\alpha_{Y_i}=\alpha_i$. +It is nef by descent of nefness through the projective surjection +$h_i$; its pullback to the next model is $\alpha_{i+1}$. +This proves that the step is crepant for $\alpha_i$. Hence it is +also a $D_i$-negative step, and the original generalized pair +remains gklt. + +Finally each $M_{j,i}$ is nef over $Z$. For a $Z$-vertical curve +$B\subset Y_i$, projectivity supplies a curve $B'\subset U_i$ +mapping onto it with some positive degree $d_B$. Then +\[ + d_B(M_{j,i}\cdot B)=L_{j,i}\cdot B'\geq0. +\] +No assertion $d_B=1$ is required. Pullback preserves this relative +nefness, so the induction applies on $U_{i+1}$. Its curve degrees +are again integers because the transported objects are line +bundles, not because any curves were lifted with degree one. +This proves the fixed bound through the whole program. +\end{proof} + +\begin{remark}[Scope of the degree grid] +The transported rational classes +\[ + \eta_{j,i}=\frac1{m_j}c_1(L_{j,i})+f_i^*\gamma_j +\] +have degrees in $m_j^{-1}\mathbb Z_{\geq0}$ on curves vertical over +the fixed smooth base $Z$. This relative assertion is exactly what +the initial projective program requires. It does not assert +nonnegativity of these individual classes on all curves of every +birational model, and uses neither rational connectedness of +resolution fibres nor Graber--Harris--Starr. +\end{remark} + +\subsubsection{The negative-part comparison for a nef adjoint} +We write $N_\sigma(\xi)$ for the divisorial negative part of a +pseudo-effective class. We use its homogeneity, subadditivity, +continuity after adding a vanishing K\"ahler perturbation, and the +identities +\[ + N_\sigma(r^*\xi+[E])=N_\sigma(r^*\xi)+E, + \qquad r_*N_\sigma(r^*\xi)=N_\sigma(\xi) +\] +for an effective $r$-exceptional divisor $E$. A nef class has zero +negative part. These are the divisorial Zariski-decomposition +properties of \cite[Section~3]{Boucksom2004} and +\cite[Appendix~A]{DHY}. No identity +$N_\sigma(r^*\xi)=r^*N_\sigma(\xi)$ for arbitrary $\xi$ is assumed. + +\begin{lemma}[The absolute negative part after a relative program] +\label{prog:negative-part-program} +Let $\mu:U\to X$ be a projective resolution, let $\alpha$ be nef on $X$, +let $c>0$, and let $F_U\geq0$ be $\mu$-exceptional. Put +\[ + D_U=c\mu^*\alpha+[F_U]. +\] +Suppose that $\phi:U\dasharrow V$ is a finite $D_U$-negative program, +possibly relative to another base, and denote its transformed class and +divisor by $D_V$ and $F_V=\phi_*F_U$. Then +\[ + N_\sigma(D_V)=F_V. +\] +\end{lemma} + +\begin{proof} +Take a common projective resolution $p:W\to U$, $q:W\to V$. +The negativity comparison for the finite program is +\[ + p^*D_U=q^*D_V+[E], + \qquad E\geq0,\quad E\text{ is }q\text{-exceptional}. +\] +This comparison is valid for a relative negative program: its proof +uses the negativity of the contracted rays and the positive adjoint on +each flipped side, not absolute nefness of the final adjoint. +Since $p^*F_U$ is effective and exceptional over $X$, the exceptional +identity and nefness of $(\mu p)^*\alpha$ give +\[ + N_\sigma(p^*D_U)=p^*F_U + =N_\sigma(q^*D_V)+E. +\] +Pushing forward by $q$ proves the assertion. In particular, the +negative part in this statement is absolute, even when the program +that produced $V$ was relative. +\end{proof} + +\subsubsection{Descent of the effective vertical divisor} + +\begin{lemma}[Vertical numerical triviality] +\label{prog:vertical-divisor} +Let $g:V\to S$ be a projective surjective morphism with connected fibres. +Assume that $S$ is globally Weil $\mathbb{Q}$-factorial. Let $F\geq0$ be +an effective real Cartier divisor on $V$, vertical over $S$, such that +\[ + F\cdot C=0 + \quad\text{for every curve }C\text{ contracted by }g. +\] +Then there is an effective real Cartier divisor $F_S$ on $S$ such that, +as actual divisors, +\[ + F=g^*F_S. +\] +\end{lemma} + +\begin{proof} +For a prime divisor $P\subset S$, write $m_Q$ for the multiplicity of +$Q$ in $g^*P$, where $Q$ ranges over the prime divisors dominating $P$. +These multiplicities may be computed over the smooth generic locus of +$P$, where $P$ is Cartier. Put +\[ + b_P=\min_{g(Q)=P}\frac{\operatorname{coeff}_Q F}{m_Q}, + \qquad F_S=\sum_P b_PP. +\] +Only finitely many $b_P$ are nonzero, since any such $P$ is the image of +a component of $F$. Thus $F_S$ is a genuine effective Weil real divisor. +Global Weil $\mathbb{Q}$-factoriality makes this finite divisor real +Cartier. + +We first check equality over codimension one in $S$. Work near a +general point of $P$ and restrict to a transverse disk. Resolving the +source, and cutting by general relative ample hypersurfaces if the +relative dimension exceeds one, reduces to a projective surface over +that disk with connected fibres. These operations can be performed +over a relatively compact Stein neighbourhood; they do not require +global sections on $S$. The intersection matrix of the components of +the special fibre is negative semidefinite, with kernel generated by +the full fibre with its multiplicities. The restriction of $F$ has +zero intersection with every fibre component. Its coefficients are +therefore proportional to those multiplicities. Equivalently, all the +ratios in the definition of $b_P$ are equal. When $g$ is birational this +assertion over the generic point of $P$ is immediate. + +Consequently +\[ + G=F-g^*F_S +\] +is a signed real Cartier divisor supported over a subset of codimension +at least two in $S$. Moreover, $G$ is numerically trivial over $S$. +We check that such a signed exceptional divisor is zero. The assertion +is local on $S$. Over a relatively compact Stein neighbourhood, take +$d=\dim V-\dim S$ general relative ample hypersurfaces and resolve their +intersection. They give a projective generically finite morphism +$H\to S$. The cuts may be chosen to meet any prescribed component of +$G$ in a nonzero divisorial trace. After normalization and Stein +factorization, $H\to S$ factors as a projective birational morphism +$H\to S'$ followed by a finite morphism $S'\to S$. The restricted +divisor $G|_H$ is exceptional for $H\to S'$ and numerically trivial +over $S'$. Applying the ordinary exceptional negativity lemma to both +$G|_H$ and $-G|_H$ gives $G|_H=0$. The choice of the cuts therefore +excludes every nonzero component of $G$. Hence $G=0$. + +The surface argument above is also the usual proof of the +degenerate-divisor negativity statement: after subtracting the minimum +multiple of the full fibre, an effective residual fibre divisor omits +a component and cannot be numerically trivial. Notice that projectivity +of $g$ was a hypothesis; it was not deduced from factoriality. +\end{proof} + +\subsubsection{The negative part on the lower-dimensional base} + +\begin{lemma}[Detecting the negative part by pullback currents] +\label{prog:negative-part-base} +Let $g:V\to S$ be a surjective morphism. Suppose that $\alpha_S$ is nef, +$F_S\geq0$ is real Cartier, $c>0$, and +\[ + D_S=c\alpha_S+[F_S],\qquad D_V=g^*D_S,\qquad + N_\sigma(D_V)=g^*F_S. +\] +Then $N_\sigma(D_S)=F_S$. +\end{lemma} + +\begin{proof} +Nefness gives $N_\sigma(D_S)\leq F_S$. Fix a prime divisor $P\subset S$ +and a prime divisor $Q\subset V$ dominating it. Write +$m=\operatorname{mult}_Q(g^*P)>0$ and $b=\operatorname{coeff}_P F_S$. +At their generic smooth points, pullback of positive currents satisfies +\[ + \nu(g^*T,Q)=m\nu(T,P). +\] +Indeed, the divisorial term $\nu(T,P)[P]$ pulls back with multiplicity +$m$, and the residual current has zero generic Lelong number there. + +Choose K\"ahler forms $\omega_S,\omega_V$ and a constant $C>0$ for which +$C\omega_V-g^*\omega_S$ is K\"ahler. For $\varepsilon>0$, let $T_\varepsilon$ +be any positive current in the big class $D_S+\varepsilon[\omega_S]$. +Minimality of the multiplicity and monotonicity under adding a nef +class give +\begin{align*} + m\nu(T_\varepsilon,P) + &=\nu(g^*T_\varepsilon,Q)\\ + &\geq\nu(D_V+\varepsilon g^*[\omega_S],Q)\\ + &\geq\nu(D_V+C\varepsilon[\omega_V],Q). +\end{align*} +Taking the infimum over $T_\varepsilon$, and then letting +$\varepsilon\downarrow0$, gives +\[ + m\nu(D_S,P)\geq\nu(D_V,Q) + =\operatorname{coeff}_Q(g^*F_S)=mb. +\] +This proves the reverse inequality for every $P$. +All multiplicities are unchanged by resolving away from the generic +points in question, so the same argument applies to the normal spaces +under consideration. + +The $\varepsilon$ perturbation is necessary: at a pseudoeffective +boundary class, the infimum over its exact positive currents need not +equal its minimal multiplicity. The argument uses that equality only +for the big perturbed classes. +\end{proof} + +\begin{lemma}[Pullback when the positive part is nef] +\label{prog:negative-part-pullback} +Suppose +\[ + \xi=P+[F],\qquad P\text{ nef},\quad F\geq0\text{ real Cartier}, + \qquad N_\sigma(\xi)=F. +\] +For every projective resolution $p:W\to S$, +\[ + N_\sigma(p^*\xi)=p^*F. +\] +\end{lemma} + +\begin{proof} +Write $N'=N_\sigma(p^*\xi)$. Since $p^*P$ is nef, $N'\leq p^*F$. +Birational invariance gives $p_*N'=F$, so +$E=p^*F-N'\geq0$ is $p$-exceptional. The positive part of $p^*\xi$ is +\[ + p^*P+[E], +\] +and is modified nef. If $E\neq0$, exceptional negativity in its +covering-curve form provides a component of $E$ covered by curves $C_t$ +contracted by $p$, with $E\cdot C_t<0$ +\cite[Appendix~A, Lemma~A.3]{DHY}. A modified nef class has nonnegative +intersection with a general member of a family of curves covering a +prime divisor: use currents with arbitrarily small negative part and +zero generic divisorial Lelong number, restrict to a general member, +and let the negative bound tend to zero. But here +\[ + (p^*P+E)\cdot C_t=E\cdot C_t<0, +\] +a contradiction. Thus $E=0$. +\end{proof} + +\begin{proposition}[Any lower-dimensional good model suffices] +\label{prog:nef-good-comparison} +Suppose the initial relative program has the data in +Lemma~\ref{prog:negative-part-program}. Assume in addition that +its class $\alpha_V$ is nef and agrees with $\mu^*\alpha$ on a common +resolution, and that there is an already projective connected-fibre +morphism $g:V\to S$ with +\[ +\alpha_V=g^*\alpha_S,\qquad D_V=g^*D_S. +\] +Suppose there is a projective small modification +$s:S^{\mathrm q}\to S$ such that $S^{\mathrm q}$ is globally Weil +$\mathbb Q$-factorial; the identity is allowed. Assume that the +lower-dimensional adjoint $s^*D_S$ has a good model: +there are a normal compact K\"ahler space $S_m$, a common projective +resolution $p:W\to S^{\mathrm q}$, $q:W\to S_m$, and a nef semiample +class $D_m$ such that +\[ + p^*s^*D_S=q^*D_m+[E], + \qquad E\geq0,\quad E\text{ is }q\text{-exceptional}. +\] +Then $\alpha$ is semiample on $X$. If the contraction defining +semiampleness of $D_m$ is Moishezon, the resulting contraction of +$\alpha$ is Moishezon as well. In particular, it is unnecessary to +lift a program on $S$ to $V$, or to require that a chosen program +producing $S_m$ preserve $\alpha_S$ at every intermediate step. +\end{proposition} + +\begin{proof} +Nefness descends under the surjective morphism $g$, so $\alpha_S$ is nef. +The class of $F_V$ is pulled back from $S$, so $F_V$ is numerically +trivial over $S$. It is therefore vertical: an effective horizontal +component would restrict to a nonzero effective divisor on a general +projective fibre, with positive intersection against a suitable power +of an ample class, contradicting numerical triviality. For a +birational $g$, verticality is automatic by dimension. +First take a projective resolution $\pi:\widetilde V\to V$ of the main +component of $V\times_S S^{\mathrm q}$. The induced map +$\widetilde g:\widetilde V\to S^{\mathrm q}$ is projective and has +connected fibres. Pull back $\alpha_V,D_V,F_V$ along $\pi$, and pull +back $\alpha_S,D_S$ along $s$. All displayed class identities are +preserved, and $\pi^*F_V$ remains an effective vertical divisor. +Lemma~\ref{prog:negative-part-program} gives +$N_\sigma(D_V)=F_V$ before this modification. Because +$D_V=c\alpha_V+[F_V]$ with $\alpha_V$ nef, +Lemma~\ref{prog:negative-part-pullback} then gives +\[ + N_\sigma(\pi^*D_V)=\pi^*F_V. +\] +We may therefore replace $(V,S,g)$ by +$(\widetilde V,S^{\mathrm q},\widetilde g)$ and suppress the new +superscripts in the rest of the proof. The good-model comparison now +reads $p^*D_S=q^*D_m+[E]$. No assertion that the crepant subboundary on +$\widetilde V$ is effective is needed: this space is used only for +classes, currents and the effective divisor $\pi^*F_V$. The +lower-dimensional generalized pair is the crepant pullback to the +small model $S^{\mathrm q}$. + +The class identity $[F_V]=g^*(D_S-c\alpha_S)$ shows that $F_V$ is +numerically trivial over $S$. Lemma~\ref{prog:vertical-divisor} +gives an actual effective real Cartier divisor $F_S$ with +$F_V=g^*F_S$. Injectivity of pullback yields +\[ + D_S=c\alpha_S+[F_S]. +\] +The established identity $N_\sigma(D_V)=F_V$ and +Lemma~\ref{prog:negative-part-base} therefore give +$N_\sigma(D_S)=F_S$, and +Lemma~\ref{prog:negative-part-pullback} gives +\[ + N_\sigma(p^*D_S)=p^*F_S. +\] +On the other hand $q^*D_m$ is nef, so the exceptional-divisor identity +applied to the good-model comparison gives +\[ + N_\sigma(p^*D_S)=E. +\] +Thus $E=p^*F_S$ as actual divisors. Subtracting them from the comparison +proves the exact class equality +\[ + q^*D_m=c\,p^*\alpha_S. +\] +This proves the needed preservation of $\alpha_S$ from the final good +model, without an assumption about its intermediate models. + +Choose a contraction $h:S_m\to T$ to a normal compact K\"ahler space +and a K\"ahler class $\kappa$ on $T$ with $D_m=h^*\kappa$. Taking a +resolution of the main component of $V\times_S W$, and then a common +projective resolution with $U$, produces a projective modification +$r:R\to X$ and a holomorphic map $\ell:R\to T$ satisfying +\[ + r^*\alpha=\frac1c\ell^*\kappa. +\] +For every curve $C$ in a fibre of $r$ this equality implies +$\ell(C)$ is a point, since a K\"ahler class has positive degree on +every nonconstant image curve. The fibres of the projective modification +$r$ are connected projective complex spaces, hence are connected by +chains of curves. Therefore $\ell$ is +constant on each fibre of $r$. The factorization theorem for a proper +map onto a normal space gives a holomorphic map $f:X\to T$ with +$\ell=f\circ r$. Pushing forward the class equality by $r$ gives +\[ + \alpha=f^*(\kappa/c). +\] +The maps from the main fibre-product component to $S_m$ have connected +general fibres; their Stein factorizations are finite birational over +the normal space $S_m$, hence have connected fibres everywhere. +Together with the connected fibres of $h$, this shows that $\ell$, +and therefore $f$, has connected fibres. This is the required +semiampleness of $\alpha$. + +Finally, $R\to S_m$ is projective: it is obtained from the projective +map $g$ by base change, followed by projective resolutions and the +projective map $q$. If $h$ is Moishezon, choose a projective surjection +$H\to S_m$ such that $H\to T$ is projective. A component of +$R\times_{S_m}H$ dominating $R$ is projective and surjective over $X$, +and its map to $T$ is projective. This is a Moishezon witness for $f$. +\end{proof} + + +\begin{lemma}[The contraction step in the dimension induction] +\label{prog:base-point-free-induction} +Assume $\mathsf C_{d-1}$, $\mathsf B_{d-1}$ and $\mathsf M_{d-1}$. +Then $\mathsf C_{d,\mathrm{big}}$ and $\mathsf B_d$ hold. +\end{lemma} +\begin{proof} +\smallskip\noindent +\emph{The big non-klt contraction.} +Apply the construction of \cite[Section~3]{HX}. On its smooth +modification $\nu:U\to X$ it writes +\[ + \nu^*\alpha=[D_U]+\eta_U, + \qquad D_U\geq0,\quad \eta_U\text{ K\"ahler}. +\] +The induction over the jumping coefficients of the multiplier +ideal reduces the new reduced stratum to dimension at most $d-1$. +The given contraction on the old non-klt subspace and +$\mathsf C_{d-1}$ provide the Moishezon maps of those strata. +Every map that is promoted to a projective map in that construction +contracts exactly the curves on which $\nu^*\alpha$ vanishes. +On each such curve the actual real line bundle $-D_U$ has degree +$\eta_U\cdot C$. On a further projective common resolution +$q:U''\to U$, choose an effective exceptional divisor $E$ with +$-E$ relatively ample and choose $\varepsilon>0$ sufficiently small. +Use the decomposition +\[ + q^*\nu^*\alpha=[D'']+\eta'',\qquad + D''=q^*D_U+\varepsilon E,\qquad + \eta''=q^*\eta_U-\varepsilon[E]. +\] +Here $\eta''$ is K\"ahler, and the actual real line bundle $-D''$ +has degree $\eta''\cdot C$ on every contracted curve. Restrict this +decomposition to the smooth reduced components in the gluing +construction. Lemma~\ref{prog:actual-line-polarization} then supplies +precisely the relative ampleness needed in place of +\cite[Lemma~2.42]{HX}, including the common-resolution step in its +Claim~3.5. The exceptional term $-\varepsilon[E]$ makes the +positive part $\eta''$ K\"ahler on the common resolution. + +The exact sequences of multiplier ideals, relative vanishing, +finite pushouts, and extension from sufficiently thickened +$\operatorname{Supp}D_U$ in that proof then apply unchanged. +They give the birational Moishezon contraction in +$\mathsf C_{d,\mathrm{big}}$. In particular the gluing step uses +neither generalized termination nor a projectivity criterion for +an undetected ray. + +\smallskip\noindent +\emph{Preparation for both cases of $\mathsf B_d$.} +The modified-big perturbation +\cite[Lemma~2.23]{HX} supplies a nef datum that dominates a +K\"ahler class on its carrier. Subtracting a sufficiently small +multiple of the pullback of a K\"ahler class $\omega_X$ still leaves +a nef datum on that carrier. Thus +$\alpha$ is a gklt adjoint plus $\varepsilon\omega_X$, and the cone +argument of \cite[Lemma~2.45]{HX} makes $\alpha$ NQC. +This preparation applies whether or not $\alpha$ is big. + +\smallskip\noindent +\emph{The nef-and-big case of $\mathsf B_d$.} +For a gklt pair the multiplier ideal is the unit ideal, so the +preceding big non-klt assertion now applies and gives a birational +contraction $h:X\to Y$. +On a projective resolution $p:W\to X$ chosen projective over $Y$, +relative Kawamata--Viehweg vanishing for the effective exceptional +divisor $-\lfloor B_W\rfloor$ gives +$R^ih_*\mathcal O_X=0$ for $i>0$. Thus $Y$ has rational +singularities. Proper birational Bott--Chern descent +\cite[Lemma~8.7]{DHP} gives $\alpha=h^*\gamma$. +The descended adjoint is gklt, nef and big, and has no trivial +curve. The criterion \cite[Theorem~4.3]{HLX} makes $\gamma$ +K\"ahler. + +For completeness, the projectivity input in that criterion is +restricted to a map $S\to Z'$ between smooth compact K\"ahler +manifolds with rationally connected general fibre: $S$ is the +chosen smooth divisor on a resolution and $Z'$ is the resolution +of the null-locus component. The smooth-source, smooth-base +argument in \cite[Theorem~3.1, Step~1]{CH} applies. Its subsequent +relative program starts with this projective map. Thus this application of the criterion uses only the +smooth-source, smooth-base projectivity argument and an +already-projective relative program. + +\smallskip\noindent +\emph{The non-big case of $\mathsf B_d$.} +Use a projective small factorialization and the modified-big +perturbation of the pair. The non-pseudo-effectivity of $K_X$ +gives an MRC fibration by \cite{Ou2025}. Choose a smooth K\"ahler +MRC base $Z$ and a smooth modification $\mu:U\to X$ such that +$U\to Z$ is projective. Only the smooth case of +\cite[Theorem~3.1, Step~1]{CH} is needed: pullback identifies +the holomorphic two-forms since the general fibre is rationally +connected. Lemma~\ref{prog:relative-hodge} identifies any $(1,1)$-class on $U$, +modulo a class pulled back from $Z$, with an actual real line bundle. +Lemma~\ref{prog:projective-cone} identifies its relative rays +with rays of the analytic cone; both statements persist along the +projective relative construction. + +Write, as in \cite[Theorem~4.1, Claim~4.1]{HX}, +\[ + D_U(a)=K_U+\Delta_U+\mathbf M_U+a\alpha_U + =(a+1)\alpha_U+F_U, + \quad \alpha_U=\mu^*\alpha,\quad + \Delta_U,F_U\geq0, +\] +where $\Delta_U$ is klt and $F_U$ is $\mu$-exceptional. +The modified-big replacement is chosen, as in the cited Claim~4.1, +so that the nef datum $\mathbf M_U$ on this smooth carrier is +K\"ahler. If a further projective resolution is needed, subtract a +sufficiently small effective exceptional divisor from its nef part +and add that divisor to the subboundary; this preserves the +adjoint and the klt inequalities. +Choose $a_0$ using the uniform bound of +Proposition~\ref{prog:relative-nqc-grid}. It makes the relative +$D_U(a_0)$-program $\alpha_U$-trivial and preserves that choice +through every step. This is a +\emph{projective} relative program from its first step: the nef +data are represented by real line bundles modulo $Z$. +Lemma~\ref{prog:relative-hodge} applies on every +intermediate model over this fixed smooth base. +More precisely, put $r:U\to Z$ and choose one actual global real +line bundle $L$ with +\[ + \mathbf M_U+a_0\alpha_U=c_1(L)+r^*\gamma. +\] +The bundle $L$ is relatively ample, because $\mathbf M_U$ is +K\"ahler and $\alpha_U$ is nef. On each of a fixed finite collection +of relatively compact Stein charts of $Z$, choose an effective +real divisor $\Theta$ representing $L$, with $(U,\Delta_U+\Theta)$ +klt. The representatives can retain a positive relatively ample +part in their boundaries. Here the equality +$K_U+\Delta_U+\Theta\sim_{\R}K_U+\Delta_U+L$ is an actual +real linear equivalence of line bundles; the equality with +$D_U(a_0)$ modulo $r^*\gamma$ is separately an equality of +Bott--Chern classes. + +Fix these ordinary pairs on the initial charts. Their actual +linear equivalences and their Bott--Chern comparisons with $D_U(a_0)$ +modulo the fixed base class persist separately throughout a global +relative program. Proposition~\ref{prog:projective-good-model} applies: +the source is smooth, its adjoint is represented by an actual global +real line modulo $Z$, its carrier nef part is relatively ample, and +$D_U(a_0)=(a_0+1)\mu^*\alpha+[F_U]$ is pseudo-effective. +It gives a finite program $U\dasharrow V$ and a good relative endpoint. +This is one global program controlled by fixed ordinary chart pairs. +Its construction uses global line algebras, not a choice of unrelated +local minimal models. Proposition~\ref{prog:relative-nqc-grid} makes +every step $\alpha_U$-trivial with the same value $a_0$. + +Here is an explicit ample-model comparison that gives the required +descent of $\alpha_V$. Both $D_V(a_0)$ and $\alpha_V$ are nef over +$Z$. Put $a_1=a_0+1$ and $a_2=a_0+2$. Before the program starts, +choose fixed ordinary representatives on the finite chart cover for +all three relatively ample nef parts $\mathbf M_U+a_j\alpha_U$, +$j=0,1,2$, on one simultaneous log resolution. The +$\alpha_U$-crepancy of the constructed program makes its endpoint +$V$ a weak model for each of these three fixed ordinary adjoints. +Their traces $D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V$ are nef. +Corollary~\ref{prog:projective-semiample} therefore makes +\[ + D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V +\] +semiample over $Z$. Their zero curves are exactly the common zero +curves of $D_V(a_0)$ and $\alpha_V$. Their projective ample-model +contractions therefore have the same fibres: these fibres are +connected by curves, and each contraction is constant on the +fibres of the other. Identify the two normal targets, writing the +common contraction as $g:V\to S$. If +$D_V(a_j)=g^*\lambda_j$, with $\lambda_j$ relatively K\"ahler over +$Z$, subtraction gives +\[ + \alpha_V=g^*(\lambda_2-\lambda_1)=g^*\alpha_S. +\] +This is an equality of Bott--Chern classes, not merely of curve +degrees. Since the initial program is $\alpha_U$-trivial, it is +also $D_U(a_1)$-negative. From now on put $a=a_1$ and suppress +the argument in $D_U(a)$ and $D_V(a)$. + +The maps $V\to Z$ and $S\to Z$ are projective. Moreover $g$ is +projective: a relatively ample bundle for $V\to Z$ restricts to +an ample bundle on every fibre of $g$, and hence is $g$-ample. +Each $D_U(a_j)$ is not big globally: pushing a big such class to +$X$ would make $(a_j+1)\alpha$ big. If it were big over the smooth +MRC base $Z$, relative-canonical positivity +\cite[Theorem~2.48]{HX}, together with the pseudo-effectivity of +$K_Z$, would make it big globally. It is therefore not big over +$Z$. This property persists to its relative semiample model, so +its relative canonical morphism has $\dim S0$, +with the map to $S$ retained throughout. +\end{lemma} + +\begin{proof} +After the smooth preparation in \cite[Proposition~5.12]{HX}, write the +adjoint as +\[ + A=D+\omega,\qquad D\geq0,\quad \omega\text{ K\"ahler}, + \qquad \operatorname{Supp}D=N_+(A). +\] +Here $N_+(A)$ denotes the reduced divisor with support +$\bigcap_{0<\epsilon\ll1}\operatorname{Supp}N_\sigma(A-\epsilon\omega)$; +this support is constant for sufficiently small positive $\epsilon$ +and is independent of the chosen Kähler class $\omega$ +\cite[Lemma~5.3]{HX}. At a nonterminal scaling step, +\[ + A_i\cdot R_i<0,\qquad + (A_i+t_i\omega_i)\cdot R_i=0,\qquad t_i>0. +\] +Consequently $\omega_i\cdot R_i>0$ and $D_i\cdot R_i<0$. +A component of the actual effective divisor $D_i$ supplies a global +rational Cartier detector. The supporting contraction follows from +$\mathsf B_d$, and its projectivity and flip follow from +Proposition~\ref{prog:detected-flip}. This argument does not require +$\omega_i$ to remain nef. + +To apply $\mathsf B_d$ to a gdlt presentation, lower its finitely many +floor coefficients on the initial smooth carrier and add the same small +class to the K\"ahler nef datum. The adjoint does not change, and the +resulting gklt presentation persists through the same negative steps. +The original gdlt presentation is retained for adjunction to the floor. +Thus the special-termination proof of \cite[Lemma~5.8]{HX} applies with +$\mathsf F_{d-1}$ on its smooth lower-dimensional carrier. Its other +inputs are adjunction, discrepancy monotonicity, and the local +Cartier-index calculation. The last neighbourhood-isomorphism step +uses the evaluation-ideal support argument in +the \hyperref[prog:special-support]{support comparison above}: the comparison of +the first detected flip has support equal to the full inverse image +of its exceptional locus, and later comparisons dominate it. +One does not infer disjointness of images merely from a vanishing +restriction of an arbitrary exceptional divisor. The theta induction of +\cite[Proposition~5.11]{HX} now gives a weak model. Applying +$\mathsf B_d$ to its nef adjoint makes it good, and +Lemma~\ref{prog:weak-to-terminal} gives the log terminal model. +All its steps are detected or belong to the already-projective +relative extraction in that lemma. + +For completeness, the negative-part identity needed in this argument +is only +\[ + N(p^*\xi+E)=N(p^*\xi)+E\quad(E\geq0\text{ $p$-exceptional}), +\] +not the stronger formula $N(p^*\xi)=p^*N(\xi)$ for an arbitrary +pseudo-effective class. The comparison with a nef endpoint identifies +its exceptional error with the negative part. The formula for +$N_+$ follows by writing a K\"ahler form upstairs as +$p^*\omega-F$, with $F$ positive on every exceptional prime, and applying +the displayed identity to +$p^*(\xi-\epsilon\omega)+E+\epsilon F$. Pushforward identifies the +nonexceptional coefficients; $\epsilon F$ supplies all exceptional +primes. These are exactly the uses in the theta argument. + +Now put $P=f^*\omega_S$ and suppose $A+cP$ is big for some $c$. +Fix $\delta>0$ smaller than the $\omega_S$-degree of every compact +integral curve on $S$ and enlarge $c$ so $c>4d/\delta$. +Use the following direct preparation, which keeps the original +unshifted globally nef datum. On a smooth projective carrier +$\nu:W\to X$ write +\[ + \nu^*(A+cP)=[G]+\omega,\qquad G\geq0,\quad\omega\text{ K\"ahler}, + \qquad [K_W+B_W]+M_W=\nu^*A, +\] +with the non-K\"ahler support property used in the big-class +preparation. Here $\operatorname{Ex}(\nu)$ below denotes the reduced +divisor formed by all $\nu$-exceptional primes. Write +$B_W=B_W^+-B_W^-$. For small $\epsilon>0$ put +\[ + \begin{split} + B'&=B_W^++\epsilon B_W^-+ + \epsilon(G+\operatorname{Ex}(\nu)),\\ + M'_W&=M_W+c\nu^*P+\epsilon\omega,\\ + b&=B_W^++\frac{\epsilon}{1+\epsilon}\operatorname{Ex}(\nu). + \end{split} +\] +The new boundary is klt and $M'_W$ is K\"ahler. Direct addition gives +\[ + A'=[K_W+B']+M'_W + =(1+\epsilon)\bigl([K_W+b]+M_W+c\nu^*P\bigr). +\] +Moreover $B'\geq b$. In every theta branch the cumulative added +boundary satisfies $0\leq C\leq1-B'$, so +\[ + b+\frac{C}{1+\epsilon} + \leq b+\frac{1-b}{1+\epsilon}<1. +\] +Thus the unshifted presentation +$[K_W+b+C/(1+\epsilon)]+M_W$ is genuinely gklt with the +\emph{original} globally nef datum. This is the presentation used +for the horizontal length bound. No base form is subtracted from +an arbitrary newly prepared nef datum. + +A horizontal negative ray would have $P$-degree at least $\delta$; +the bound $0<-A_{\rm unshifted}\cdot C\leq4d$ would contradict +$c\delta>4d$. Every constructed step is therefore over $S$. +Its projective connected fibres are contracted by $f$, so $f$ factors +through its base and through the flip. The final finite extraction is +also over $S$. + +The shifted nef endpoint is semiample by $\mathsf B_d$. Its zero face +contains no horizontal curve: decompose such a curve by the unshifted +adjoint's cone theorem; nefness of the shifted class forces all +summands into the zero face, and a summand of positive $P$-degree again +contradicts the length bound. The shifted class is big, so the +connected semiampleness map is bimeromorphic. Its connected fibres are Moishezon and are connected +by chains of compact curves. The preceding zero-face argument makes +the map to $S$ constant on each such curve, hence on every fibre. +The analytic factorization lemma therefore factors the map to $S$ +through the semiampleness map. Subtracting $c\omega_S$ from the +normalized class on its target gives the required relative K\"ahler class for the unshifted adjoint. The effective +resolution error from the preparation is removed by the usual +negativity comparison, giving a model of the original data. + +The forward traces and exceptional splittings persist by +Lemma~\ref{prog:detected-cohomology}. In the last extraction, +the newly extracted prime classes move from the old exceptional span +into the pullback span; they are actual rational Cartier classes. +Both-sign negativity proves directness as before. +\end{proof} + +\begin{lemma}[Changing the base of a relatively trivial adjoint] +\label{prog:graph-good-model} +Suppose $g:X\to Y$ is projective, its general fibre is rationally +connected, and a gklt adjoint satisfies $A_X=g^*A_Y$. Assume the +prepared nef datum is K\"ahler on a smooth carrier. Let +$Y\dashrightarrow Y_m$ be the chosen nonextracting good model +supplied by $\mathsf M_{d-1}$ over $S$, with its exceptional +cohomology splitting, and let $\dim Y2d/\delta$, where every +integral curve on $S$ has $\omega_S$-degree greater than +$\delta>0$. Set +\[ + A_c=A_0+cP=D+N,\qquad D=K_U+B_U,\quad N=M_U+cP. +\] +Since $A_c$ is not big globally, relative-canonical positivity +\cite[Theorem~2.48(2)]{HX}, together with pseudo-effectivity of $K_Z$, +shows that $A_c$ is not big over $Z$, whether or not it is globally +pseudo-effective. The class $N$ is big over $Z$. Consequently $D$ +cannot be pseudo-effective over $Z$: otherwise $D+N=A_c$ would be +big over $Z$. The smooth projective RC Hodge projection represents +every class modulo $Z$ by an actual real line. The ordinary global klt adjoint $D$ is therefore relatively +non-pseudo-effective. The projective ordinary Mori construction of +Corollary~\ref{prog:projective-mori}, with scaling of the actual +relatively ample real line represented by $N$, terminates in a Mori +fibre space. Lemma~\ref{prog:relative-hodge} and +Lemma~\ref{prog:projective-cone} identify every relative ray +with a full analytic ray. Its nef b-data retain the fixed globally +nef carrier $U$. + +First suppose $A_c$ is not pseudo-effective globally. The retained +relative-canonical positivity theorem on $h$, together with +pseudo-effectivity of $K_Z$, then says that $A_c$ is not +pseudo-effective over $Z$. The terminating relative program ends +in a Mori contraction at a threshold $\tau>1$. At every step its +scaled zero ray has $A_c$-degree negative. Inductively the map to +$S$ is retained: if such a ray were horizontal, its $P$-degree +would exceed $\delta$, while the gklt cone theorem for $A_0$ gives +a rational generator of $-A_0$-degree at most $2d$, contradicting +$A_c\cdot R\leq0$. Once $P\cdot R=0$, the step is negative or +crepant for $A_0$, so its gklt presentation persists. The same +argument applies to the final Mori ray. This would be an +$A_c$-negative Mori contraction over $S$, contradicting the +preserved pseudo-effectivity over $S$. Hence $A_c$ is +pseudo-effective globally. + +It is not big globally. Relative-canonical positivity now implies +that it is not big over $Z$. Its pseudo-effective threshold for +$D+tN$ over $Z$ is exactly one: for $t>1$ the class is big, while +pseudo-effectivity for $t<1$ would make $A_c$ big because $N$ is +big over $Z$. The same finite relative program therefore ends in +a Mori contraction $g:U'\to Y$ at threshold one. +All birational steps, including any steps at threshold one, and +the final zero ray are over $S$ by the preceding length argument +with $A_c\cdot R\leq0$. For a zero ray with positive $P$-degree, +$A_0$ would again have degree less than $-c\delta$. +Thus $Y$ carries a map to $S$ and $\dim Y0$ below the +$\omega_c$-degree of every compact integral curve in $X_c$. Fix $\bar\lambda<\delta/(\delta+2d)$ and for +$0<\lambda\leq\bar\lambda$ put +\[ + A_{\lambda,i}=(1-\lambda)H_i+\lambda A_{u,i}. +\] +The same cone estimate shows more than nefness: for a sufficiently +small $\epsilon>0$, uniform on the finite list and the chosen radial +interval, $A_{\lambda,i}-\epsilon H_i$ is nef. Indeed on a +horizontal $A_{u,i}$-negative rational ray its degree is greater than +$(1-\lambda-\epsilon)\delta-2d\lambda>0$; on the nonnegative +part of the cone and on the vertical cone it is nonnegative. +It follows for every class $\xi\in\overline{\mathrm{NA}}(Y_i)$ that +\[ + A_{\lambda,i}\cdot\xi=0 + \quad\Longleftrightarrow\quad + H_i\cdot\xi=A_{u,i}\cdot\xi=0. +\] +In particular its null curves are exactly those contracted by +$\psi_i$. The central data are crepant through these maps over $X_c$; +thus the convex data defining $A_{\lambda,i}$ remain gklt with +globally modified-big boundary. Apply $\mathsf B_d$ to $A_{\lambda,i}$. Its connected +semiampleness map and $\psi_i$ both have Moishezon fibres, and each +connected such fibre is connected by chains of compact curves. +Equality of their contracted curves makes each map constant on the +fibres of the other. The analytic factorization lemma identifies +the two maps. Thus the selected relative canonical model is also +the canonical model over $S$ on this radial interval. +This proves local polyhedrality and the finite local list; compactness +gives the global finite list of canonical models and chosen terminal +models. + +\paragraph{Uniform preparation preserving every weak model.} +\label{prog:uniform-preparation} +Write $A_u=[K_X+B_u+\mathbf M_{u,X}]$ for the adjoints in the +parameter polytope. We first record why a weak model of these gklt data +extracts no divisor. On a common resolution $p:U\to X$, $q:U\to Y$ +put $D=p^*A_u-q^*A_{u,Y}$, using the actual structure-boundary +comparison. The weak-model discrepancy inequalities give $p_*D\geq0$, +and $-D$ is $p$-nef because $A_{u,Y}$ is nef over the fixed base. +Negativity therefore gives $D\geq0$. An extracted prime on $Y$ has +coefficient one in the weak-model boundary, whereas its coefficient in +the structure boundary of the gklt source is less than one; it would +give a negative coefficient of $D$. Hence there is no such prime. +Consequently $D$ is $q$-exceptional and the target is itself gklt. + +Fix a parameter $u_0$. The boundary plus nef part is globally +modified big by hypothesis. The modified-big preparation +\cite[Lemma~2.23]{HX} gives, on a fixed projective smooth carrier +$a:W\to X$, an effective klt boundary $\Gamma$, a K\"ahler class +$\gamma_0$, and an effective $a$-exceptional real divisor $F$ such that +\begin{equation}\label{prog:eq:uniform-preparation} + [K_W+\Gamma]+\gamma_0=a^*A_{u_0}+[F]. +\end{equation} +Here is the effectivity detail in this use of the preparation. Its new +nef datum dominates a K\"ahler form on a carrier. After resolving the +new structure boundary, choose an effective exceptional divisor whose +negative is relatively ample and subtract a sufficiently small multiple +from the pulled-back datum to make it K\"ahler. Add the same multiple +to the structure boundary, +then replace its negative coefficients by zero and, if needed, add +small positive coefficients on the remaining exceptional primes. +All coefficients stay below one. Only exceptional coefficients have +been increased; their increase is the divisor $F\geq0$ in +\eqref{prog:eq:uniform-preparation}. + +On a sufficiently small closed polyhedral neighbourhood of $u_0$ put +\[ + \gamma_u=\gamma_0+a^*(A_u-A_{u_0}). +\] +These classes remain K\"ahler by openness, and the same $\Gamma$ and +the same $F$ satisfy +\[ + [K_W+\Gamma]+\gamma_u=a^*A_u+[F]. +\] +Thus this is one affine family of gklt pairs on one smooth carrier, +with a fixed effective boundary and K\"ahler nef data. + +Let $\psi:X\dashrightarrow Y$ be \emph{any} weak model at a parameter +$u$ in this neighbourhood, and resolve the induced map $W\dashrightarrow +Y$ by $r:T\to W$, $q:T\to Y$. Its original comparison is +\[ + (ar)^*A_u=q^*A_{u,Y}+E_u, + \qquad E_u\geq0\text{ is }q\text{-exceptional}. +\] +Since $\psi$ is nonextracting, every $a$-exceptional prime is exceptional +over $Y$. Hence $r^*F$ is also $q$-exceptional, and +\begin{equation}\label{prog:eq:prepared-weak} + r^*\bigl([K_W+\Gamma]+\gamma_u\bigr) + =q^*A_{u,Y}+E_u+r^*F. +\end{equation} +This proves that $W\dashrightarrow Y$ is a weak model of the prepared +pair. In detail, subtract the effective exceptional error on the right +from its structure boundary on $T$ and push down to $Y$. The resulting +boundary is precisely the transform of $\Gamma$, it is effective, the +adjoint is the already existing Bott--Chern class $A_{u,Y}$, and all +discrepancies are at least those of the prepared gklt pair. This uses generalized data at the level of currents. On $T$ keep +the nef datum $r^*\gamma_u$, and subtract the displayed effective +$q$-exceptional error from the prepared structure boundary. Pushing +forward by $q$ gives the boundary $\Gamma_Y$ and the trace of this +fixed nef b-class. Their sum with the canonical current represents +the already existing locally exact adjoint $A_{u,Y}$; the resolution +identity defines its structure boundary. The generalized discrepancies +have not decreased. This argument neither requires $K_Y+\Gamma_Y$ to +be real Cartier nor asserts that the nef trace is separately a +Bott--Chern class. It therefore applies to every K\"ahler weak target, +including the adjunction strata used in special termination. + +\paragraph{Every weak model occurs in the finite canonical list.} +Choose a small box in $H^{1,1}(W,\mathbb R)$ which spans that vector +space and whose addition to the compact family $\{\gamma_u\}$ stays +inside the K\"ahler cone. For any $Y$ above choose a K\"ahler class +$\eta_Y$ and define $\eta_W=r_*q^*\eta_Y$. Smooth-source cohomology +puts $\eta_W$ in $H^{1,1}(W,\mathbb R)$ and gives an actual +$r$-exceptional real divisor $D_\eta$ with +\[ + r^*\eta_W-q^*\eta_Y=[D_\eta]. +\] +Its negative is $r$-nef, so $D_\eta\geq0$ by exceptional negativity. +As $W\dashrightarrow Y$ extracts no divisors, $D_\eta$ is also +$q$-exceptional. Scale $\eta_Y$ by a sufficiently small positive number +so that $\eta_W$ lies in the fixed box. Adding this identity to +\eqref{prog:eq:prepared-weak} yields an effective +$q$-exceptional comparison with target class $A_{u,Y}+\eta_Y$, which +is K\"ahler over the original base. Thus $W\dashrightarrow Y$ is the +canonical model of a member of this one enlarged prepared polytope. +Its canonical-model list is finite by the first part of the geography +proof. The induced list of maps from $X$ is therefore finite as well. +A finite cover of the original compact parameter polytope completes +the proof for all weak models. + +\end{proof} + + +\begin{theorem}[The restricted package in every dimension] +\label{prog:restricted-package} +The assertions $\mathsf B_d$, $\mathsf C_d$, $\mathsf M_d$ and +$\mathsf F_d$ hold in every finite dimension, for the globally nef +carrier data and globally modified-big boundary traces specified above. +All birational steps used to prove them are detected steps or belong +to an already-projective relative construction with the stated +relative-line algebraicity. A general semiample morphism is asserted +to be Moishezon; projectivity is asserted only when the construction +provides an actual relatively ample line. +\end{theorem} +\begin{proof} +The assertions in dimensions zero and one follow from degrees and +factorization of maps of compact curves. Suppose the package holds +in dimensions less than $d$. Lemma~\ref{prog:base-point-free-induction} +first proves $\mathsf C_{d,\mathrm{big}}$ and then $\mathsf B_d$. +The former uses $\mathsf C_{d-1}$; the nonbig part of the latter +uses only a chosen $\mathsf M_{d-1}$ model and the explicit +negative-part comparison. In particular, no lower program is lifted. + +With $\mathsf B_d$ available, Proposition~\ref{prog:ordinary-step} +constructs detected negative steps in dimension $d$. +Lemma~\ref{prog:big-good-model}, using the already known +$\mathsf F_{d-1}$, gives the big-shift case of $\mathsf M_d$. +Lemma~\ref{prog:relative-model} gives its remaining case, using +$\mathsf M_{d-1}$ and the projective graph construction. This proves +$\mathsf M_d$ over every proper compact K\"ahler base, with its +chosen-model invariant. Lemma~\ref{prog:finite-geography}, whose +hypotheses are \emph{both} $\mathsf B_d$ and $\mathsf M_d$, now +proves $\mathsf F_d$. + +It remains to prove $\mathsf C_d$ when its nef NQC adjoint $\alpha$ +is not big. First use +Lemma~\ref{prog:base-presentation-preparation}, preserving the +adjoint $\alpha$ and making the nef datum K\"ahler on a smooth +carrier. Its multiplier-ideal inclusion allows the given contraction +to restrict to the prepared non-klt closed subspace by +Lemma~\ref{prog:base-emc}. Next take the projective dlt preparation +in \cite[Section~6]{HX}, keeping its crepant structure data. The +working space is klt and globally strongly $\mathbb Q$-factorial. +The retained crepant structure boundary need not be lc; it is +effective, including the extracted primes of coefficient at least one. On a common carrier the nef datum is +nef and big. Consequently the total boundary trace is modified big, +and $\alpha-c_1(K_X)$ is a big Bott--Chern class. The contraction on +the non-klt closed subspace transports through this preparation by +Lemma~\ref{prog:multiplier-transport}. Choose an NQC expression with +positive coefficients and integral multiples of its rational +degree-two summands. Corollary~\ref{prog:uniform-canonical-parameter} +provides a single number $t$ such that each negative ray of +$K_X+t\alpha$ is $\alpha$-trivial. Choose a K\"ahler scaling class +$\omega$ with $K_X+t\alpha+\omega$ nef. Every selected negative +ray is ordinary $K_X$-negative; the actual rational canonical line +is its detector. Proposition~\ref{prog:ordinary-step} supplies its +projective contraction or flip, and the fixed integral grid persists. +The descended $\alpha$ remains nef by projective descent of nefness. + +The driving class is non-pseudo-effective throughout, by the +corollary. Its initial pseudo-effective scaling threshold is positive; +choose a cutoff $\varepsilon>0$ below that threshold. On the fixed +initial carrier the compact segment of globally nef data +\[ + t\alpha+[\varepsilon,1]\omega +\] +has modified-big total boundary. At its scaling parameter each working +model is a marked weak model of a member of this segment. Each selected +step is negative for the fixed driving adjoint, so its discrepancies +do not decrease and increase at a valuation affected by that step. +An infinite sequence would repeat a marked model by $\mathsf F_d$; +the cumulative nonzero effective comparison for the fixed adjoint +would then contradict equality of that marking. Hence this program +terminates. Non-pseudo-effectivity excludes a nef endpoint, leaving +an $\alpha$-trivial ordinary Mori contraction $f:X'\to Z$. + +The structure data of the prepared contraction problem remain +crepant throughout this $\alpha$-trivial program. Their effective +traces and globally nef and big carrier datum therefore persist. +Lemma~\ref{prog:multiplier-transport} transports the contraction on +the non-klt closed subspace through each birational step. +Lemma~\ref{prog:bc-descent} gives +\[ + \alpha_{X'}=f^*\gamma +\] +in Bott--Chern cohomology. The class $\gamma$ is nef by projective +descent; Lemma~\ref{prog:integral-h2} descends the fixed rational +NQC summands, and projective multisections preserve their +nonnegative degrees. + +Apply Corollary~\ref{prog:base-emc-induction} to this projective +Mori contraction. In its notation the prepared non-klt subspace +$N_1$ has Stein space $V(f_*J_1)$ over $Z$. If $N_1$ dominates +$Z$, this Stein space is $Z$ itself and its inherited contraction +already supplies the contraction of $\gamma$. Otherwise the base +presentation has multiplier ideal $J_Z\supset f_*J_1$, so the +contraction restricts to $V(J_Z)$ as a closed subspace of +$V(f_*J_1)$. Since $\dim Z0$, is finite. If the driving adjoint is not +pseudo-effective, the scaling terminates with a Mori fibre space. +\end{corollary} +\begin{proof} +Proposition~\ref{prog:ordinary-step} constructs each step and +Lemma~\ref{prog:detected-cohomology} supplies the forward traces from +the fixed carrier. Each reached model at a scaling parameter is a +marked weak model of that parameter's adjoint. The compact parameter +segment satisfies $\mathsf F_d$, so there are finitely many such +markings. Strict increase for the fixed driving adjoint rules out +repetition. If that adjoint is not pseudo-effective, the positive +pseudo-effective scaling threshold gives a fixed positive cutoff +containing every parameter of a continuing program. The finite +program cannot end nef, hence ends with the claimed Mori map. +\end{proof} diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/04-boundary.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/04-boundary.tex index 0a3a68955c7ea25b5dc62fe3db2001bf48505dcb..64933743a22ff8ef8fc258686390cc5985c6667d 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/04-boundary.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/04-boundary.tex @@ -1,1424 +1,1424 @@ -\section{Generation along the dlt boundary}\label{sec:boundary} - -We prove that the restrictions of a nef adjoint to the separate log -canonical strata fit together to generate its restriction to the whole -reduced boundary. Semiampleness on the normal components does not by itself -supply compatible sections on their union. The proof has three -parts. A Mori contraction reduces the obstruction to restriction from a -stratum to one comparison of residues. Crepant transport respects all -lower residues. Finally a finiteness argument permits us to impose every -comparison simultaneously by taking products of sections. - -Throughout this section, \(n\geq1\) and \(\Ggood_j\) is assumed for every -\(j0\), contrary to the preceding inequalities. -The strict transform is finite birational over the smooth marked curve, -and hence isomorphic to it. Adjunction to that smooth strict transform -thus has nonnegative coefficient. This is the original coefficient by the -surface-slice residue calculation. - -The full-weight pair is dlt and has the asserted floor. On an SNC model, -at the general point of the center of a divisorial valuation \(v\), take -normal parameters \(x_1,\ldots,x_a\), giving an unmarked parameter boundary -weight zero. The SNC discrepancy inequality is -\[ - a(v;\widehat Z,\widehat B_{\widehat Z}) - \geq \sum_{i=1}^a(1-b_i)v(x_i). -\] -Every \(v(x_i)>0\). Thus a zero-discrepancy center is an intersection -of unit components. Those intersections meet the isomorphism locus, -while every exceptional component has coefficient below one. To produce -a dlt resolution for the restricted map, principalize its nonisomorphism -locus and resolve together with the ordered SNC boundary. That locus -contains no entire log canonical center, so the displayed discrepancy -inequality makes every new exceptional coefficient strictly below one. -This is the dlt resolution criterion. It also identifies the unit primes -downstairs with the next incident strata. - -Finally let \(R_{\rm all}\) be the whole unit union on \(\widehat Z\). Sections -on its smooth components which agree on all their SNC intersections glue -by the elementary equalizer for the coordinate ideals of an SNC union. -Even iterated residues make the equalities independent of the chosen chain. -Apply the floor-connectedness lemma to the full-weight pair and this full -unit union. It gives -\((p_Z)_*\mathcal O_{R_{\rm all}}=\mathcal O_{C_Z}\). -Projection formula therefore descends the glued section uniquely to the -restricted invertible line on the reduced \(C_Z\). The proof with \(Z=V\) -gives the assertion for \(S\). -\end{proof} - -Each boundary stratum has dimension less than \(n\). The class of -\(J_Z:=K_Z+B_Z\) is nef, since it is the restriction of the nef adjoint -class. Proposition~\ref{bir:resolution-transfer}, applied using -\(\Ggood_{\dim Z}\) on a log resolution, makes the actual rational line -\(J_Z\) semiample. Increase \(q\) once -for all so that -\(L_Z:=\mathcal O_Z(qJ_Z)=\mathcal O_V(qJ)|_Z\) is generated for every -boundary stratum. Its map and Stein factorization give -\begin{equation}\label{floor:semiample-map} - f_Z:Z\longrightarrow Y_Z,\qquad L_Z=f_Z^*N_Z, -\end{equation} -where \(Y_Z\) is normal projective, \(f_Z\) has connected fibers, and -\(N_Z\) is ample. These are actual line identities. We call \(C_Z\) -\emph{vertical} if \(f_Z(C_Z)\ne Y_Z\); the empty floor is vertical. -If \(C_Z\) dominates \(Y_Z\), then for every \(k\geq1\) the restriction -\[ - H^0(Z,L_Z^k)\longrightarrow H^0(C_Z,L_Z^k|_{C_Z}) -\] -is injective. Indeed, every section comes from \(Y_Z\); if its pullback -vanishes on \(C_Z\), surjectivity makes it vanish at every point of \(Y_Z\). - -Fix a K\"ahler class \(c\) on \(V\), and let \(c_Z\) be its restriction -to each stratum. For boundary strata \(Z,Z'\) of the same dimension, an -\emph{arrow} is a proper bimeromorphic comparison with a projective -resolution whose source is smooth and compact K\"ahler, -\[ - \begin{tikzcd}[column sep=large] - & T\arrow[dl,"p"']\arrow[dr,"p'"]&\\ - Z&&Z' - \end{tikzcd} -\] -such that the two pulled-back adjoint lines are equal as invertible -subsheaves of meromorphic \(q\)-pluricanonical forms on \(T\), and -\begin{equation}\label{floor:class-comparison} - p^*c_Z-(p')^*c_{Z'}\in\NS(T)_{\R}. -\end{equation} -Here \(\NS(T)_{\R}\) is the real span of first Chern classes of -holomorphic lines, with torsion removed. The degree-two formula for a -smooth modification says that its new classes are exceptional divisor -classes. It follows that Equation~\eqref{floor:class-comparison} is independent -of further resolution and is preserved by composition. These arrows -therefore form a groupoid on the finite set of strata of each dimension. -An arrow \(\gamma:Z\dashrightarrow Z'\) transports sections by -\[ - (p')^*(\gamma_*s)=p^*s. -\] -Normality gives existence and uniqueness. Transport commutes with -multiplication and identifies the complete systems and their Stein targets -in Equation~\eqref{floor:semiample-map}. -The equality of meromorphic lines defines this transport of adjoint sections; the class -congruence will control self-arrows when the floor is vertical. - -\subsection{One residue comparison suffices for restriction} - -The only obstruction to extending a section from \(C_Z\) is its possible -variation among the connected components of a general floor fiber. We show -that all these components meet one general Mori fiber, whose floor is -connected or consists of two points. Thus at most one residue comparison -is needed. - -\begin{lemma}[Restriction and a Mori link]\label{floor:restriction} -For each positive-dimensional boundary stratum \(Z\), there is either no -comparison or one arrow between two components of \(C_Z\), allowing a -component to be compared with itself, with the following property. In all -sufficiently large divisible degrees \(k\), a section of \(L_Z^k|_{C_Z}\) -extends to \(Z\) if its two residues agree under that arrow. With no -arrow, every such section extends. The degree bound is uniform over -sections. When present, the arrow compares the two coefficient-one points -on general \(\PP^1\) fibers of a projective Mori contraction on a -bimeromorphic compact K\"ahler model of \(Z\). -\end{lemma} - -\begin{proof} -We use the dimension-free torsion-free restriction lemma -\cite[Lemma~7.1]{BL}. In its notation, for an effective compact K\"ahler -lc pair \((T,G)\), klt away from its reduced floor \(C\), and a connected -map \(f:T\to P\) to a normal compact base from which an actual positive -adjoint multiple is pulled back, it asserts that -\(R^1f_*\mathcal O_T(-C)\) is torsion-free. The ideal sequence gives -\begin{equation}\label{floor:restriction-obstruction} - \mathscr Q:=\coker(\mathcal O_P\longrightarrow f_*\mathcal O_C) - \lhook\joinrel\longrightarrow R^1f_*\mathcal O_T(-C). -\end{equation} -For \(T=Z\) and \(P=Y_Z\), if \(C_Z\) is vertical, \(\mathscr Q\) has -proper support and is zero. Serre vanishing for the kernel of -\(\mathcal O_P\to f_*\mathcal O_C\), tensored with \(N_Z^k\), extends -every restriction in a uniform large-degree tail. If \(C_Z\) dominates, -the same display says that a section extends as soon as its values are -constant on the reduced floor fiber over a dense open of \(Y_Z\): its -class in the torsion-free \(\mathscr Q\otimes N_Z^k\) then vanishes, and -the local lifts glue. They are unique because \(C_Z\) dominates \(Y_Z\), -so \(\mathcal O_{Y_Z}\to(f_Z)_*\mathcal O_{C_Z}\) is injective. -This uses neither reduced special scheme -fibers nor base change at their points. - -Suppose now that \(C_Z\) dominates \(Y_Z\). There is an effective rational -Cartier divisor \(D\) with \(\Supp D=\Supp C_Z\): restrict to \(Z\) the -globally \(\Q\)-Cartier floor primes of \(V\) not containing \(Z\). -For small rational -\(\epsilon>0\), \((Z,B_Z-\epsilon D)\) is effective klt. Apply -Corollary~\ref{prog:small-model} to obtain a projective small strong -\(\Q\)-factorialization \(\rho:Z_0\to Z\), crepant also for the full pair. -Write \(\Delta_0\) for its full boundary and \(D_0=\rho^*D\). -The full adjoint -\[ - P=K_{Z_0}+\Delta_0 -\] -is the actual semiample rational line pulled back from \(Y_Z\), and the -lowered klt adjoint is \(J_{\rm low}=P-\epsilon D_0\). Put -\(d=\dim Z\), choose \(m>0\) with \(mP\) Cartier and generated, and take -\(b>4md\). Consider -\[ - H=J_{\rm low}+bP=(1+b)P-\epsilon D_0. -\] -On a general fiber of the map to \(Y_Z\), its class is -\(-\epsilon\{D_0\}\). This is the negative of a nonzero effective -divisor because the floor dominates \(Y_Z\), so its pairing with a -K\"ahler power is negative. If \(H\) were pseudo-effective, a positive -current representing it would restrict to a positive current on almost -every resolved smooth fiber, contradicting this pairing. Thus \(H\) -is not pseudo-effective. - -Apply Lemma~\ref{prog:nef-trivial} to the lowered klt pair and the nef -rational line \(P\). Its \(H\)-program with K\"ahler scaling has ordinary -\(J_{\rm low}\)-negative steps, is finite, and ends in a projective Mori -contraction. Every step, including the final Mori ray, is \(P\)-trivial. -The same actual Cartier line \(mP\) descends at each birational step. -Consequently its semiample map to the fixed base \(Y_Z\) persists on -every working model. That map is constant on each connected final Mori -fiber and hence factors through the Mori contraction. We obtain -\[ - T\xrightarrow{u}W\xrightarrow{g}Y_Z. -\] -Both maps have connected fibers. The full adjoint on each model is the -trace of \(P\). On a common resolution of a birational step, its natural -meromorphic canonical comparison is an exceptional divisor with zero -class, since the two actual lines descend from the same line on the -contraction base. Exceptional negativity applied to both signs makes -that divisor zero. Thus the full pair remains crepant, including its -meromorphic adjoint identifications. The full transformed pair -\((T,G)\) is effective lc and is klt away from its full floor \(C^+\): -there it agrees with the transformed lowered klt pair. The program is -nonextracting, so the transformed divisor \(D^+\) has -\(\Supp D^+=\Supp C^+\). Since the full adjoint is pulled back from -\(Y_Z\), the lowered adjoint on a \(u\)-fiber is -\(-\epsilon D^+\). Hence \(D^+\) is relatively ample over \(W\). - -Compare the connected components of a general floor fiber on the original -small model and on \(T\). On a common projective log resolution the -full crepant subboundary is the same. Its unit union maps to both floors -with connected fibers by \cite[Lemma~2.1]{BL}. Proper closed surjections -with connected fibers preserve connected components, also after restricting -over a point of \(Y_Z\). Thus the connected components of the two -underlying floor fibers correspond. - -Every connected component of the floor fiber on \(T\) meets one common -general Mori fiber. Indeed \(C^+\to W\) is surjective because \(C^+\) -is the support of the relatively ample \(D^+\). Apply -Equation~\eqref{floor:restriction-obstruction} to \(u\). Surjectivity of -the floor makes \(\mathcal O_W\to u_*\mathcal O_{C^+}\) injective; the -displayed torsion-free cokernel then makes \(u_*\mathcal O_{C^+}\) -torsion-free. Let -\[ - C^+\longrightarrow\overline C\longrightarrow W -\] -be the Stein factorization. The finite space \(\overline C\) is reduced, -and every irreducible component of -\(\overline C\) dominates \(W\). To see the -last implication algebraically, a vertical minimal prime of a finite -reduced algebra over a domain would give, by prime avoidance, a nonzero -element killed by a nonzero element of the domain. After shrinking \(Y_Z\) -to a dense open, the fiber-dimension theorem gives every component of -\(\overline C_y\) dimension \(\dim W-\dim Y_Z\). The fiber \(W_y\) is irreducible of that -dimension: a resolution of \(W\) still has connected fibers over \(Y_Z\), -and a general one is smooth and connected and surjects onto \(W_y\). -Finiteness now makes each component of \(\overline C_y\) dominate \(W_y\). -Consequently each connected component of \(C^+_y\) meets \(u^{-1}(w)\) -for the same general \(w\in W_y\). - -The reduced general fiber \(F\) of \(u\) is irreducible by the same -resolution argument. If \(\dim F\geq2\), the support of the effective -ample Cartier multiple of \(D^+|_F\), whose support is \(C^+\cap F\), -is connected. One elementary -proof, valid even if \(F_{\rm red}\) is not normal, cuts by general -hyperplanes to an integral projective surface. If the ample divisor -split into disjoint nonzero Cartier parts \(D_1,D_2\), their pullbacks -to a resolution would be orthogonal, while the projection formula gives -\(v^*(D_1+D_2)\cdot v^*D_i>0\) for both \(i\). Thus both -\((v^*D_i)^2>0\), contradicting the surface Hodge index theorem. -If \(\dim F=1\), choose the fiber also off the singular, boundary -intersection, and ramification loci. It is a smooth connected curve and -\[ - 0=\deg(K_F+G|_F)=2g(F)-2+\deg(G|_F). -\] -There is a unit point, so \(F\simeq\PP^1\). Its floor has one or two -points; if there are two they exhaust the horizontal boundary. In every -connected case the preceding common-fiber observation makes the general -floor fiber connected, so no comparison is needed. In the remaining case -the two unit points represent all its connected components. - -Normalize the horizontal floor primes and take their Stein factorizations -over \(W\). If there are two primes, each finite Stein map has degree one -and is an isomorphism over normal \(W\); their main fiber product gives -a proper bimeromorphic comparison. If there is one prime, its normal Stein -space is a double cover of \(W\). The exchange on its general fiber -extends over the branch locus: the reduced horizontal non-diagonal -component of the double fiber product has finite birational projections -to the normal cover and hence both projections are isomorphisms. Its graph -lifts to the normalized prime. Composing with the program graphs transports -the comparison to two original normal floor primes, possibly the same one. -All graph projections are projective, and the primes survive birationally -because the program extracts no divisors. - -This comparison is crepant as a meromorphic residue identity. On a smooth -ruled open, order the markings after an \'etale local cover, choose a base -volume \(\xi\), and put the markings at \(z=0,\infty\). A pulled-back -\(q\)-pluriadjoint frame has the form -\[ - a(w)(dz/z)^{\otimes q}\otimes\xi^{\otimes q}. -\] -Its two residues are \(a(w)\xi^{\otimes q}\) and -\((-1)^q a(w)\xi^{\otimes q}\), which agree. This is the two-marking -Poincar\'e-residue comparison of -\cite[Section~3, Definition~13 and Proposition~14]{Kollar2013Sources}. -On a resolved graph both restricted actual lines are pulled back from -\(Y_Z\); their meromorphic embeddings agree on this dense open and hence -everywhere, including along exceptional primes. Equal residues therefore -give a constant value on the entire general floor fiber. The horizontal -consequence of Equation~\eqref{floor:restriction-obstruction} extends the section. -Normality descends it from the small model. Its restriction is the -prescribed section on the original reduced floor, since every original -floor prime is covered birationally and the two restrictions agree after -that pullback. - -It remains to check Equation~\eqref{floor:class-comparison}. Let \(\widehat T\) -be a common smooth resolution of the parent and Mori models and let -\(a,b:H\to\widehat T\) be the two maps from a resolved branch graph. -The resolution of the Mori model is an isomorphism near a general entire -\(\PP^1\) fiber: its centers have codimension at least two, hence cannot -dominate the ruling base. Every holomorphic two-form on \(\widehat T\) -comes from the base on that ruled open. In fact its vertical one-form -terms vanish on \(\PP^1\), and its remaining coefficients are constant -on that compact fiber. Thus \(a^*-b^*\) kills \(H^{2,0}\), and also -\(H^{0,2}\) by conjugation. This rational Hodge map on degree two has -image of type \((1,1)\). The Lefschetz \((1,1)\) theorem puts its real -image in \(\NS(H)_{\R}\). Apply this to the pulled-back parent class -\(c_Z\). The link is therefore an arrow as defined above. -\end{proof} - -\subsection{Transport respects every lower residue} - -An arrow can contract a divisor to a higher-codimension stratum. The next -local observation explains how to compare residues even in that case. -All valuations in its proof are local monomial valuations in an SNC chart. - -\begin{lemma}[Unit strata above an SNC stratum]\label{floor:unit-strata} -Let \(p:T\to Z\) be a projective log resolution of a dlt pair, with -crepant SNC subboundary on the smooth \(T\). Let \(R\subseteq T\) be a -unit stratum and let \(D=p(R)\) be its center. Then -\(D\) is a dlt stratum. There is a unit stratum \(R'\subseteq R\) such -that \(p(R')=D\) and \(R'\to D\) is birational. In the iterated residue -comparison along \(R'\), the normal logarithmic Jacobian is \(\pm1\). -\end{lemma} - -\begin{proof} -The center is a dlt stratum by the SNC discrepancy calculation in the -proof of Lemma~\ref{floor:adjunction}. Work over the generic SNC part of \(D\), -of codimension \(a\), with normal coordinates \(x_1,\ldots,x_a\) for -the unit divisors. The zero-discrepancy divisorial valuations centered -there are exactly the primitive rational monomial rays in -\(\R_{>0}^a\). Here is a local verification that also fixes their -normalization. The discrepancy inequality in the proof of -Lemma~\ref{floor:adjunction} says on every log-smooth model that the center -of a zero-discrepancy valuation is exactly a unit stratum: an additional -unmarked normal parameter, or a parameter with coefficient below one, would -give positive discrepancy. Blow up that stratum. If the positive normal -orders of the valuation are \(w_1,\ldots,w_b\), the chart indexed by a -smallest order \(w_k\) replaces these by \(w_k\) and the positive members -of \(w_i-w_k\). The exceptional divisor is again unit, and the center -is exactly the new unit stratum. While there is more than one positive -order their sum decreases. Eventually the center is the generic point of -one unit divisor, where a normalized divisorial valuation has order one. -Reversing these ordinary toric blowups proves monomiality and primitive -normalization. Conversely this subtraction algorithm for any primitive -positive integer vector terminates at order one, and reversing it extracts -that monomial valuation. Comparison on a common graph gives uniqueness -of the valuation with those weights and of its center on any model. - -For a unit component \(E\) meeting the inverse image of the generic SNC -part of \(D\), put -\[ - v_E=(\ord_E x_1,\ldots,\ord_E x_a)\in\Z_{\geq0}^a. -\] -If the unit components through a unit stratum \(Q\) meeting that inverse image are -\(E_1,\ldots,E_b\), define -\[ - \phi_Q:\R_{\geq0}^b\longrightarrow\R_{\geq0}^a,\qquad - (t_1,\ldots,t_b)\longmapsto\sum_{j=1}^b t_jv_{E_j}, - \qquad \sigma_Q=\phi_Q(\R_{\geq0}^b). -\] -The source carries its standard lattice \(\Z^b\). For the fixed -resolution \(T\), these cones subdivide the orthant over its interior. -On rational rays, \(\phi_Q\) sends a -monomial valuation on \(T\) to the same normalized valuation on \(Z\), -so it is injective on each cone. Interiors of different cones are -disjoint, because the center and the weights of a monomial valuation on -\(T\) are unique. Every rational ray in the orthant occurs: view its -valuation on \(T\) and use the same SNC description there. There are -only finitely many cones over the generic part of \(D\); density of -rational rays therefore gives coverage. Taking subsets of unit components -gives the faces, and uniqueness of valuations also on faces identifies -intersections as common faces. Consequently \(\sigma_R\) is a face -of an \(a\)-dimensional cone \(\sigma_{R'}\) with the same downstairs -center, for a unit stratum \(R'\subseteq R\). - -Let \(E_1,\ldots,E_a\) be the unit components through \(R'\). -The integer matrix \(A=(\ord_{E_j}x_i)\) of this full cone is unimodular. -Indeed every positive -primitive integral vector in the source cone is the normalized vector of -a divisorial valuation, so its image is primitive. If a prime divided -\(\det A\), a nonzero kernel vector modulo that prime, lifted to a -positive primitive integral vector, would have nonprimitive image. Hence -\(\det A=\pm1\). The strata \(R'\) and \(D\) have the same dimension. -We next prove that their generically finite map has degree one. In local coordinates at a -general point of \(R'\), -\[ - x_i=u_i(y,z)\prod_{j=1}^a y_j^{A_{ij}}, -\] -where the \(u_i\) are units, the \(y_j\) are normal coordinates, and -\(z\) are coordinates along the stratum. The tangential map is -generically \'etale in characteristic zero. Since \(A^{-1}\) is integral, -replace \(y_j\) by -\(\widetilde y_j=y_j\prod_i u_i^{(A^{-1})_{ji}}\). Then -\(x_i=\prod_j\widetilde y_j^{A_{ij}}\), and the map consisting of -\(\widetilde y\) and the downstairs tangential coordinates is locally -biholomorphic at a general \'etale point of \(R'\to D\). Two distinct -points over a general point of \(D\) would therefore give two lifts of -nearby target points whose normals approach with any fixed positive rate -vector in the interior of this cone. Varying the nonzero leading -coefficients fills an open set of nearby torus points, so such a target can -be chosen in the locus where \(p\) is an isomorphism. This contradiction -proves that \(R'\to D\) is birational. Finally, after taking residue -along \(R'\), the normal coefficient of -\(\bigwedge_i dx_i/x_i\) is \(\det A\) times that of -\(\bigwedge_jdy_j/y_j\); terms differentiating the units vanish in that -residue. Since \(\det A=\pm1\), the even pluriresidue removes the sign. -\end{proof} - -\begin{lemma}[Preservation of lower restrictions]\label{floor:transport} -Fix \(0\leq d0\), choose a common log resolution -\(p:T\to Z\), \(p':T\to Z'\) of the arrow. For a target floor prime -\(D'\), start with its strict transform \(R\). It is a unit stratum and -maps birationally to \(D'\). Its source center \(p(R)\) is a stratum by -Lemma~\ref{floor:unit-strata}. If \(R\to p(R)\) is not birational, that -lemma replaces \(R\) by a unit stratum inside it which maps birationally -onto the same source center. The target center may then shrink. If the -map to that center is not birational, apply the lemma on the target side, -and continue alternately. Each necessary replacement strictly decreases -dimension. The process therefore ends with one unit stratum mapping -birationally to lower strata \(D_0,D_0'\) on the two sides. - -The image in the common system base remains unchanged throughout. Indeed -the two maps from the resolved graph to the Stein targets are identified -by the arrow, and a replacement keeps its entire image on the side then -being treated. Initially the image was \(f_{Z'}(D')\). The terminal -comparison of \(D_0\) and \(D_0'\) is an arrow: restricting the actual -meromorphic identification to iterated residues gives crepancy, with the -logarithmic determinant sign removed by Lemma~\ref{floor:unit-strata}; -restricting the line Chern classes in Equation~\eqref{floor:class-comparison} -gives the same condition modulo \(\NS_{\R}\). Lower invariance thus -identifies the transported and assigned residues on \(D_0'\). - -This equality determines the entire section on \(D'\). The Stein map -of \(L_{D'}=L_{Z'}|_{D'}\) is the Stein factor of -\(f_{Z'}|_{D'}\), because the line induced on that Stein space is ample; -its target is finite over \(f_{Z'}(D')\). The image of \(D_0'\) in this -target is a closed irreducible subset of full dimension, since it still -maps onto \(f_{Z'}(D')\), and therefore is the whole target. Both -sections on \(D'\) come from it. Equality after restriction to -\(D_0'\) proves equality on \(D'\). - -The same construction shows that the image of every target floor prime -in the system base is the image of a lower source stratum, and the reverse -statement follows from the inverse arrow. A floor dominates the system -base on one side exactly when it does on the other. This proves the last -assertion. -\end{proof} - -\subsection{Finite actions on vertical strata} - -Once lower-dimensional sections are compatible and invariant, -Lemma~\ref{floor:transport} preserves their restrictions under every arrow. -If \(C_Z\) dominates \(Y_Z\), injectivity of restriction forces any such -transported extension to equal the assigned extension. Thus only vertical -strata require the following argument for finite image. - -\begin{proposition}[Finite image of self-arrows]\label{floor:finite-action} -Let \(Z\) be a boundary stratum with \(C_Z\) vertical. For all sufficiently -large divisible \(k\), the self-arrows of \(Z\) have finite image on -\(H^0(Z,L_Z^k)\). -\end{proposition} - -For \(m=qk\), a pluricanonical eigenform \(s\in H^0(Z,L_Z^k)\) with -\(\int_Z|s|^{2/m}<\infty\) becomes the \(m\)-th power of a holomorphic top -form on a smooth resolution of a cyclic root cover, so the auxiliary proposition below makes its -eigenvalue a root of unity. A uniform bound on the middle Betti numbers of -selected resolutions of these covers in the fixed degree will then bound -the possible orders; boundedness of the representation will give finite -image. The auxiliary proposition uses the polarization congruence in -Equation~\eqref{floor:class-comparison}. - -\subsubsection{Top-form scalars} - -\begin{proposition}[A polarized top-form scalar]\label{floor:top-form-scalar} -Let \(U\) be a smooth connected compact K\"ahler manifold of dimension -\(d0\) sufficiently divisible that the connected semiample map -and its Stein target satisfy -\[ - f:U'\longrightarrow Y,\qquad - \mathcal O_{U'}(\ell P)=f^*N, -\] -where \(Y\) is normal projective and \(N\) is ample, and that -Lemma~\ref{bir:sections} identifies every graded piece in -\[ - R_\ell(U):=\bigoplus_{j\geq0}H^0(U,j\ell K_U) - \simeq \bigoplus_{j\geq0}H^0(Y,N^j). -\] -This is the full section ring of the ample line \(N\), so -\(Y\simeq\Proj R_\ell(U)\). Passing to a further divisible Veronese gives -the same Proj. We call \(Y\) the \emph{canonical Iitaka base}. A -bimeromorphic self-map of \(U\) acts on this graded ring and therefore -induces an automorphism of \(Y\). - -\begin{lemma}[Finite action on the canonical Iitaka base]\label{floor:finite-base} -Let \(U\) be a smooth connected compact K\"ahler manifold of dimension -\(d0\) in the interior, since \(\chi_R\) diverges at the boundary -and \(\chi\) is bounded on the closed disk. At that point, -\[ - 0\ge\partial_z\partial_{\bar z}\log(\chi/\chi_R) - \ge\frac{\kappa}{2}(\chi-\chi_R)>0, -\] -a contradiction. Thus the Ahlfors--Schwarz comparison gives -\(\chi(0)\le4/(\kappa R^2)\) \cite{Ahlfors1938}. -Letting \(R\to\infty\) forces the metric to vanish -on \(\C\). The argument is local and allows zeros of the -induced metric, so the local representations just constructed suffice. -Consequently their period maps are locally constant along each such orbit. -In any holomorphic volume frame the logarithm of the coefficient in -Equation~\eqref{floor:hodge-density} is harmonic along each orbit: the top Hodge line is -flat there, and \(\sigma\) is a nonzero holomorphic multiple of a flat -generator. - -This is incompatible with finite mass. Diagonalize the one-parameter -action in projective coordinates, and choose two coordinates nonzero at a -general point with different weights, of difference \(a\ne0\). A small -smooth transverse slice \(Q\) on which their ratio is one gives a -holomorphic local biholomorphism onto its image -\[ - \Phi:\C^*\times Q\longrightarrow Y^* -\] -with fibers of size at most \(|a|\): the ratio at \(\Phi(z,t)\) is -\(z^a\). The entire \(\C^*\) factor is allowed because \(Y^*\) is -invariant. In the product frame \((dz/z)\wedge dt_1\wedge\cdots\wedge -dt_{\dim Q}\), write \(\Phi^*\nu\) with coefficient \(\rho(z,t)>0\). -For fixed \(t\), \(\log\rho(z,t)\) is harmonic on \(\C^*\). Its -circular mean at \(|z|=e^u\) is \(A(t)u+B(t)\). Jensen's inequality -therefore gives, with an irrelevant positive normalization, -\[ - \int_{\C^*}\rho(z,t)\frac{i\,dz\wedge d\overline z}{|z|^2} - \ \geq\ c\int_{\R}e^{A(t)u+B(t)}\,du=\infty. -\] -Tonelli's theorem contradicts the finite mass of \(\Phi^*\nu\), which -is at most \(|a|\nu(Y)\) by the bounded covering degree. This proves -that \(h\) has finite order. -\end{proof} - -After a power, the scalar problem can thus be restricted to a fiber with -Kodaira dimension zero. The polarization congruence then passes to that -fiber; the next two lattice arguments control its nonprojective factors. -In the next lemma, the hypothesis -on the origin of the model is what provides the degree-two exceptional -splitting; rational singularities alone are not used for that splitting. - -\begin{lemma}[The scalar on a symplectic factor]\label{floor:symplectic-scalar} -Let \(Y\) be a terminal compact K\"ahler primitive symplectic space of -dimension \(2e0\). Since \(\beta\) -is nef over \(Y\), exceptional negativity applied to the divisor \(E\), -with \(\{E\}=p^*c-[\beta]\), gives \(E=\sum_i e_iE_i\geq0\). Pair with the positive -normalization \(W_R=(p^*\eta\,p^*\overline\eta)^{e-1}\). Exceptional -classes pair to zero with \(p^*c\) by projection formula, and hence for -some constant \(b_e>0\) -\begin{equation}\label{floor:positive-bb} - q_Y(c)=b_e\left(\int_R\beta^2 W_R+\sum_i e_i\int_{E_i}\beta W_R\right)>0. -\end{equation} -Every term is nonnegative. The first is strictly positive on the common -open where \(\beta\) is positive and the symplectic form is nondegenerate. - -Set \(H_\Q=H^2(Y,\Q)\) and -\(D_\Q=H_\Q\cap H^{1,1}(Y,\C)\). Lefschetz \((1,1)\) on \(R\), together -with the line-trace assertion of Lemma~\ref{prog:cohomology-transport}, -identifies \(D_\Q\) with the rational span of line Chern classes. -It is \(G\)-invariant because \(G\) is a rational Hodge automorphism. -The same line-trace assertion applied to the graph congruence gives -\(Gc-c\in D_\R\). The line \(\C\eta\) survives in \(H_\C/D_\C\); -our immediate aim is to prove that all eigenvalues on this quotient have -absolute value one. Write \(\Sigma=(H^{2,0}\oplus H^{0,2})_\R\), the -positive real two-plane. There are three possibilities for the restriction -of the Lorentz form to \(D_\R\). - -If \(D_\R\) is negative definite, its orthogonal projection \(c_0\) of -\(c\) to \(D_\R^\perp\) is fixed by \(G\) and has positive square by -Equation~\eqref{floor:positive-bb}. On \(D_\R^\perp\), the invariant space -\(\Sigma\oplus\R c_0\) is positive definite and its complement is -negative definite. All eigenvalues there have absolute value one. If -\(D_\R\) contains a positive vector, it is nondegenerate with one -positive direction. Its orthogonal complement is the positive plane -\(\Sigma\) plus a negative-definite space, giving the same conclusion. -In either case \(H_\R/D_\R\simeq D_\R^\perp\). - -In the remaining case \(D_\R\) is negative semidefinite with radical a -rational isotropic line \(\ell\). The lattice action on \(\ell\) is -\(\pm1\). On \(\ell^\perp/\ell\) the form is the positive plane -\(\Sigma\) plus a negative-definite \((1,1)\) space, and the quotient -\(H_\R/\ell^\perp\) is dual to \(\ell\). The invariant flag -\(0\subset\ell\subset\ell^\perp\subset H_\R\) therefore shows that -all eigenvalues on \(H_\R\) have absolute value one; this argument also -allows unipotent parts. In all cases the eigenvalues on -\(H_\C/D_\C\) have absolute value one. The rational quotient -\(H_\Q/D_\Q\) carries the preserved lattice -\[ - \overline\Lambda_Y=\Lambda_Y/(\Lambda_Y\cap D_\Q), -\] -where the intersection is saturated. Thus -\(\lambda\) is an algebraic integer and all its algebraic conjugates have -absolute value one. Kronecker's theorem makes it a root of unity. -\end{proof} - -\begin{lemma}[The scalar on a torus]\label{floor:torus-scalar} -Let \(A\) be the integral Hodge automorphism of \(H^1(T,\Z)\) induced -by an automorphism of a compact complex torus \(T\). Suppose a positive -Hermitian class \(c\in H^{1,1}(T,\R)\) satisfies -\((\bigwedge^2 A)c-c\in\NS(T)_\R\). Then -\(\det(A|H^{1,0}(T))\) is a root of unity. -\end{lemma} - -\begin{proof} -Put \(V=H^1(T,\Q)\) and \(D_\Q=\NS(T)_\Q\). For each eigenvalue -\(\theta\) of \(A_\C\), let \(V_\theta\) be its generalized eigenspace and set -\(d_\theta=\dim V_\theta\), \(p_\theta=\dim(V_\theta\cap H^{1,0}(T))\). -Complex conjugation gives -\(p_\theta+p_{\overline\theta}=d_\theta=d_{\overline\theta}\). - -For \(|\theta|\ne1\), project \(c\) to the block -\(V_\theta\wedge V_{\overline\theta}\), with \(\bigwedge^2V_\theta\) understood for -real \(\theta\). This projection preserves \(D_{\overline\Q}\). Indeed -the Chinese-remainder projectors onto the generalized spaces are -polynomials in \(A\) with algebraic coefficients. Apply them to the -two tensor slots separately and then alternate. The resulting maps are -algebraic linear combinations of rational Hodge maps on \(\bigwedge^2V\); -these preserve rational \((1,1)\) classes, hence \(D_\Q\) by Lefschetz -\((1,1)\). Separate slot projections avoid any collision of products of -eigenvalues in degree two. - -On this block \(A_2=\bigwedge^2 A\) has the single generalized -eigenvalue \(\theta\overline\theta=|\theta|^2\ne1\). The inverse of \(A_2-1\) -there is a finite polynomial in its nilpotent part. Project the congruence -for \(c\) to the block and apply this inverse. It follows that its block -\(c_{\theta,\overline\theta}\) belongs to \(D_\C\). Positivity says that this -block is a perfect tensor between \(V_\theta\) and \(V_{\overline\theta}\), -pairing their opposite Hodge types: in a basis of \(H^{1,0}\) the relevant -matrices are positive-definite principal Hermitian blocks. For real -\(\theta\) the corresponding alternating tensor is nondegenerate. - -The intersection of this block with \(D_\C\) is defined over -\(\overline\Q\). Nondegeneracy is the nonvanishing of a determinant, -so it contains a nondegenerate tensor with algebraic coefficients. For -any Galois automorphism \(\gamma\), the conjugate tensor remains in -\(D_{\overline\Q}\), hence is still of type \((1,1)\) for the given -Hodge decomposition, and is perfect between \(V_{\gamma(\theta)}\) and -\(V_{\gamma(\overline\theta)}\). Opposite-type perfection yields -\begin{equation}\label{floor:torus-hodge-count} - p_{\gamma(\theta)}+p_{\gamma(\overline\theta)} - =d_\theta\qquad(|\theta|\ne1). -\end{equation} -This does not replace \(\gamma(\overline\theta)\) by -\(\overline{\gamma(\theta)}\), and assumes no Galois invariance of the Hodge -decomposition. - -The top-form scalar is the algebraic integer -\(z=\prod_\theta \theta^{p_\theta}=\det(A|H^{1,0})\), also when generalized eigenspaces -are nontrivial. For any Galois automorphism \(\delta\), reindex by the -image roots and pair complex conjugates: -\[ - \begin{aligned} - \log|\delta z| - &=\frac12\sum_\theta - \bigl(p_{\delta^{-1}(\theta)} - +p_{\delta^{-1}(\overline\theta)}\bigr)\log|\theta|\\ - &=\frac12\sum_\theta d_\theta\log|\theta| - =\tfrac12\log|\det A|=0. - \end{aligned} -\] -For \(|\theta|\ne1\) the second equality uses -Equation~\eqref{floor:torus-hodge-count} with \(\gamma=\delta^{-1}\); the other -terms vanish. The last equality is unimodularity of the integral action. -Kronecker's theorem now proves the assertion. -\end{proof} - -\begin{proof}[Proof of Proposition~\ref{floor:top-form-scalar}] -Apply Lemma~\ref{floor:finite-base}. After a power, \(g\) acts trivially -on the canonical Iitaka base. Use the modification \(\pi:U'\to U\) and -the map \(f:U'\to Y\) in its construction, and restrict over a general -point where the graph and the fibration are smooth and the fiber meets the -isomorphism locus of \(\pi\). Dividing \(\alpha\) by a local base volume -gives a nonzero top form on the smooth compact fiber \(F\), which has -\(\kappa(F)=0\); its returning map scales the form by the corresponding -power of \(\mu\). A point fiber gives scalar one. For a positive-dimensional -fiber, let \(\omega_U\) be a K\"ahler form representing the class in the -proposition and put \(\beta_F=(\pi|_F)^*\omega_U\). This is a smooth -semipositive form, strictly positive on the dense open where \(\pi\) is -an isomorphism. Pulling the original graph congruence to \(U'\) and -restricting to the fiber shows that the two pullbacks of \([\beta_F]\) on -a common resolved graph of the returning map of \(F\) differ by real line classes. -This weaker positivity suffices below. - -Since \(\dim F\leq d0\) and a compatible invariant generating system -of tuples through dimension \(n-1\). -\end{proposition} - -\begin{proof} -At the point strata use the same scalar in the canonical zero-form -generator \(1\). The even residue convention identifies these generators -along every path and under every point arrow. This gives the required -system in dimension zero; if there are no points the empty system is -understood. - -Suppose the system has been constructed through dimension \(d-1\). -Replace it by powers and their span in a common degree \(k\) chosen so that -Lemma~\ref{floor:restriction} applies on -every \(d\)-stratum, Proposition~\ref{floor:finite-action} applies on every -vertical \(d\)-stratum, and -\(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) is generated for every -\(d\)-stratum \(Z\). These conditions hold in a common divisible tail: -there are finitely many strata, and the last condition is Serre's theorem. -For a \(d\)-stratum \(Z\), Lemma~\ref{floor:adjunction} glues -the lower tuple to a section on its whole \(C_Z\). Its residues satisfy -the one link in Lemma~\ref{floor:restriction}, if that link is present, -because the lower tuple is invariant. That lemma extends the section to -\(Z\). - -Let \(\mathcal P_d(k)\) be the vector space of all tuples through -dimension \(d\) whose lower part is in \(\mathcal V_{d-1}(k)\) and whose -new components restrict to that lower part. Call its elements -\emph{pre-tuples}. Its projection to -\(\mathcal V_{d-1}(k)\) is surjective, since the finitely many extensions -can be chosen independently. This system already generates. To check a -point \(z\in Z\), put \(y=f_Z(z)\). If \(y\in f_Z(C_Z)\), choose a -floor point over \(y\) and a lower tuple nonzero there. Every extension is -the pullback of a section of \(N_Z^k\), so it is nonzero at every point -over \(y\), including \(z\). If \(y\notin f_Z(C_Z)\), the chosen generation -of \(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) supplies a base section nonzero -at \(y\). It pulls back to a section -vanishing on \(C_Z\); combine it with the zero lower tuple and zero -components on the other new strata. This also treats an empty floor. -Surjectivity to the lower system preserves generation at lower points. - -We impose invariance in dimension \(d\). For a dominating floor, -injectivity of restriction and Lemma~\ref{floor:transport} already make -every pre-tuple invariant on that stratum. -Verticality is invariant under arrows, so the remaining orbits consist -entirely of vertical strata. - -For each such orbit choose a representative \(Z\), and let \(G_Z\) be -the finite isotropy image on \(H^0(Z,L_Z^k)\) given by -Proposition~\ref{floor:finite-action}. For a pre-tuple with component -\(s_Z\), form the norm -\[ - P_Z(s_Z)=\prod_{g\in G_Z}g(s_Z) - \in H^0(Z,L_Z^{k|G_Z|}). -\] -Choose a common positive integer \(a\) divisible by all \(|G_Z|\), use -\(P_Z(s_Z)^{a/|G_Z|}\), and transport it to each member of that orbit. -This is independent of the chosen arrow: changing that arrow by an -isotropy arrow merely permutes the factors in degree \(k\). No finiteness -assertion in degree \(ak\) is needed. On strata with dominating floor -use \(s_Z^a\), and on all lower strata use the componentwise \(a\)-th -power of the assigned tuple. - -The resulting tuple is compatible. By Lemma~\ref{floor:transport}, -every transported factor in a norm has exactly the prescribed lower -restriction. The norm and its indicated power therefore restrict to -the \(a\)-th power of that restriction, the same value used on all lower -strata and in all other orbits. It is invariant in dimension \(d\) by -construction, and remains invariant below it. - -These tuples still generate. Fix a point \(z\) in an orbit member. -For each of the finitely many transported factors, choose a point over -\(z\) on a resolution of its comparison graph. Equality of the -pulled-back invertible adjoint lines identifies the fiber values. The -factor is nonzero at \(z\) exactly when the representative component -\(s_Z\) is nonzero at the corresponding point of \(Z\). Evaluation at each -such point is a nonzero linear functional on the generating space -\(\mathcal P_d(k)\). -A finite union of their proper kernels cannot cover a complex vector -space. One pre-tuple makes all factors nonzero and gives a norm nonzero -at \(z\). At a lower point a nonzero assigned value stays nonzero after -its \(a\)-th power. Taking the linear span of all the constructed tuples -therefore gives a compatible invariant generating system in degree \(ak\). -This completes the induction on \(d\). -\end{proof} - -\begin{proof}[Proof of Theorem~\ref{floor:generation}] -Take the system in Proposition~\ref{floor:tuples}. Its components on the -prime components of \(S\) agree on every common lower stratum, so -Lemma~\ref{floor:adjunction} descends each tuple uniquely to a section of -the actual line \(\mathcal O_V(qkJ)|_S\). At any \(x\in S\), choose a -prime component through \(x\) and a tuple nonzero there. It is the pullback -of the descended value in the same invertible line fiber at \(x\), so -that value is nonzero. The finite-dimensional span of these descended -sections generates at every point of the reduced \(S\). This is the -claimed global generation of an actual Cartier multiple. -\end{proof} +\section{Generation along the dlt boundary}\label{sec:boundary} + +We prove that the restrictions of a nef adjoint to the separate log +canonical strata fit together to generate its restriction to the whole +reduced boundary. Semiampleness on the normal components does not by itself +supply compatible sections on their union. The proof has three +parts. A Mori contraction reduces the obstruction to restriction from a +stratum to one comparison of residues. Crepant transport respects all +lower residues. Finally a finiteness argument permits us to impose every +comparison simultaneously by taking products of sections. + +Throughout this section, \(n\geq1\) and \(\Ggood_j\) is assumed for every +\(j0\), contrary to the preceding inequalities. +The strict transform is finite birational over the smooth marked curve, +and hence isomorphic to it. Adjunction to that smooth strict transform +thus has nonnegative coefficient. This is the original coefficient by the +surface-slice residue calculation. + +The full-weight pair is dlt and has the asserted floor. On an SNC model, +at the general point of the center of a divisorial valuation \(v\), take +normal parameters \(x_1,\ldots,x_a\), giving an unmarked parameter boundary +weight zero. The SNC discrepancy inequality is +\[ + a(v;\widehat Z,\widehat B_{\widehat Z}) + \geq \sum_{i=1}^a(1-b_i)v(x_i). +\] +Every \(v(x_i)>0\). Thus a zero-discrepancy center is an intersection +of unit components. Those intersections meet the isomorphism locus, +while every exceptional component has coefficient below one. To produce +a dlt resolution for the restricted map, principalize its nonisomorphism +locus and resolve together with the ordered SNC boundary. That locus +contains no entire log canonical center, so the displayed discrepancy +inequality makes every new exceptional coefficient strictly below one. +This is the dlt resolution criterion. It also identifies the unit primes +downstairs with the next incident strata. + +Finally let \(R_{\rm all}\) be the whole unit union on \(\widehat Z\). Sections +on its smooth components which agree on all their SNC intersections glue +by the elementary equalizer for the coordinate ideals of an SNC union. +Even iterated residues make the equalities independent of the chosen chain. +Apply the floor-connectedness lemma to the full-weight pair and this full +unit union. It gives +\((p_Z)_*\mathcal O_{R_{\rm all}}=\mathcal O_{C_Z}\). +Projection formula therefore descends the glued section uniquely to the +restricted invertible line on the reduced \(C_Z\). The proof with \(Z=V\) +gives the assertion for \(S\). +\end{proof} + +Each boundary stratum has dimension less than \(n\). The class of +\(J_Z:=K_Z+B_Z\) is nef, since it is the restriction of the nef adjoint +class. Proposition~\ref{bir:resolution-transfer}, applied using +\(\Ggood_{\dim Z}\) on a log resolution, makes the actual rational line +\(J_Z\) semiample. Increase \(q\) once +for all so that +\(L_Z:=\mathcal O_Z(qJ_Z)=\mathcal O_V(qJ)|_Z\) is generated for every +boundary stratum. Its map and Stein factorization give +\begin{equation}\label{floor:semiample-map} + f_Z:Z\longrightarrow Y_Z,\qquad L_Z=f_Z^*N_Z, +\end{equation} +where \(Y_Z\) is normal projective, \(f_Z\) has connected fibers, and +\(N_Z\) is ample. These are actual line identities. We call \(C_Z\) +\emph{vertical} if \(f_Z(C_Z)\ne Y_Z\); the empty floor is vertical. +If \(C_Z\) dominates \(Y_Z\), then for every \(k\geq1\) the restriction +\[ + H^0(Z,L_Z^k)\longrightarrow H^0(C_Z,L_Z^k|_{C_Z}) +\] +is injective. Indeed, every section comes from \(Y_Z\); if its pullback +vanishes on \(C_Z\), surjectivity makes it vanish at every point of \(Y_Z\). + +Fix a K\"ahler class \(c\) on \(V\), and let \(c_Z\) be its restriction +to each stratum. For boundary strata \(Z,Z'\) of the same dimension, an +\emph{arrow} is a proper bimeromorphic comparison with a projective +resolution whose source is smooth and compact K\"ahler, +\[ + \begin{tikzcd}[column sep=large] + & T\arrow[dl,"p"']\arrow[dr,"p'"]&\\ + Z&&Z' + \end{tikzcd} +\] +such that the two pulled-back adjoint lines are equal as invertible +subsheaves of meromorphic \(q\)-pluricanonical forms on \(T\), and +\begin{equation}\label{floor:class-comparison} + p^*c_Z-(p')^*c_{Z'}\in\NS(T)_{\R}. +\end{equation} +Here \(\NS(T)_{\R}\) is the real span of first Chern classes of +holomorphic lines, with torsion removed. The degree-two formula for a +smooth modification says that its new classes are exceptional divisor +classes. It follows that Equation~\eqref{floor:class-comparison} is independent +of further resolution and is preserved by composition. These arrows +therefore form a groupoid on the finite set of strata of each dimension. +An arrow \(\gamma:Z\dashrightarrow Z'\) transports sections by +\[ + (p')^*(\gamma_*s)=p^*s. +\] +Normality gives existence and uniqueness. Transport commutes with +multiplication and identifies the complete systems and their Stein targets +in Equation~\eqref{floor:semiample-map}. +The equality of meromorphic lines defines this transport of adjoint sections; the class +congruence will control self-arrows when the floor is vertical. + +\subsection{One residue comparison suffices for restriction} + +The only obstruction to extending a section from \(C_Z\) is its possible +variation among the connected components of a general floor fiber. We show +that all these components meet one general Mori fiber, whose floor is +connected or consists of two points. Thus at most one residue comparison +is needed. + +\begin{lemma}[Restriction and a Mori link]\label{floor:restriction} +For each positive-dimensional boundary stratum \(Z\), there is either no +comparison or one arrow between two components of \(C_Z\), allowing a +component to be compared with itself, with the following property. In all +sufficiently large divisible degrees \(k\), a section of \(L_Z^k|_{C_Z}\) +extends to \(Z\) if its two residues agree under that arrow. With no +arrow, every such section extends. The degree bound is uniform over +sections. When present, the arrow compares the two coefficient-one points +on general \(\PP^1\) fibers of a projective Mori contraction on a +bimeromorphic compact K\"ahler model of \(Z\). +\end{lemma} + +\begin{proof} +We use the dimension-free torsion-free restriction lemma +\cite[Lemma~7.1]{BL}. In its notation, for an effective compact K\"ahler +lc pair \((T,G)\), klt away from its reduced floor \(C\), and a connected +map \(f:T\to P\) to a normal compact base from which an actual positive +adjoint multiple is pulled back, it asserts that +\(R^1f_*\mathcal O_T(-C)\) is torsion-free. The ideal sequence gives +\begin{equation}\label{floor:restriction-obstruction} + \mathscr Q:=\coker(\mathcal O_P\longrightarrow f_*\mathcal O_C) + \lhook\joinrel\longrightarrow R^1f_*\mathcal O_T(-C). +\end{equation} +For \(T=Z\) and \(P=Y_Z\), if \(C_Z\) is vertical, \(\mathscr Q\) has +proper support and is zero. Serre vanishing for the kernel of +\(\mathcal O_P\to f_*\mathcal O_C\), tensored with \(N_Z^k\), extends +every restriction in a uniform large-degree tail. If \(C_Z\) dominates, +the same display says that a section extends as soon as its values are +constant on the reduced floor fiber over a dense open of \(Y_Z\): its +class in the torsion-free \(\mathscr Q\otimes N_Z^k\) then vanishes, and +the local lifts glue. They are unique because \(C_Z\) dominates \(Y_Z\), +so \(\mathcal O_{Y_Z}\to(f_Z)_*\mathcal O_{C_Z}\) is injective. +This uses neither reduced special scheme +fibers nor base change at their points. + +Suppose now that \(C_Z\) dominates \(Y_Z\). There is an effective rational +Cartier divisor \(D\) with \(\Supp D=\Supp C_Z\): restrict to \(Z\) the +globally \(\Q\)-Cartier floor primes of \(V\) not containing \(Z\). +For small rational +\(\epsilon>0\), \((Z,B_Z-\epsilon D)\) is effective klt. Apply +Corollary~\ref{prog:small-model} to obtain a projective small strong +\(\Q\)-factorialization \(\rho:Z_0\to Z\), crepant also for the full pair. +Write \(\Delta_0\) for its full boundary and \(D_0=\rho^*D\). +The full adjoint +\[ + P=K_{Z_0}+\Delta_0 +\] +is the actual semiample rational line pulled back from \(Y_Z\), and the +lowered klt adjoint is \(J_{\rm low}=P-\epsilon D_0\). Put +\(d=\dim Z\), choose \(m>0\) with \(mP\) Cartier and generated, and take +\(b>4md\). Consider +\[ + H=J_{\rm low}+bP=(1+b)P-\epsilon D_0. +\] +On a general fiber of the map to \(Y_Z\), its class is +\(-\epsilon\{D_0\}\). This is the negative of a nonzero effective +divisor because the floor dominates \(Y_Z\), so its pairing with a +K\"ahler power is negative. If \(H\) were pseudo-effective, a positive +current representing it would restrict to a positive current on almost +every resolved smooth fiber, contradicting this pairing. Thus \(H\) +is not pseudo-effective. + +Apply Lemma~\ref{prog:nef-trivial} to the lowered klt pair and the nef +rational line \(P\). Its \(H\)-program with K\"ahler scaling has ordinary +\(J_{\rm low}\)-negative steps, is finite, and ends in a projective Mori +contraction. Every step, including the final Mori ray, is \(P\)-trivial. +The same actual Cartier line \(mP\) descends at each birational step. +Consequently its semiample map to the fixed base \(Y_Z\) persists on +every working model. That map is constant on each connected final Mori +fiber and hence factors through the Mori contraction. We obtain +\[ + T\xrightarrow{u}W\xrightarrow{g}Y_Z. +\] +Both maps have connected fibers. The full adjoint on each model is the +trace of \(P\). On a common resolution of a birational step, its natural +meromorphic canonical comparison is an exceptional divisor with zero +class, since the two actual lines descend from the same line on the +contraction base. Exceptional negativity applied to both signs makes +that divisor zero. Thus the full pair remains crepant, including its +meromorphic adjoint identifications. The full transformed pair +\((T,G)\) is effective lc and is klt away from its full floor \(C^+\): +there it agrees with the transformed lowered klt pair. The program is +nonextracting, so the transformed divisor \(D^+\) has +\(\Supp D^+=\Supp C^+\). Since the full adjoint is pulled back from +\(Y_Z\), the lowered adjoint on a \(u\)-fiber is +\(-\epsilon D^+\). Hence \(D^+\) is relatively ample over \(W\). + +Compare the connected components of a general floor fiber on the original +small model and on \(T\). On a common projective log resolution the +full crepant subboundary is the same. Its unit union maps to both floors +with connected fibers by \cite[Lemma~2.1]{BL}. Proper closed surjections +with connected fibers preserve connected components, also after restricting +over a point of \(Y_Z\). Thus the connected components of the two +underlying floor fibers correspond. + +Every connected component of the floor fiber on \(T\) meets one common +general Mori fiber. Indeed \(C^+\to W\) is surjective because \(C^+\) +is the support of the relatively ample \(D^+\). Apply +Equation~\eqref{floor:restriction-obstruction} to \(u\). Surjectivity of +the floor makes \(\mathcal O_W\to u_*\mathcal O_{C^+}\) injective; the +displayed torsion-free cokernel then makes \(u_*\mathcal O_{C^+}\) +torsion-free. Let +\[ + C^+\longrightarrow\overline C\longrightarrow W +\] +be the Stein factorization. The finite space \(\overline C\) is reduced, +and every irreducible component of +\(\overline C\) dominates \(W\). To see the +last implication algebraically, a vertical minimal prime of a finite +reduced algebra over a domain would give, by prime avoidance, a nonzero +element killed by a nonzero element of the domain. After shrinking \(Y_Z\) +to a dense open, the fiber-dimension theorem gives every component of +\(\overline C_y\) dimension \(\dim W-\dim Y_Z\). The fiber \(W_y\) is irreducible of that +dimension: a resolution of \(W\) still has connected fibers over \(Y_Z\), +and a general one is smooth and connected and surjects onto \(W_y\). +Finiteness now makes each component of \(\overline C_y\) dominate \(W_y\). +Consequently each connected component of \(C^+_y\) meets \(u^{-1}(w)\) +for the same general \(w\in W_y\). + +The reduced general fiber \(F\) of \(u\) is irreducible by the same +resolution argument. If \(\dim F\geq2\), the support of the effective +ample Cartier multiple of \(D^+|_F\), whose support is \(C^+\cap F\), +is connected. One elementary +proof, valid even if \(F_{\rm red}\) is not normal, cuts by general +hyperplanes to an integral projective surface. If the ample divisor +split into disjoint nonzero Cartier parts \(D_1,D_2\), their pullbacks +to a resolution would be orthogonal, while the projection formula gives +\(v^*(D_1+D_2)\cdot v^*D_i>0\) for both \(i\). Thus both +\((v^*D_i)^2>0\), contradicting the surface Hodge index theorem. +If \(\dim F=1\), choose the fiber also off the singular, boundary +intersection, and ramification loci. It is a smooth connected curve and +\[ + 0=\deg(K_F+G|_F)=2g(F)-2+\deg(G|_F). +\] +There is a unit point, so \(F\simeq\PP^1\). Its floor has one or two +points; if there are two they exhaust the horizontal boundary. In every +connected case the preceding common-fiber observation makes the general +floor fiber connected, so no comparison is needed. In the remaining case +the two unit points represent all its connected components. + +Normalize the horizontal floor primes and take their Stein factorizations +over \(W\). If there are two primes, each finite Stein map has degree one +and is an isomorphism over normal \(W\); their main fiber product gives +a proper bimeromorphic comparison. If there is one prime, its normal Stein +space is a double cover of \(W\). The exchange on its general fiber +extends over the branch locus: the reduced horizontal non-diagonal +component of the double fiber product has finite birational projections +to the normal cover and hence both projections are isomorphisms. Its graph +lifts to the normalized prime. Composing with the program graphs transports +the comparison to two original normal floor primes, possibly the same one. +All graph projections are projective, and the primes survive birationally +because the program extracts no divisors. + +This comparison is crepant as a meromorphic residue identity. On a smooth +ruled open, order the markings after an \'etale local cover, choose a base +volume \(\xi\), and put the markings at \(z=0,\infty\). A pulled-back +\(q\)-pluriadjoint frame has the form +\[ + a(w)(dz/z)^{\otimes q}\otimes\xi^{\otimes q}. +\] +Its two residues are \(a(w)\xi^{\otimes q}\) and +\((-1)^q a(w)\xi^{\otimes q}\), which agree. This is the two-marking +Poincar\'e-residue comparison of +\cite[Section~3, Definition~13 and Proposition~14]{Kollar2013Sources}. +On a resolved graph both restricted actual lines are pulled back from +\(Y_Z\); their meromorphic embeddings agree on this dense open and hence +everywhere, including along exceptional primes. Equal residues therefore +give a constant value on the entire general floor fiber. The horizontal +consequence of Equation~\eqref{floor:restriction-obstruction} extends the section. +Normality descends it from the small model. Its restriction is the +prescribed section on the original reduced floor, since every original +floor prime is covered birationally and the two restrictions agree after +that pullback. + +It remains to check Equation~\eqref{floor:class-comparison}. Let \(\widehat T\) +be a common smooth resolution of the parent and Mori models and let +\(a,b:H\to\widehat T\) be the two maps from a resolved branch graph. +The resolution of the Mori model is an isomorphism near a general entire +\(\PP^1\) fiber: its centers have codimension at least two, hence cannot +dominate the ruling base. Every holomorphic two-form on \(\widehat T\) +comes from the base on that ruled open. In fact its vertical one-form +terms vanish on \(\PP^1\), and its remaining coefficients are constant +on that compact fiber. Thus \(a^*-b^*\) kills \(H^{2,0}\), and also +\(H^{0,2}\) by conjugation. This rational Hodge map on degree two has +image of type \((1,1)\). The Lefschetz \((1,1)\) theorem puts its real +image in \(\NS(H)_{\R}\). Apply this to the pulled-back parent class +\(c_Z\). The link is therefore an arrow as defined above. +\end{proof} + +\subsection{Transport respects every lower residue} + +An arrow can contract a divisor to a higher-codimension stratum. The next +local observation explains how to compare residues even in that case. +All valuations in its proof are local monomial valuations in an SNC chart. + +\begin{lemma}[Unit strata above an SNC stratum]\label{floor:unit-strata} +Let \(p:T\to Z\) be a projective log resolution of a dlt pair, with +crepant SNC subboundary on the smooth \(T\). Let \(R\subseteq T\) be a +unit stratum and let \(D=p(R)\) be its center. Then +\(D\) is a dlt stratum. There is a unit stratum \(R'\subseteq R\) such +that \(p(R')=D\) and \(R'\to D\) is birational. In the iterated residue +comparison along \(R'\), the normal logarithmic Jacobian is \(\pm1\). +\end{lemma} + +\begin{proof} +The center is a dlt stratum by the SNC discrepancy calculation in the +proof of Lemma~\ref{floor:adjunction}. Work over the generic SNC part of \(D\), +of codimension \(a\), with normal coordinates \(x_1,\ldots,x_a\) for +the unit divisors. The zero-discrepancy divisorial valuations centered +there are exactly the primitive rational monomial rays in +\(\R_{>0}^a\). Here is a local verification that also fixes their +normalization. The discrepancy inequality in the proof of +Lemma~\ref{floor:adjunction} says on every log-smooth model that the center +of a zero-discrepancy valuation is exactly a unit stratum: an additional +unmarked normal parameter, or a parameter with coefficient below one, would +give positive discrepancy. Blow up that stratum. If the positive normal +orders of the valuation are \(w_1,\ldots,w_b\), the chart indexed by a +smallest order \(w_k\) replaces these by \(w_k\) and the positive members +of \(w_i-w_k\). The exceptional divisor is again unit, and the center +is exactly the new unit stratum. While there is more than one positive +order their sum decreases. Eventually the center is the generic point of +one unit divisor, where a normalized divisorial valuation has order one. +Reversing these ordinary toric blowups proves monomiality and primitive +normalization. Conversely this subtraction algorithm for any primitive +positive integer vector terminates at order one, and reversing it extracts +that monomial valuation. Comparison on a common graph gives uniqueness +of the valuation with those weights and of its center on any model. + +For a unit component \(E\) meeting the inverse image of the generic SNC +part of \(D\), put +\[ + v_E=(\ord_E x_1,\ldots,\ord_E x_a)\in\Z_{\geq0}^a. +\] +If the unit components through a unit stratum \(Q\) meeting that inverse image are +\(E_1,\ldots,E_b\), define +\[ + \phi_Q:\R_{\geq0}^b\longrightarrow\R_{\geq0}^a,\qquad + (t_1,\ldots,t_b)\longmapsto\sum_{j=1}^b t_jv_{E_j}, + \qquad \sigma_Q=\phi_Q(\R_{\geq0}^b). +\] +The source carries its standard lattice \(\Z^b\). For the fixed +resolution \(T\), these cones subdivide the orthant over its interior. +On rational rays, \(\phi_Q\) sends a +monomial valuation on \(T\) to the same normalized valuation on \(Z\), +so it is injective on each cone. Interiors of different cones are +disjoint, because the center and the weights of a monomial valuation on +\(T\) are unique. Every rational ray in the orthant occurs: view its +valuation on \(T\) and use the same SNC description there. There are +only finitely many cones over the generic part of \(D\); density of +rational rays therefore gives coverage. Taking subsets of unit components +gives the faces, and uniqueness of valuations also on faces identifies +intersections as common faces. Consequently \(\sigma_R\) is a face +of an \(a\)-dimensional cone \(\sigma_{R'}\) with the same downstairs +center, for a unit stratum \(R'\subseteq R\). + +Let \(E_1,\ldots,E_a\) be the unit components through \(R'\). +The integer matrix \(A=(\ord_{E_j}x_i)\) of this full cone is unimodular. +Indeed every positive +primitive integral vector in the source cone is the normalized vector of +a divisorial valuation, so its image is primitive. If a prime divided +\(\det A\), a nonzero kernel vector modulo that prime, lifted to a +positive primitive integral vector, would have nonprimitive image. Hence +\(\det A=\pm1\). The strata \(R'\) and \(D\) have the same dimension. +We next prove that their generically finite map has degree one. In local coordinates at a +general point of \(R'\), +\[ + x_i=u_i(y,z)\prod_{j=1}^a y_j^{A_{ij}}, +\] +where the \(u_i\) are units, the \(y_j\) are normal coordinates, and +\(z\) are coordinates along the stratum. The tangential map is +generically \'etale in characteristic zero. Since \(A^{-1}\) is integral, +replace \(y_j\) by +\(\widetilde y_j=y_j\prod_i u_i^{(A^{-1})_{ji}}\). Then +\(x_i=\prod_j\widetilde y_j^{A_{ij}}\), and the map consisting of +\(\widetilde y\) and the downstairs tangential coordinates is locally +biholomorphic at a general \'etale point of \(R'\to D\). Two distinct +points over a general point of \(D\) would therefore give two lifts of +nearby target points whose normals approach with any fixed positive rate +vector in the interior of this cone. Varying the nonzero leading +coefficients fills an open set of nearby torus points, so such a target can +be chosen in the locus where \(p\) is an isomorphism. This contradiction +proves that \(R'\to D\) is birational. Finally, after taking residue +along \(R'\), the normal coefficient of +\(\bigwedge_i dx_i/x_i\) is \(\det A\) times that of +\(\bigwedge_jdy_j/y_j\); terms differentiating the units vanish in that +residue. Since \(\det A=\pm1\), the even pluriresidue removes the sign. +\end{proof} + +\begin{lemma}[Preservation of lower restrictions]\label{floor:transport} +Fix \(0\leq d0\), choose a common log resolution +\(p:T\to Z\), \(p':T\to Z'\) of the arrow. For a target floor prime +\(D'\), start with its strict transform \(R\). It is a unit stratum and +maps birationally to \(D'\). Its source center \(p(R)\) is a stratum by +Lemma~\ref{floor:unit-strata}. If \(R\to p(R)\) is not birational, that +lemma replaces \(R\) by a unit stratum inside it which maps birationally +onto the same source center. The target center may then shrink. If the +map to that center is not birational, apply the lemma on the target side, +and continue alternately. Each necessary replacement strictly decreases +dimension. The process therefore ends with one unit stratum mapping +birationally to lower strata \(D_0,D_0'\) on the two sides. + +The image in the common system base remains unchanged throughout. Indeed +the two maps from the resolved graph to the Stein targets are identified +by the arrow, and a replacement keeps its entire image on the side then +being treated. Initially the image was \(f_{Z'}(D')\). The terminal +comparison of \(D_0\) and \(D_0'\) is an arrow: restricting the actual +meromorphic identification to iterated residues gives crepancy, with the +logarithmic determinant sign removed by Lemma~\ref{floor:unit-strata}; +restricting the line Chern classes in Equation~\eqref{floor:class-comparison} +gives the same condition modulo \(\NS_{\R}\). Lower invariance thus +identifies the transported and assigned residues on \(D_0'\). + +This equality determines the entire section on \(D'\). The Stein map +of \(L_{D'}=L_{Z'}|_{D'}\) is the Stein factor of +\(f_{Z'}|_{D'}\), because the line induced on that Stein space is ample; +its target is finite over \(f_{Z'}(D')\). The image of \(D_0'\) in this +target is a closed irreducible subset of full dimension, since it still +maps onto \(f_{Z'}(D')\), and therefore is the whole target. Both +sections on \(D'\) come from it. Equality after restriction to +\(D_0'\) proves equality on \(D'\). + +The same construction shows that the image of every target floor prime +in the system base is the image of a lower source stratum, and the reverse +statement follows from the inverse arrow. A floor dominates the system +base on one side exactly when it does on the other. This proves the last +assertion. +\end{proof} + +\subsection{Finite actions on vertical strata} + +Once lower-dimensional sections are compatible and invariant, +Lemma~\ref{floor:transport} preserves their restrictions under every arrow. +If \(C_Z\) dominates \(Y_Z\), injectivity of restriction forces any such +transported extension to equal the assigned extension. Thus only vertical +strata require the following argument for finite image. + +\begin{proposition}[Finite image of self-arrows]\label{floor:finite-action} +Let \(Z\) be a boundary stratum with \(C_Z\) vertical. For all sufficiently +large divisible \(k\), the self-arrows of \(Z\) have finite image on +\(H^0(Z,L_Z^k)\). +\end{proposition} + +For \(m=qk\), a pluricanonical eigenform \(s\in H^0(Z,L_Z^k)\) with +\(\int_Z|s|^{2/m}<\infty\) becomes the \(m\)-th power of a holomorphic top +form on a smooth resolution of a cyclic root cover, so the auxiliary proposition below makes its +eigenvalue a root of unity. A uniform bound on the middle Betti numbers of +selected resolutions of these covers in the fixed degree will then bound +the possible orders; boundedness of the representation will give finite +image. The auxiliary proposition uses the polarization congruence in +Equation~\eqref{floor:class-comparison}. + +\subsubsection{Top-form scalars} + +\begin{proposition}[A polarized top-form scalar]\label{floor:top-form-scalar} +Let \(U\) be a smooth connected compact K\"ahler manifold of dimension +\(d0\) sufficiently divisible that the connected semiample map +and its Stein target satisfy +\[ + f:U'\longrightarrow Y,\qquad + \mathcal O_{U'}(\ell P)=f^*N, +\] +where \(Y\) is normal projective and \(N\) is ample, and that +Lemma~\ref{bir:sections} identifies every graded piece in +\[ + R_\ell(U):=\bigoplus_{j\geq0}H^0(U,j\ell K_U) + \simeq \bigoplus_{j\geq0}H^0(Y,N^j). +\] +This is the full section ring of the ample line \(N\), so +\(Y\simeq\Proj R_\ell(U)\). Passing to a further divisible Veronese gives +the same Proj. We call \(Y\) the \emph{canonical Iitaka base}. A +bimeromorphic self-map of \(U\) acts on this graded ring and therefore +induces an automorphism of \(Y\). + +\begin{lemma}[Finite action on the canonical Iitaka base]\label{floor:finite-base} +Let \(U\) be a smooth connected compact K\"ahler manifold of dimension +\(d0\) in the interior, since \(\chi_R\) diverges at the boundary +and \(\chi\) is bounded on the closed disk. At that point, +\[ + 0\ge\partial_z\partial_{\bar z}\log(\chi/\chi_R) + \ge\frac{\kappa}{2}(\chi-\chi_R)>0, +\] +a contradiction. Thus the Ahlfors--Schwarz comparison gives +\(\chi(0)\le4/(\kappa R^2)\) \cite{Ahlfors1938}. +Letting \(R\to\infty\) forces the metric to vanish +on \(\C\). The argument is local and allows zeros of the +induced metric, so the local representations just constructed suffice. +Consequently their period maps are locally constant along each such orbit. +In any holomorphic volume frame the logarithm of the coefficient in +Equation~\eqref{floor:hodge-density} is harmonic along each orbit: the top Hodge line is +flat there, and \(\sigma\) is a nonzero holomorphic multiple of a flat +generator. + +This is incompatible with finite mass. Diagonalize the one-parameter +action in projective coordinates, and choose two coordinates nonzero at a +general point with different weights, of difference \(a\ne0\). A small +smooth transverse slice \(Q\) on which their ratio is one gives a +holomorphic local biholomorphism onto its image +\[ + \Phi:\C^*\times Q\longrightarrow Y^* +\] +with fibers of size at most \(|a|\): the ratio at \(\Phi(z,t)\) is +\(z^a\). The entire \(\C^*\) factor is allowed because \(Y^*\) is +invariant. In the product frame \((dz/z)\wedge dt_1\wedge\cdots\wedge +dt_{\dim Q}\), write \(\Phi^*\nu\) with coefficient \(\rho(z,t)>0\). +For fixed \(t\), \(\log\rho(z,t)\) is harmonic on \(\C^*\). Its +circular mean at \(|z|=e^u\) is \(A(t)u+B(t)\). Jensen's inequality +therefore gives, with an irrelevant positive normalization, +\[ + \int_{\C^*}\rho(z,t)\frac{i\,dz\wedge d\overline z}{|z|^2} + \ \geq\ c\int_{\R}e^{A(t)u+B(t)}\,du=\infty. +\] +Tonelli's theorem contradicts the finite mass of \(\Phi^*\nu\), which +is at most \(|a|\nu(Y)\) by the bounded covering degree. This proves +that \(h\) has finite order. +\end{proof} + +After a power, the scalar problem can thus be restricted to a fiber with +Kodaira dimension zero. The polarization congruence then passes to that +fiber; the next two lattice arguments control its nonprojective factors. +In the next lemma, the hypothesis +on the origin of the model is what provides the degree-two exceptional +splitting; rational singularities alone are not used for that splitting. + +\begin{lemma}[The scalar on a symplectic factor]\label{floor:symplectic-scalar} +Let \(Y\) be a terminal compact K\"ahler primitive symplectic space of +dimension \(2e0\). Since \(\beta\) +is nef over \(Y\), exceptional negativity applied to the divisor \(E\), +with \(\{E\}=p^*c-[\beta]\), gives \(E=\sum_i e_iE_i\geq0\). Pair with the positive +normalization \(W_R=(p^*\eta\,p^*\overline\eta)^{e-1}\). Exceptional +classes pair to zero with \(p^*c\) by projection formula, and hence for +some constant \(b_e>0\) +\begin{equation}\label{floor:positive-bb} + q_Y(c)=b_e\left(\int_R\beta^2 W_R+\sum_i e_i\int_{E_i}\beta W_R\right)>0. +\end{equation} +Every term is nonnegative. The first is strictly positive on the common +open where \(\beta\) is positive and the symplectic form is nondegenerate. + +Set \(H_\Q=H^2(Y,\Q)\) and +\(D_\Q=H_\Q\cap H^{1,1}(Y,\C)\). Lefschetz \((1,1)\) on \(R\), together +with the line-trace assertion of Lemma~\ref{prog:cohomology-transport}, +identifies \(D_\Q\) with the rational span of line Chern classes. +It is \(G\)-invariant because \(G\) is a rational Hodge automorphism. +The same line-trace assertion applied to the graph congruence gives +\(Gc-c\in D_\R\). The line \(\C\eta\) survives in \(H_\C/D_\C\); +our immediate aim is to prove that all eigenvalues on this quotient have +absolute value one. Write \(\Sigma=(H^{2,0}\oplus H^{0,2})_\R\), the +positive real two-plane. There are three possibilities for the restriction +of the Lorentz form to \(D_\R\). + +If \(D_\R\) is negative definite, its orthogonal projection \(c_0\) of +\(c\) to \(D_\R^\perp\) is fixed by \(G\) and has positive square by +Equation~\eqref{floor:positive-bb}. On \(D_\R^\perp\), the invariant space +\(\Sigma\oplus\R c_0\) is positive definite and its complement is +negative definite. All eigenvalues there have absolute value one. If +\(D_\R\) contains a positive vector, it is nondegenerate with one +positive direction. Its orthogonal complement is the positive plane +\(\Sigma\) plus a negative-definite space, giving the same conclusion. +In either case \(H_\R/D_\R\simeq D_\R^\perp\). + +In the remaining case \(D_\R\) is negative semidefinite with radical a +rational isotropic line \(\ell\). The lattice action on \(\ell\) is +\(\pm1\). On \(\ell^\perp/\ell\) the form is the positive plane +\(\Sigma\) plus a negative-definite \((1,1)\) space, and the quotient +\(H_\R/\ell^\perp\) is dual to \(\ell\). The invariant flag +\(0\subset\ell\subset\ell^\perp\subset H_\R\) therefore shows that +all eigenvalues on \(H_\R\) have absolute value one; this argument also +allows unipotent parts. In all cases the eigenvalues on +\(H_\C/D_\C\) have absolute value one. The rational quotient +\(H_\Q/D_\Q\) carries the preserved lattice +\[ + \overline\Lambda_Y=\Lambda_Y/(\Lambda_Y\cap D_\Q), +\] +where the intersection is saturated. Thus +\(\lambda\) is an algebraic integer and all its algebraic conjugates have +absolute value one. Kronecker's theorem makes it a root of unity. +\end{proof} + +\begin{lemma}[The scalar on a torus]\label{floor:torus-scalar} +Let \(A\) be the integral Hodge automorphism of \(H^1(T,\Z)\) induced +by an automorphism of a compact complex torus \(T\). Suppose a positive +Hermitian class \(c\in H^{1,1}(T,\R)\) satisfies +\((\bigwedge^2 A)c-c\in\NS(T)_\R\). Then +\(\det(A|H^{1,0}(T))\) is a root of unity. +\end{lemma} + +\begin{proof} +Put \(V=H^1(T,\Q)\) and \(D_\Q=\NS(T)_\Q\). For each eigenvalue +\(\theta\) of \(A_\C\), let \(V_\theta\) be its generalized eigenspace and set +\(d_\theta=\dim V_\theta\), \(p_\theta=\dim(V_\theta\cap H^{1,0}(T))\). +Complex conjugation gives +\(p_\theta+p_{\overline\theta}=d_\theta=d_{\overline\theta}\). + +For \(|\theta|\ne1\), project \(c\) to the block +\(V_\theta\wedge V_{\overline\theta}\), with \(\bigwedge^2V_\theta\) understood for +real \(\theta\). This projection preserves \(D_{\overline\Q}\). Indeed +the Chinese-remainder projectors onto the generalized spaces are +polynomials in \(A\) with algebraic coefficients. Apply them to the +two tensor slots separately and then alternate. The resulting maps are +algebraic linear combinations of rational Hodge maps on \(\bigwedge^2V\); +these preserve rational \((1,1)\) classes, hence \(D_\Q\) by Lefschetz +\((1,1)\). Separate slot projections avoid any collision of products of +eigenvalues in degree two. + +On this block \(A_2=\bigwedge^2 A\) has the single generalized +eigenvalue \(\theta\overline\theta=|\theta|^2\ne1\). The inverse of \(A_2-1\) +there is a finite polynomial in its nilpotent part. Project the congruence +for \(c\) to the block and apply this inverse. It follows that its block +\(c_{\theta,\overline\theta}\) belongs to \(D_\C\). Positivity says that this +block is a perfect tensor between \(V_\theta\) and \(V_{\overline\theta}\), +pairing their opposite Hodge types: in a basis of \(H^{1,0}\) the relevant +matrices are positive-definite principal Hermitian blocks. For real +\(\theta\) the corresponding alternating tensor is nondegenerate. + +The intersection of this block with \(D_\C\) is defined over +\(\overline\Q\). Nondegeneracy is the nonvanishing of a determinant, +so it contains a nondegenerate tensor with algebraic coefficients. For +any Galois automorphism \(\gamma\), the conjugate tensor remains in +\(D_{\overline\Q}\), hence is still of type \((1,1)\) for the given +Hodge decomposition, and is perfect between \(V_{\gamma(\theta)}\) and +\(V_{\gamma(\overline\theta)}\). Opposite-type perfection yields +\begin{equation}\label{floor:torus-hodge-count} + p_{\gamma(\theta)}+p_{\gamma(\overline\theta)} + =d_\theta\qquad(|\theta|\ne1). +\end{equation} +This does not replace \(\gamma(\overline\theta)\) by +\(\overline{\gamma(\theta)}\), and assumes no Galois invariance of the Hodge +decomposition. + +The top-form scalar is the algebraic integer +\(z=\prod_\theta \theta^{p_\theta}=\det(A|H^{1,0})\), also when generalized eigenspaces +are nontrivial. For any Galois automorphism \(\delta\), reindex by the +image roots and pair complex conjugates: +\[ + \begin{aligned} + \log|\delta z| + &=\frac12\sum_\theta + \bigl(p_{\delta^{-1}(\theta)} + +p_{\delta^{-1}(\overline\theta)}\bigr)\log|\theta|\\ + &=\frac12\sum_\theta d_\theta\log|\theta| + =\tfrac12\log|\det A|=0. + \end{aligned} +\] +For \(|\theta|\ne1\) the second equality uses +Equation~\eqref{floor:torus-hodge-count} with \(\gamma=\delta^{-1}\); the other +terms vanish. The last equality is unimodularity of the integral action. +Kronecker's theorem now proves the assertion. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{floor:top-form-scalar}] +Apply Lemma~\ref{floor:finite-base}. After a power, \(g\) acts trivially +on the canonical Iitaka base. Use the modification \(\pi:U'\to U\) and +the map \(f:U'\to Y\) in its construction, and restrict over a general +point where the graph and the fibration are smooth and the fiber meets the +isomorphism locus of \(\pi\). Dividing \(\alpha\) by a local base volume +gives a nonzero top form on the smooth compact fiber \(F\), which has +\(\kappa(F)=0\); its returning map scales the form by the corresponding +power of \(\mu\). A point fiber gives scalar one. For a positive-dimensional +fiber, let \(\omega_U\) be a K\"ahler form representing the class in the +proposition and put \(\beta_F=(\pi|_F)^*\omega_U\). This is a smooth +semipositive form, strictly positive on the dense open where \(\pi\) is +an isomorphism. Pulling the original graph congruence to \(U'\) and +restricting to the fiber shows that the two pullbacks of \([\beta_F]\) on +a common resolved graph of the returning map of \(F\) differ by real line classes. +This weaker positivity suffices below. + +Since \(\dim F\leq d0\) and a compatible invariant generating system +of tuples through dimension \(n-1\). +\end{proposition} + +\begin{proof} +At the point strata use the same scalar in the canonical zero-form +generator \(1\). The even residue convention identifies these generators +along every path and under every point arrow. This gives the required +system in dimension zero; if there are no points the empty system is +understood. + +Suppose the system has been constructed through dimension \(d-1\). +Replace it by powers and their span in a common degree \(k\) chosen so that +Lemma~\ref{floor:restriction} applies on +every \(d\)-stratum, Proposition~\ref{floor:finite-action} applies on every +vertical \(d\)-stratum, and +\(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) is generated for every +\(d\)-stratum \(Z\). These conditions hold in a common divisible tail: +there are finitely many strata, and the last condition is Serre's theorem. +For a \(d\)-stratum \(Z\), Lemma~\ref{floor:adjunction} glues +the lower tuple to a section on its whole \(C_Z\). Its residues satisfy +the one link in Lemma~\ref{floor:restriction}, if that link is present, +because the lower tuple is invariant. That lemma extends the section to +\(Z\). + +Let \(\mathcal P_d(k)\) be the vector space of all tuples through +dimension \(d\) whose lower part is in \(\mathcal V_{d-1}(k)\) and whose +new components restrict to that lower part. Call its elements +\emph{pre-tuples}. Its projection to +\(\mathcal V_{d-1}(k)\) is surjective, since the finitely many extensions +can be chosen independently. This system already generates. To check a +point \(z\in Z\), put \(y=f_Z(z)\). If \(y\in f_Z(C_Z)\), choose a +floor point over \(y\) and a lower tuple nonzero there. Every extension is +the pullback of a section of \(N_Z^k\), so it is nonzero at every point +over \(y\), including \(z\). If \(y\notin f_Z(C_Z)\), the chosen generation +of \(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) supplies a base section nonzero +at \(y\). It pulls back to a section +vanishing on \(C_Z\); combine it with the zero lower tuple and zero +components on the other new strata. This also treats an empty floor. +Surjectivity to the lower system preserves generation at lower points. + +We impose invariance in dimension \(d\). For a dominating floor, +injectivity of restriction and Lemma~\ref{floor:transport} already make +every pre-tuple invariant on that stratum. +Verticality is invariant under arrows, so the remaining orbits consist +entirely of vertical strata. + +For each such orbit choose a representative \(Z\), and let \(G_Z\) be +the finite isotropy image on \(H^0(Z,L_Z^k)\) given by +Proposition~\ref{floor:finite-action}. For a pre-tuple with component +\(s_Z\), form the norm +\[ + P_Z(s_Z)=\prod_{g\in G_Z}g(s_Z) + \in H^0(Z,L_Z^{k|G_Z|}). +\] +Choose a common positive integer \(a\) divisible by all \(|G_Z|\), use +\(P_Z(s_Z)^{a/|G_Z|}\), and transport it to each member of that orbit. +This is independent of the chosen arrow: changing that arrow by an +isotropy arrow merely permutes the factors in degree \(k\). No finiteness +assertion in degree \(ak\) is needed. On strata with dominating floor +use \(s_Z^a\), and on all lower strata use the componentwise \(a\)-th +power of the assigned tuple. + +The resulting tuple is compatible. By Lemma~\ref{floor:transport}, +every transported factor in a norm has exactly the prescribed lower +restriction. The norm and its indicated power therefore restrict to +the \(a\)-th power of that restriction, the same value used on all lower +strata and in all other orbits. It is invariant in dimension \(d\) by +construction, and remains invariant below it. + +These tuples still generate. Fix a point \(z\) in an orbit member. +For each of the finitely many transported factors, choose a point over +\(z\) on a resolution of its comparison graph. Equality of the +pulled-back invertible adjoint lines identifies the fiber values. The +factor is nonzero at \(z\) exactly when the representative component +\(s_Z\) is nonzero at the corresponding point of \(Z\). Evaluation at each +such point is a nonzero linear functional on the generating space +\(\mathcal P_d(k)\). +A finite union of their proper kernels cannot cover a complex vector +space. One pre-tuple makes all factors nonzero and gives a norm nonzero +at \(z\). At a lower point a nonzero assigned value stays nonzero after +its \(a\)-th power. Taking the linear span of all the constructed tuples +therefore gives a compatible invariant generating system in degree \(ak\). +This completes the induction on \(d\). +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{floor:generation}] +Take the system in Proposition~\ref{floor:tuples}. Its components on the +prime components of \(S\) agree on every common lower stratum, so +Lemma~\ref{floor:adjunction} descends each tuple uniquely to a section of +the actual line \(\mathcal O_V(qkJ)|_S\). At any \(x\in S\), choose a +prime component through \(x\) and a tuple nonzero there. It is the pullback +of the descended value in the same invertible line fiber at \(x\), so +that value is nonzero. The finite-dimensional span of these descended +sections generates at every point of the reduced \(S\). This is the +claimed global generation of an actual Cartier multiple. +\end{proof} diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-projective-program-input.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-projective-program-input.tex index 791dded29fbf0357f50ca72abf2e8608ba7d3440..29fa116c007561188bd8e7d048600b5d20777b04 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-projective-program-input.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-projective-program-input.tex @@ -1,206 +1,206 @@ -\subsection{Projective programs under the assumption} -\label{rank:projective-program-input} - -The argument for a projective base uses a terminating program to reach a nef -adjoint and, after decreasing the nef data, a program ending in a Mori fiber -space. The conditional input is termination in the pseudo-effective case. -We first state the exact program result and then derive that case from the -projective good models in Proposition~\ref{proj:good-models}. - -\begin{proposition}[A conditional generalized program] -\label{proj:generalized-mmp} -Assume Assumption~\ref{asm:log-iitaka}. Let -\((V,B+\boldsymbol M)\) be a projective generalized dlt rational pair, -where \(V\) is \(\Q\)-factorial, \(B\geq0\), and the nef b-divisor -\(\boldsymbol M\) is determined by a nef rational Cartier divisor on a -projective birational model. Put \(D=K_V+B+M_V\). - -There is a choice of \(D\)-MMP with ample scaling which terminates. If -\(D\) is pseudo-effective, its endpoint \(V_m\) has nef adjoint \(D_m\). -If \(D\) is not pseudo-effective, its endpoint has a Mori fiber contraction -to a projective variety of smaller dimension. The birational steps and their -endpoints stay \(\Q\)-factorial generalized dlt, and the forward birational -map extracts no divisors. -In the pseudo-effective case, on a common resolution one has an actual -rational divisor identity -\[ - u^*D=v^*D_m+F,\qquad F\geq0\text{ and }v\text{-exceptional}. -\] -\end{proposition} - -For the pseudo-effective alternative, we first derive the relative weak -Zariski decompositions used by the minimal-model existence theorem. - -For a projective morphism, first replace its image by its normal Stein -factor. We use the relative convention of \cite[Section~2]{TsakanikasXie}: -a divisor is pseudo-effective over the base when its restriction to a very -general fiber of this surjective morphism is pseudo-effective. An NQC weak Zariski decomposition of \(D\) over the -base is a birational equality \(h^*D\equiv_Z P+N\), where \(P\) is a -nonnegative real combination of relatively nef rational Cartier divisors -and \(N\geq0\). The decomposition below has rational Cartier parts. - -\begin{lemma}[Relative canonical decomposition]\label{proj:relative-wzd} -Assume Assumption~\ref{asm:log-iitaka}. Let \(V\) be a smooth -quasi-projective complex variety and let \(f:V\to Z\) be projective, with -\(Z\) normal and quasi-projective. If \(K_V\) is pseudo-effective over -\(Z\), then on a normal variety \(Y\) with a projective birational morphism \(h:Y\to V\) there -are rational Cartier divisors \(P,N\) such that -\[ - h^*K_V\equiv_Z P+N,\qquad P\text{ nef over }Z,\qquad N\geq0. -\] -\end{lemma} - -\begin{proof} -We use the construction in the proof of -\cite[Proposition~3.3]{External037}. Its absolute input is a nef-plus- -effective rational decomposition for the canonical divisor of a smooth -projective variety with pseudo-effective canonical class. This is supplied -here by Proposition~\ref{proj:good-models} and its common-resolution -comparison. - -Replace the image of \(f\) by its normal Stein factor. This does not change -relative numerical classes or contracted curves, and now \(f\) is -surjective with connected fibers. Its very general fiber is smooth -projective with pseudo-effective canonical divisor, hence is non-uniruled -by \cite[Theorem~0.2 and Corollary~0.3]{BDPP}. Compactify the projective -morphism and resolve away from \(V\). We obtain -\(\bar f:\bar V\to\bar Z\), with \(\bar V\) smooth projective and -\(\bar Z\) normal projective, unchanged over \(Z\). If \(\dim Z=0\), -\(V\) is already projective and the absolute input proves the lemma. - -Suppose \(\dim Z>0\), and let \(r:\widetilde Z\to\bar Z\) be a smooth -projective resolution. Choose a sufficiently positive very ample divisor -\(A_Z\) on \(\bar Z\) and a general \(B\in|2A_Z|\). The pulled-back systems -on both \(\widetilde Z\) and \(\bar V\) are base point free. Bertini makes -\(r^*B\) and \(\bar f^*B\) smooth nonempty reduced divisors. Their double -covers -\[ - \pi_Z:Z^\sharp\to\widetilde Z, - \qquad \pi:V^\sharp\to\bar V -\] -are smooth integral projective varieties. The canonical formulas are -\[ - K_{Z^\sharp}\sim_{\Q}\pi_Z^*(K_{\widetilde Z}+r^*A_Z), - \qquad - K_{V^\sharp}\sim_{\Q}\pi^*(K_{\bar V}+\bar f^*A_Z). -\] -The first line is big when \(A_Z\) is sufficiently positive. The rational -map \(V^\sharp\dashrightarrow Z^\sharp\) has the same very general fibers -as \(f\). A covering family of rational curves on \(V^\sharp\) would -either dominate a covering family of rational curves on \(Z^\sharp\), or -cover its very general fibers by vertical rational curves. Both are -impossible. Thus \(V^\sharp\) is non-uniruled and \(K_{V^\sharp}\) is -pseudo-effective by BDPP. - -Apply the absolute decomposition to \(V^\sharp\). Moving the rational -principal difference in the canonical formula into the nef part gives an -actual rational divisor decomposition for a pullback of -\(\pi^*(K_{\bar V}+\bar f^*A_Z)\). Take a common equivariant resolution of -that model and its conjugate under the covering involution. On this smooth -projective resolution \(g:T\to V^\sharp\), average the two decompositions. -We obtain -\[ - g^*\pi^*(K_{\bar V}+\bar f^*A_Z)=\bar P+\bar N, - \qquad \bar P\text{ nef},\quad \bar N\geq0, -\] -where \(\bar P\) and \(\bar N\) are invariant rational Cartier divisors. - -Let \(q:T\to\bar Y\) be the finite quotient by the involution. The invariant -composite \(\pi g\) induces a projective birational morphism -\(\bar h:\bar Y\to\bar V\), with \(\bar Y\) normal and projective. An invariant -rational divisor descends by dividing its coefficient at an upstairs prime -by the ramification index. The descended divisor is rational Cartier when -the original divisor is: after clearing denominators, choose one Cartier -equation on a semilocal neighborhood of the finite orbit above a point -(a Cartier divisor is principal on a semilocal ring), and multiply all its -translates. The product is invariant and has divisor -equal to the group order times the original divisor. It descends to a local -equation for a multiple of the downstairs divisor. This argument includes -stabilizers and ramification primes. - -Accordingly \(\bar P=q^*P\) and \(\bar N=q^*N\) for rational Cartier -divisors on \(\bar Y\). Lifting curves through the finite map shows that -\(P\) is nef, and coefficientwise descent shows that \(N\) is effective. -Finite pullback is injective on rational divisors, so the displayed identity -descends. Restrict it to \(Y=\bar h^{-1}(V)\). The term pulled -back from \(A_Z\) is numerically trivial over \(Z\), giving -\(h^*K_V\equiv_ZP+N\), as required. -\end{proof} - -\begin{proof}[Proof of Proposition~\ref{proj:generalized-mmp}] -Lemma~\ref{proj:relative-wzd} and -\cite[Theorem~5.2]{TsakanikasXie}, applied to the generalized pair -\((U/Z,0+0)\) with \(U\) smooth, prove relative minimal-model existence for smooth varieties -by dimension induction. The hypothesis of that theorem is relative smooth -minimal-model existence one dimension lower; the lemma supplies its other -hypothesis, an NQC weak Zariski decomposition. The induction begins in -dimension zero. Then \cite[Theorem~5.4]{TsakanikasXie} supplies minimal -models for every pseudo-effective NQC generalized lc pair in the relevant -dimension. Their Theorem~2.7 gives existence in the Birkar--Shokurov sense. - -The nef data in the statement are NQC, because a nef rational Cartier -divisor is a positive multiple of a nef Cartier divisor. In the -pseudo-effective case the preceding paragraph therefore gives the -Birkar--Shokurov model required by -\cite[Theorem~4.2]{TsakanikasXie}. In the other case, the non-pseudo-effective -alternative of that theorem applies directly. To meet its scaling -hypothesis, take an effective sufficiently positive ample rational divisor -\(A\) whose general components have sufficiently small coefficients. On -a fixed log resolution carrying \(\boldsymbol M\), these components are -transverse to the boundary and preserve generalized log canonicity, while -their ample class can be chosen large enough that \(D+A\) is nef. The -theorem supplies a terminating MMP starting on \(V\). - -We verify the step types inductively. Suppose \(V_i\) is -\(\Q\)-factorial generalized dlt. By -\cite[Remark~2.15]{TsakanikasXie}, each birational step has the form -\[ - V_i\xrightarrow{g_i}Z_i\xleftarrow{h_i}V_{i+1}, -\] -where \(g_i\) contracts a negative extremal ray and \(h_i\) is small and projective. -Suppose \(g_i\) is divisorial, and write \(R\) for its ray. An exceptional -prime \(E\) has \(E\cdot R<0\): otherwise it would be effective, -exceptional, and \(g_i\)-nef, contrary to negativity. There is only one -exceptional prime. Indeed, a combination of two distinct exceptional -primes can be chosen to have degree zero on \(R\); negativity applied to -both signs of that combination would make it zero. - -We show directly that \(Z_i\) is \(\Q\)-factorial. For a prime divisor -\(T\) on \(Z_i\), let \(\widetilde T\) be its strict transform, choose a -curve \(C\) spanning \(R\), and put -\[ - a=-\frac{\widetilde T\cdot C}{E\cdot C}\in\Q. -\] -Choose a positive integer \(m\) for which \(m(\widetilde T+aE)\) is -Cartier. Its line has degree zero on \(R\), so -\cite[Theorem~1.5]{Xie2022} gives an actual Cartier line \(\mathcal L_T\) -on \(Z_i\) and an isomorphism -\[ - \OO_{V_i}\bigl(m(\widetilde T+aE)\bigr)\simeq g_i^*\mathcal L_T. -\] -On an open set trivializing \(\mathcal L_T\), the Cartier divisor -\(m(\widetilde T+aE)\) is principal upstairs. Pushing this principal -divisor identity down through the birational \(g_i\) gives a principal -divisor \(mT\) on that open set. -Thus \(T\) is rational Cartier. It follows that \(h_i\) is an isomorphism: -the pushforward of an \(h_i\)-ample Cartier divisor is rational Cartier -on \(Z_i\), and its pullback is the original divisor because \(h_i\) -is small. It has degree zero on every contracted curve, so relative -ampleness rules out positive-dimensional fibers; a finite birational -map to the normal \(Z_i\) is an isomorphism. If \(g_i\) is small, -the relatively ample canonical model \(h_i\) is the flip of -\cite[Definition~3.2]{HanLi}. Thus the chosen birational steps are -divisorial contractions and flips. The underlying variety of a -generalized dlt pair is klt, and these steps preserve the -\(\Q\)-factorial generalized dlt category -\cite[Remark~2.3 and Lemma~3.7]{HanLi}. They introduce no prime -divisors on the new models. For completeness, on a common resolution -of one step \(V_i\dashrightarrow V_{i+1}\) carrying \(\boldsymbol M\), -taking the difference of the two generalized discrepancy formulas cancels -their common nef divisor. The resulting difference of the adjoint pullbacks is exceptional over the new -model and anti-nef over it. The negativity lemma makes this difference -effective. Composing these actual rational divisor comparisons gives -\(u^*D=v^*D_m+F\) with \(F\) effective and exceptional over the endpoint. -The endpoints in the two cases are the nef and Mori fiber endpoints supplied -by the terminating program theorem. -\end{proof} +\subsection{Projective programs under the assumption} +\label{rank:projective-program-input} + +The argument for a projective base uses a terminating program to reach a nef +adjoint and, after decreasing the nef data, a program ending in a Mori fiber +space. The conditional input is termination in the pseudo-effective case. +We first state the exact program result and then derive that case from the +projective good models in Proposition~\ref{proj:good-models}. + +\begin{proposition}[A conditional generalized program] +\label{proj:generalized-mmp} +Assume Assumption~\ref{asm:log-iitaka}. Let +\((V,B+\boldsymbol M)\) be a projective generalized dlt rational pair, +where \(V\) is \(\Q\)-factorial, \(B\geq0\), and the nef b-divisor +\(\boldsymbol M\) is determined by a nef rational Cartier divisor on a +projective birational model. Put \(D=K_V+B+M_V\). + +There is a choice of \(D\)-MMP with ample scaling which terminates. If +\(D\) is pseudo-effective, its endpoint \(V_m\) has nef adjoint \(D_m\). +If \(D\) is not pseudo-effective, its endpoint has a Mori fiber contraction +to a projective variety of smaller dimension. The birational steps and their +endpoints stay \(\Q\)-factorial generalized dlt, and the forward birational +map extracts no divisors. +In the pseudo-effective case, on a common resolution one has an actual +rational divisor identity +\[ + u^*D=v^*D_m+F,\qquad F\geq0\text{ and }v\text{-exceptional}. +\] +\end{proposition} + +For the pseudo-effective alternative, we first derive the relative weak +Zariski decompositions used by the minimal-model existence theorem. + +For a projective morphism, first replace its image by its normal Stein +factor. We use the relative convention of \cite[Section~2]{TsakanikasXie}: +a divisor is pseudo-effective over the base when its restriction to a very +general fiber of this surjective morphism is pseudo-effective. An NQC weak Zariski decomposition of \(D\) over the +base is a birational equality \(h^*D\equiv_Z P+N\), where \(P\) is a +nonnegative real combination of relatively nef rational Cartier divisors +and \(N\geq0\). The decomposition below has rational Cartier parts. + +\begin{lemma}[Relative canonical decomposition]\label{proj:relative-wzd} +Assume Assumption~\ref{asm:log-iitaka}. Let \(V\) be a smooth +quasi-projective complex variety and let \(f:V\to Z\) be projective, with +\(Z\) normal and quasi-projective. If \(K_V\) is pseudo-effective over +\(Z\), then on a normal variety \(Y\) with a projective birational morphism \(h:Y\to V\) there +are rational Cartier divisors \(P,N\) such that +\[ + h^*K_V\equiv_Z P+N,\qquad P\text{ nef over }Z,\qquad N\geq0. +\] +\end{lemma} + +\begin{proof} +We use the construction in the proof of +\cite[Proposition~3.3]{External037}. Its absolute input is a nef-plus- +effective rational decomposition for the canonical divisor of a smooth +projective variety with pseudo-effective canonical class. This is supplied +here by Proposition~\ref{proj:good-models} and its common-resolution +comparison. + +Replace the image of \(f\) by its normal Stein factor. This does not change +relative numerical classes or contracted curves, and now \(f\) is +surjective with connected fibers. Its very general fiber is smooth +projective with pseudo-effective canonical divisor, hence is non-uniruled +by \cite[Theorem~0.2 and Corollary~0.3]{BDPP}. Compactify the projective +morphism and resolve away from \(V\). We obtain +\(\bar f:\bar V\to\bar Z\), with \(\bar V\) smooth projective and +\(\bar Z\) normal projective, unchanged over \(Z\). If \(\dim Z=0\), +\(V\) is already projective and the absolute input proves the lemma. + +Suppose \(\dim Z>0\), and let \(r:\widetilde Z\to\bar Z\) be a smooth +projective resolution. Choose a sufficiently positive very ample divisor +\(A_Z\) on \(\bar Z\) and a general \(B\in|2A_Z|\). The pulled-back systems +on both \(\widetilde Z\) and \(\bar V\) are base point free. Bertini makes +\(r^*B\) and \(\bar f^*B\) smooth nonempty reduced divisors. Their double +covers +\[ + \pi_Z:Z^\sharp\to\widetilde Z, + \qquad \pi:V^\sharp\to\bar V +\] +are smooth integral projective varieties. The canonical formulas are +\[ + K_{Z^\sharp}\sim_{\Q}\pi_Z^*(K_{\widetilde Z}+r^*A_Z), + \qquad + K_{V^\sharp}\sim_{\Q}\pi^*(K_{\bar V}+\bar f^*A_Z). +\] +The first line is big when \(A_Z\) is sufficiently positive. The rational +map \(V^\sharp\dashrightarrow Z^\sharp\) has the same very general fibers +as \(f\). A covering family of rational curves on \(V^\sharp\) would +either dominate a covering family of rational curves on \(Z^\sharp\), or +cover its very general fibers by vertical rational curves. Both are +impossible. Thus \(V^\sharp\) is non-uniruled and \(K_{V^\sharp}\) is +pseudo-effective by BDPP. + +Apply the absolute decomposition to \(V^\sharp\). Moving the rational +principal difference in the canonical formula into the nef part gives an +actual rational divisor decomposition for a pullback of +\(\pi^*(K_{\bar V}+\bar f^*A_Z)\). Take a common equivariant resolution of +that model and its conjugate under the covering involution. On this smooth +projective resolution \(g:T\to V^\sharp\), average the two decompositions. +We obtain +\[ + g^*\pi^*(K_{\bar V}+\bar f^*A_Z)=\bar P+\bar N, + \qquad \bar P\text{ nef},\quad \bar N\geq0, +\] +where \(\bar P\) and \(\bar N\) are invariant rational Cartier divisors. + +Let \(q:T\to\bar Y\) be the finite quotient by the involution. The invariant +composite \(\pi g\) induces a projective birational morphism +\(\bar h:\bar Y\to\bar V\), with \(\bar Y\) normal and projective. An invariant +rational divisor descends by dividing its coefficient at an upstairs prime +by the ramification index. The descended divisor is rational Cartier when +the original divisor is: after clearing denominators, choose one Cartier +equation on a semilocal neighborhood of the finite orbit above a point +(a Cartier divisor is principal on a semilocal ring), and multiply all its +translates. The product is invariant and has divisor +equal to the group order times the original divisor. It descends to a local +equation for a multiple of the downstairs divisor. This argument includes +stabilizers and ramification primes. + +Accordingly \(\bar P=q^*P\) and \(\bar N=q^*N\) for rational Cartier +divisors on \(\bar Y\). Lifting curves through the finite map shows that +\(P\) is nef, and coefficientwise descent shows that \(N\) is effective. +Finite pullback is injective on rational divisors, so the displayed identity +descends. Restrict it to \(Y=\bar h^{-1}(V)\). The term pulled +back from \(A_Z\) is numerically trivial over \(Z\), giving +\(h^*K_V\equiv_ZP+N\), as required. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{proj:generalized-mmp}] +Lemma~\ref{proj:relative-wzd} and +\cite[Theorem~5.2]{TsakanikasXie}, applied to the generalized pair +\((U/Z,0+0)\) with \(U\) smooth, prove relative minimal-model existence for smooth varieties +by dimension induction. The hypothesis of that theorem is relative smooth +minimal-model existence one dimension lower; the lemma supplies its other +hypothesis, an NQC weak Zariski decomposition. The induction begins in +dimension zero. Then \cite[Theorem~5.4]{TsakanikasXie} supplies minimal +models for every pseudo-effective NQC generalized lc pair in the relevant +dimension. Their Theorem~2.7 gives existence in the Birkar--Shokurov sense. + +The nef data in the statement are NQC, because a nef rational Cartier +divisor is a positive multiple of a nef Cartier divisor. In the +pseudo-effective case the preceding paragraph therefore gives the +Birkar--Shokurov model required by +\cite[Theorem~4.2]{TsakanikasXie}. In the other case, the non-pseudo-effective +alternative of that theorem applies directly. To meet its scaling +hypothesis, take an effective sufficiently positive ample rational divisor +\(A\) whose general components have sufficiently small coefficients. On +a fixed log resolution carrying \(\boldsymbol M\), these components are +transverse to the boundary and preserve generalized log canonicity, while +their ample class can be chosen large enough that \(D+A\) is nef. The +theorem supplies a terminating MMP starting on \(V\). + +We verify the step types inductively. Suppose \(V_i\) is +\(\Q\)-factorial generalized dlt. By +\cite[Remark~2.15]{TsakanikasXie}, each birational step has the form +\[ + V_i\xrightarrow{g_i}Z_i\xleftarrow{h_i}V_{i+1}, +\] +where \(g_i\) contracts a negative extremal ray and \(h_i\) is small and projective. +Suppose \(g_i\) is divisorial, and write \(R\) for its ray. An exceptional +prime \(E\) has \(E\cdot R<0\): otherwise it would be effective, +exceptional, and \(g_i\)-nef, contrary to negativity. There is only one +exceptional prime. Indeed, a combination of two distinct exceptional +primes can be chosen to have degree zero on \(R\); negativity applied to +both signs of that combination would make it zero. + +We show directly that \(Z_i\) is \(\Q\)-factorial. For a prime divisor +\(T\) on \(Z_i\), let \(\widetilde T\) be its strict transform, choose a +curve \(C\) spanning \(R\), and put +\[ + a=-\frac{\widetilde T\cdot C}{E\cdot C}\in\Q. +\] +Choose a positive integer \(m\) for which \(m(\widetilde T+aE)\) is +Cartier. Its line has degree zero on \(R\), so +\cite[Theorem~1.5]{Xie2022} gives an actual Cartier line \(\mathcal L_T\) +on \(Z_i\) and an isomorphism +\[ + \OO_{V_i}\bigl(m(\widetilde T+aE)\bigr)\simeq g_i^*\mathcal L_T. +\] +On an open set trivializing \(\mathcal L_T\), the Cartier divisor +\(m(\widetilde T+aE)\) is principal upstairs. Pushing this principal +divisor identity down through the birational \(g_i\) gives a principal +divisor \(mT\) on that open set. +Thus \(T\) is rational Cartier. It follows that \(h_i\) is an isomorphism: +the pushforward of an \(h_i\)-ample Cartier divisor is rational Cartier +on \(Z_i\), and its pullback is the original divisor because \(h_i\) +is small. It has degree zero on every contracted curve, so relative +ampleness rules out positive-dimensional fibers; a finite birational +map to the normal \(Z_i\) is an isomorphism. If \(g_i\) is small, +the relatively ample canonical model \(h_i\) is the flip of +\cite[Definition~3.2]{HanLi}. Thus the chosen birational steps are +divisorial contractions and flips. The underlying variety of a +generalized dlt pair is klt, and these steps preserve the +\(\Q\)-factorial generalized dlt category +\cite[Remark~2.3 and Lemma~3.7]{HanLi}. They introduce no prime +divisors on the new models. For completeness, on a common resolution +of one step \(V_i\dashrightarrow V_{i+1}\) carrying \(\boldsymbol M\), +taking the difference of the two generalized discrepancy formulas cancels +their common nef divisor. The resulting difference of the adjoint pullbacks is exceptional over the new +model and anti-nef over it. The negativity lemma makes this difference +effective. Composing these actual rational divisor comparisons gives +\(u^*D=v^*D_m+F\) with \(F\) effective and exceptional over the endpoint. +The endpoints in the two cases are the nef and Mori fiber endpoints supplied +by the terminating program theorem. +\end{proof} diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-rank-one.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-rank-one.tex index f030d9a901a13c7d6d86b986c78b5eba9df5f207..f262f4c3cad7d85942be5109db3fc376c7b7c43c 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-rank-one.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-rank-one.tex @@ -1,1173 +1,1173 @@ -\section{Descent along a fibration} -\label{sec:rank-one} - -Throughout this section we assume Assumption~\ref{asm:log-iitaka}. - -We prove Proposition~\ref{rank:fibration-case}. Its geometric reduction -leads to a fibration whose very general fiber has logarithmic Kodaira -dimension zero. In sufficiently divisible degrees the log pluricanonical -systems on that fiber are one-dimensional, and a relative generating form -determines an adjoint line on the base. We prove the resulting rank-one -proposition by a secondary induction on the positive base dimension: after proving -pseudo-effectivity on the base, we either lower its dimension or lift a nef -line and its entire fixed divisor. The last step proves generation of the -remaining nef line. - -\subsection{Reduction to the rank-one case} - -\begin{proposition}[The rank-one case] -\label{rank:rank-one} -Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume -\(\Ggood_d\) for every \(d0\). If \(a(X)=n\), then \(X\) is Moishezon and -Kähler, hence projective, and Proposition~\ref{proj:good-models} applies. -Otherwise resolve the algebraic reduction as a fibration -\(g:X'\to W\) to a smooth projective variety of dimension \(a(X)\). -Put \(J'=K_{X'}+B'\) for the modified log adjoint. Its restriction to -a very general smooth fiber is pseudo-effective. Indeed, one may choose -a sequence of positive currents in -\(c_1(J')+\varepsilon_j[\omega]\), with analytic singularities and -\(\varepsilon_j\downarrow0\), and restrict all of them outside a -countable union of proper analytic subsets of the base. By -\(\Ggood_{\dim F}\), the restricted adjoint has nonnegative Kodaira -dimension. - -That dimension must be zero. If it were positive, generic coherent -base change would give a divisible \(q\) for which -\(\mathcal E=g_*\OO_{X'}(qJ')\) has a fiber system with a -positive-dimensional image on very general fibers. A sufficiently -large ample twist of the coherent sheaf \(\mathcal E\) is generated at -the generic point of the projective \(W\). Evaluation would therefore -give two sections of -\(\OO_{X'}(qJ')\otimes g^*\OO_W(A)\), for one ample \(A\), whose ratio -is nonconstant on a general \(g\)-fiber. This ratio is a meromorphic -function on \(X'\). Every meromorphic function on \(X'\) factors -through its algebraic reduction, a contradiction. -Proposition~\ref{rank:rank-one} now applies. - -Suppose next that \(a(X)=0\), and resolve the fibration in the -hypothesis as \(f:(X',B')\to Y\). The base may be replaced by a smooth -compact Kähler model, and \(a(Y)=0\), since its meromorphic functions -pull back to \(X'\). Write \(d=\dim X'-\dim Y>0\). -The same restriction argument and \(\Ggood_d\) show that -\[ - \kappa(F,K_F+B'_F)\geq0 -\] -on very general smooth fibers. If this integer is \(j0\), then \(K_S+a_0P_S\) is big for some rational -\(a_0>0\); if \(S\) is a point, \(M\sim_{\Q}0\). -There is a rational divisor \(A_*\) on \(X\) giving an actual rational -line identity -\begin{equation} - J\sim_{\Q}g^*H+A_*. -\label{rank:line-comparison} -\end{equation} -These data satisfy the following properties. - -\begin{enumerate} -\item The horizontal part \(A_{*,\mathrm{hor}}\) is effective. -On a very general smooth fiber it is the zero divisor of a relative -generating section of a divisible multiple of \(K_F+B_F\), divided -by that multiple. - -\item For each prime \(D\subset W\), let \(E\) run over the primes of -\(X\) dominating \(D\), and put \(a_E=\ord_E(g^*D)\). -Then -\begin{equation} - (A_*)_E\geq0,\qquad - \min_{E\to D}\frac{(A_*)_E}{a_E}=0. -\label{rank:zero-minimum} -\end{equation} -There is no assertion here about the sign at a prime whose image has -codimension at least two. - -\item For all degrees \(q\) divisible by one positive integer, -multiplication by the relative generating form identifies -\[ - H^0(W,qH)\simeq H^0(X,qJ) -\] -and preserves ratios of sections. In the same degrees, for every -proper holomorphic \(h:W\to V\), there is a natural injection -\begin{equation} - (hg)_*\OO_X(qJ)\ \lhook\joinrel\longrightarrow\ - h_*\OO_W(qH). -\label{rank:relative-injection} -\end{equation} - -\item Let \(\tau:W'\to W\) be a further smooth modification included -in another diagram with the preceding properties over the same reference -\(X_0\), using the same source-boundary convention. Let \(H'\) be the -base line produced by that diagram. Then, as actual rational lines, -\begin{equation} - H'\sim_{\Q}\tau^*H+E_H, - \qquad E_H\geq0\ \text{\(\tau\)-exceptional}. -\label{rank:higher-base} -\end{equation} -\end{enumerate} -\end{lemma} - -\begin{proof} -The flattening construction of \cite[Lemma~2.4]{CI} gives the stated -reference property, and preserves it after any finite sequence of -further base modifications. Its source boundary is the one in -\cite[Lemma~2.1]{CI}. We may resolve the discriminant and the -horizontal divisor of the relative generator so that all relevant -strata are smooth over the dense smooth log locus. Proposition~2.7 -of \cite{CI} then gives \(T\) and a rational parabolic Hodge line -\(M\). A positive multiple of that line is the highest rank-one -Hodge step of a complex summand of a pure real-polarizable variation -with an integral lattice. Theorem~3.1 of \cite{CI} gives the -factorization \(M=p^*P_S\) and the stated positivity, after a further -modification and a positive rational rescaling. These are identities -in \(\Pic\otimes\Q\), not merely identities of classes. - -Here is the order computation that gives -Equations~\eqref{rank:line-comparison}--\eqref{rank:relative-injection}. -Choose a degree \(m\) for which -\(g_*\OO_X(m(K_{X/W}+B))\) has generic rank one and all boundary -denominators are cleared. Its reflexive hull is a line on the smooth -\(W\). Let \(s\) be a meromorphic relative generating form in a -local frame of this line. At the generic point of a prime \(D\), -choose a nonvanishing ordinary base volume \(\omega_W\), and set -\[ - l_E=\frac1m\ord_E(s\wedge g^*\omega_W^m),\qquad - \delta_E=B_E,\qquad - \alpha_D=\min_{E\to D}\frac{l_E+\delta_E}{a_E}. -\] -The relative-times-base identification in this formula is taken in -the \(m\)-th canonical tensor power. The source is resolved over -the generic point of \(D\), so all the primes needed for the -divisorial test occur in this minimum. In the notation of the -proof of \cite[Proposition~2.7]{CI}, the parabolic order is -\[ - \beta_D=\min_{E\to D}\frac{l_E+1-a_E}{a_E}, - \qquad T_D=\alpha_D-\beta_D. -\] -Thus the corresponding local frame of \(T+M\) has order -\(\alpha_D\). Comparing its pullback with the log adjoint form -defines \(A_*\), with -\begin{equation} - (A_*)_E=l_E+\delta_E-a_E\alpha_D . -\label{rank:vertical-order} -\end{equation} -Changing the generating form multiplies it by a meromorphic base -function. This changes both \(\alpha_D\) and \(\beta_D\) by its -normalized base order and leaves the comparison invariant. The local -comparisons therefore give the rational divisor and the line identity -globally. Equation~\eqref{rank:vertical-order} proves -Equation~\eqref{rank:zero-minimum}. Horizontally the comparison is -precisely the zero divisor of the fiberwise log section, so it is -effective and has the asserted restriction. - -The same order computation proves the relative injection. -In a divisible degree \(q\), a meromorphic base coefficient -\(\varphi\) gives a regular base section at \(D\) exactly when -\[ - \ord_D(\varphi)+q\alpha_D\geq0. -\] -Its corresponding upstairs orders at the primes \(E\to D\) are -\[ - a_E\ord_D(\varphi)+q(l_E+\delta_E). -\] -Their simultaneous nonnegativity is equivalent to the preceding -inequality. Conversely, an upstairs section restricts to a multiple -of the generator on general connected fibers, so its ratio with the -generator descends meromorphically to \(W\). The order inequalities -make it regular outside codimension two on \(W\), and normality -extends it. Consequently -\[ - g_*\OO_X(qJ)\ \lhook\joinrel\longrightarrow\ - \OO_W(qH). -\] -Applying the left exact functor \(h_*\) proves -Equation~\eqref{rank:relative-injection}. The reverse global -comparison follows from the reference property: its only initially -untested poles are on \(g\)-contracted primes, all exceptional over -\(X_0\); pushing to \(X_0\), extending in codimension two, and pulling -back with the reduced-exceptional boundary removes them. This is -the global comparison in \cite[Proposition~2.7]{CI}. - -Finally consider \(\tau:W'\to W\). The parabolic rational line pulls -back under SNC modifications, as in the construction preceding -\cite[Theorem~3.1]{CI}. The coefficient \(T_D\) at an old strict -transform is unchanged. Indeed the sheaf of regular relative -adjoint forms inside the common meromorphic generator line is -unchanged over the generic point of \(D\) by the log modification -formula. Equivalently, the old inequalities above already test all -vertical primes there; in SNC coordinates logarithmic pullback -preserves those inequalities, so new source exceptional primes -impose no stronger one. This keeps \(\alpha_D\) unchanged in -compatible frames, while parabolic pullback keeps \(\beta_D\) -unchanged. Over a new \(\tau\)-exceptional prime every source prime -is exceptional over \(X_0\): a pre-existing one had image of -codimension at least two on \(W\), and a new one is exceptional -by construction. All therefore have boundary coefficient one. -The formula for \(T_D=\alpha_D-\beta_D\) then gives coefficient one -on that new base prime. Thus -\[ - T'=\tau_*^{-1}T+\operatorname{Exc}(\tau)_{\mathrm{red}}. -\] -The usual log discrepancy formula for the SNC pair \((W,T)\) -now gives \(K_{W'}+T'=\tau^*(K_W+T)+E_H\), with \(E_H\) -effective and exceptional. Adding the pulled-back parabolic line -proves Equation~\eqref{rank:higher-base}. -\end{proof} - -\begin{lemma}[Pseudo-effectivity of the base line] -\label{rank:base-psef} -In the setting of Lemma~\ref{rank:comparison}, suppose -\(\dim X=n\), \(0<\dim W0\), Equation~\eqref{rank:first-nef-endpoint} preserves -the divisible section spaces. For \(\kappa(H)=k\) the nef -\(H_{\mathrm{nef}}\) is big, and the second alternative holds with -\(W_m=W_{\mathrm{nef}}\) and \(H_m=H_{\mathrm{nef}}\). If -\(0<\kappa(H)0\). For a rational \(c\in(0,1)\), put -\(H_c=K_W+T+cM\). We claim -\begin{equation} - H_c\ \text{is not pseudo-effective} - \quad(00\) for which \(H-\delta p^*U\) has an -effective representative, forcing \(\kappa(H)\geq\dim S>0\). -Choose a rational \(a>\max\{1,a_0\}\) and a small -rational \(\lambda>0\) such that \(K_S+aP_S-\lambda U\) is big. -This is possible because \(P_S\) is nef and -\(K_S+a_0P_S\) is big. Choose -\[ - 0<\eta<\lambda\frac{1-c}{a-1} - \quad\text{rational}. -\] -The rational line \(cP_S+\eta U\) is ample. A sufficiently high -general member divided by its degree pulls back to a rational -boundary transverse to \(T\) with subunit coefficient. Applying -the nonvanishing part of Proposition~\ref{proj:good-models} to -this ordinary projective log pair gives -\[ - E_c\sim_{\Q}H_c+\eta p^*U,\qquad E_c\geq0. -\] -A nonzero section defining \(E_c\) restricts nontrivially to very -general \(p\)-fibers in a fixed divisible degree. The hypotheses -of the weak-effectivity statement \cite[Lemma~3.3]{CI} are now -satisfied: the source is smooth in class \(\mathcal C\), \(T\) -is rational SNC, the base \(S\) is smooth projective, and a -relative log system is nonzero. Applied to the big line -\(K_S+aP_S-\lambda U\), it gives -\[ - E_a\sim_{\Q}K_W+T+aM-\lambda p^*U,\qquad E_a\geq0. -\] -For -\[ - \theta=\frac{1-c}{a-c},\qquad - \delta=\theta\lambda-(1-\theta)\eta>0, -\] -the rational interpolation is -\[ - (1-\theta)E_c+\theta E_a - \sim_{\Q}H-\delta p^*U. -\] -Multiplying pullbacks of sections of \(U\) by a section of the -effective left-hand side yields -\(\kappa(H)\geq\kappa(p^*U)=\dim S>0\). -This contradicts Equation~\eqref{rank:nontrivial-nef-data} and -proves Equation~\eqref{rank:lowered-nonpsef}. - -We next run a program to fiber type while descending the actual nef line -\(H_{\mathrm{nef}}\) through every contraction. Choose the common -resolution in Equation~\eqref{rank:first-nef-endpoint} to determine the nef -data. The negativity lemma gives -\[ - F_M:=\sigma^*M_{\mathrm{nef}}-\mu^*M\geq0 - \quad\text{\(\sigma\)-exceptional}. -\] -Indeed its negative is \(\sigma\)-nef and exceptional. For every -rational \(00\) with -\(rH_{\mathrm{nef}}\) Cartier, and choose an integer \(d_0>2kr\). -This coefficient will force every ray of the second program to have zero -degree for the descended nef line. Decreasing the nef data to -\((1-\varepsilon)\boldsymbol M\) preserves generalized dlt -singularities: on the displayed resolution the crepant boundary -changes by \(-\varepsilon F_M\), so discrepancies do not decrease. -Add \(d_0H_{\mathrm{nef}}\), determined as nef data on -\(W_{\mathrm{nef}}\). It changes no discrepancies, and the resulting -adjoint is -\[ - (1+d_0)H_{\mathrm{nef}}-\varepsilon M_{\mathrm{nef}} - =(1+d_0)\left( - H_{\mathrm{nef}}-\frac{\varepsilon}{1+d_0}M_{\mathrm{nef}}\right). -\] -It is not pseudo-effective by the preceding paragraph. The -non-pseudo-effective case of Proposition~\ref{proj:generalized-mmp} -therefore gives a chosen terminating program to a Mori fiber -contraction in the \(\Q\)-factorial generalized dlt category. - -At any stage of this second program, let \(H_i\) be the descended rational -line, assuming inductively that it is nef and \(rH_i\) is Cartier, and write -\[ - G_i=K_{W_i}+T_i+(1-\varepsilon)M_i. -\] -An extremal ray negative for \(G_i+d_0H_i\) is negative for \(G_i\). -The pair with adjoint \(G_i\) is still generalized dlt: the added -nef b-divisor \(d_0H_i\) descends to the current model, so removing -it changes no discrepancy. The generalized length bound -\cite[Proposition~3.13]{HanLi} gives a curve \(C\) -on that ray with \(0<-G_i\cdot C\leq2k\). If \(H_i\cdot C>0\), -Cartier integrality would give -\[ - \frac{d_0}{r}\leq d_0H_i\cdot C - <-G_i\cdot C\leq2k, -\] -a contradiction. Thus \(H_i\cdot C=0\). -The exact contraction statement \cite[Theorem~1.5]{Xie2022} -descends the Cartier line \(rH_i\) along this contraction as an -actual Cartier line. In a flip we pull that same line to the -flipped side. The descended line is nef: every curve on the -contraction base has a curve above it mapping with positive degree, -so its degree is nonnegative; pullback preserves nefness. -Consequently \(r\) and nefness persist, and the argument applies -inductively to all steps and to the final contraction. Each step -is crepant for \(H_i\). - -Let \(W_m\) be the last birational model of this second program and let -\(H_m\) be the descended line on it. Crepancy for the \(H_i\), together -with the absence of extraction, turns -Equation~\eqref{rank:first-nef-endpoint} into -Equation~\eqref{rank:base-endpoint} on a common resolution, with an -effective divisor exceptional over \(W_m\). -The final contraction \(h:W_m\to V\) has connected fibers and, -by the same exact descent, -\[ - H_m\sim_{\Q}h^*H_V -\] -for a rational line \(H_V\) on the normal projective \(V\). -If \(V\) is a point, the second alternative holds with -\(H_m\sim_{\Q}0\). - -Assume \(\dim V>0\). To obtain the first alternative, it remains to show -that the log adjoint on the new very general fibers has Kodaira dimension -zero. Resolve the base program by -\(\mu:W''\to W\) and \(\nu:W''\to W_m\), and prepare the source -over \(W''\), calling it \(g'':(X'',B'')\to W''\). The diagram -we use is -\[ -\begin{tikzcd} - X'' \arrow[r,"g''"] & W'' \arrow[r,"\nu"] \arrow[d,"\mu"'] & - W_m \arrow[r,"h"] & V\\ - & W -\end{tikzcd} -\] -with composite \(f= h\nu g''\). -Equation~\eqref{rank:higher-base} and -Equation~\eqref{rank:base-endpoint} give -\[ - H''\sim_{\Q}\nu^*H_m+E'', - \qquad E''\geq0\ \text{\(\nu\)-exceptional}. -\] -To see the exceptionality, the error in -Equation~\eqref{rank:base-endpoint} is already exceptional over -\(W_m\), and every \(\mu\)-exceptional prime is exceptional -over \(W_m\) as well: otherwise \(W_m\) would extract a divisor -over \(W\). -For every degree divisible by one fixed integer, -Equation~\eqref{rank:relative-injection}, exceptional descent, -and projection formula now yield -\begin{align*} - f_*\OO_{X''}(q(K_{X''}+B'')) - &\lhook\joinrel\longrightarrow - (h\nu)_*\OO_{W''}(qH'')\\ - &=h_*\OO_{W_m}(qH_m)=\OO_V(qH_V). -\end{align*} -Here \(\nu_*\OO_{W''}(qE'')=\OO_{W_m}\) by normality and -\(h_*\OO_{W_m}=\OO_V\). Generic coherent base change, -simultaneously for the countably many divisible degrees, bounds -the dimension of every such system on a very general \(f\)-fiber -by one. The log adjoint on that fiber is pseudo-effective by -restriction of a countable sequence of perturbed currents with -analytic singularities. Its dimension is less than \(n\), so -the lower-dimensional \(\Ggood\) supplies a nonzero section in -some divisible degree. Taking a common multiple shows that the -fiber log Kodaira dimension is zero. All maps in the composite -have connected fibers; resolving \(V\) and the source keeps this -property and gives a smooth projective base of dimension -\(0<\dim V0\), since at least one component of -\(Q\) has image dimension \(l\). Thus \(Q\) is not in the radical, -and its square is strictly negative, contrary to -Equation~\eqref{rank:nonnegative-square}. - -It remains to treat \(l=k-1\). Terms from primes whose image has -smaller dimension vanish in this form, as do cross terms from -different divisorial images. Fix a prime divisor \(D\subset Y\) -and take all the primes \(E_1,\ldots,E_s\subset X\) dominating it. -Write \(a_i=\ord_{E_i}(b^*D)>0\) and -\(C_{ij}=\langle E_i,E_j\rangle_{k-1}\). -The divisor \(D\) is rational Cartier, also when \(Y=W_m\). -Consequently -\[ - C_{ij}\geq0\ (i\ne j),\qquad - \sum_j C_{ij}a_j=0. -\] -The first assertion is positivity of the proper intersection of -distinct effective primes. The second pairs \(E_i\) with the -pulled-back divisor \(D\); the additional base factor makes the -intersection vanish by dimension. Components of \(b^*D\) -mapping into a proper subset of \(D\) make no contribution. -The graph with edges \(C_{ij}>0\) is connected. Indeed over a -very general point of \(D\) the fiber is connected, all its points -lie in the components \(E_i\), and an intersection contributes a -positive entry precisely when it dominates \(D\). - -For any real vector \(x=(x_i)\), the two displayed properties give -the exact identity -\[ - x^{\mathsf T}Cx - =-\sum_{i1\) for which -\(uL|_{D_Z}\) is generated, and consider the integral line -\begin{equation} - \mathscr N - :=\OO_U(u\ell+E-C) - =\OO_U\bigl(K_U+\Delta_F+(u-1)\ell\bigr). -\label{rank:vanishing-line} -\end{equation} -Although the expression on the right uses rational divisors, its -sum is the integral line on the left. The restriction sequence is -\[ - 0\longrightarrow\mathscr N - \longrightarrow\OO_U(u\ell+E) - \longrightarrow\OO_C(u\ell+E)\longrightarrow0. -\] -The direct image of the last sheaf is supported on the proper analytic -subset \(f(C)\), so it is torsion. Consequently, the two assertions -\begin{equation} - R^1f_*\mathscr N\ \text{torsion-free}, - \qquad H^1(Y,f_*\mathscr N)=0 -\label{rank:vanishing} -\end{equation} -will give surjectivity on global sections: torsion-freeness makes the -connecting map to \(R^1f_*\mathscr N\) zero, and the \(H^1\)-vanishing -then removes the obstruction to lifting a global section. - -We prove Equation~\eqref{rank:vanishing} using analytic injectivity. -Equation~\eqref{rank:big-pullback} identifies -\(\mathscr N\otimes K_U^{-1}\) with -\(\Delta_F+(u-1)f^*H_Y\) in \(\Pic(U)\otimes\Q\); this comparison -need not be an isomorphism of the corresponding integral lines. Fix a -very ample line \(A_Y\) on \(Y\) and its Fubini--Study metric. -A Kodaira decomposition of the big nef \(H_Y\), given arbitrarily -small weight as in the proof of Lemma~\ref{rank:lift}, writes -\((u-1)H_Y\) as a positive rational multiple of \(A_Y\) plus -an effective rational divisor whose fixed singular contribution -is arbitrarily small and whose remaining contribution is a -general divided ample member. Choose the small weight below -the log canonical threshold of its pullback relative to the -klt SNC boundary \(\Delta_F\), and choose the ample member generally. -Then the resulting boundary on \(U\) is klt. Choose a positive -integer \(q_0\) clearing all its denominators and realizing both the -chosen Kodaira decomposition and the preceding rational-line comparison -as line isomorphisms, in particular -\[ - (\mathscr N\otimes K_U^{-1})^{\otimes q_0} - \simeq \OO_U(q_0\Delta_F)\otimes - f^*\OO_Y\bigl(q_0(u-1)H_Y\bigr). -\] -The divisor metrics and the Fubini--Study metric -define a singular Hermitian metric on -\((\mathscr N\otimes K_U^{-1})^{\otimes q_0}\). Its \(q_0\)-th -root is a metric on the actual line \(\mathscr N\otimes K_U^{-1}\), -with multiplier ideal \(\OO_U\) and curvature dominating a positive -multiple of the pullback Fubini--Study form. The same statements hold after every -nonnegative twist by \(f^*A_Y\). - -The injectivity theorem \cite[Theorem~A]{FujinoMatsumura} applies -on the compact Kähler \(U\): multiplication by the pullback of -any nonzero section of a positive power of \(A_Y\) is injective -on -\[ - H^1(U,\mathscr N\otimes f^*A_Y^j) - \quad(j\geq0) -\] -into the correspondingly higher twist. Its assumptions are -exactly the semipositive smooth metric on the multiplying line, -the preceding positive lower curvature bound, and the trivial -multiplier ideal. -This injectivity implies Equation~\eqref{rank:vanishing}. -For the second assertion, the Leray edge map injects -\(H^1(Y,f_*\mathscr N)\) into \(H^1(U,\mathscr N)\). -Multiplication by a section of a sufficiently high \(A_Y\)-power -sends this subspace to zero, since Serre vanishing kills -\(H^1(Y,f_*\mathscr N\otimes A_Y^j)\) for large \(j\). -Injectivity upstairs forces the original subspace to be zero. -For the first assertion, suppose \(R^1f_*\mathscr N\) has a -nonzero torsion subsheaf. A section of a sufficiently high -power of \(A_Y\) annihilates a nonzero such subsheaf. -After a further sufficiently large twist it has a nonzero -global section, and Serre vanishing and Leray identify that -section with a nonzero class in -\(H^1(U,\mathscr N\otimes f^*A_Y^j)\). -Multiplication annihilates this class, again contradicting -injectivity. This proves both assertions. - -The resulting surjectivity extends the floor sections as follows. -Pull any section of \(uL|_{D_Z}\) -to \(C\), multiply by the canonical section of \(E|_C\), and -extend by this surjectivity. The extended section descends to -\(Z\), since \(r_*\OO_U(E)=\OO_Z\) by exceptionality and -normality. On the strict transforms of the components of -\(D_Z\), division by the canonical section of \(E\) recovers -the prescribed section. The descended restriction therefore -equals it on the whole reduced \(D_Z\), since equality holds -generically on every component. Since \(uL|_{D_Z}\) is generated, -this proves Equation~\eqref{rank:floor-avoidance}. - -We finish by showing that this stable base locus is empty. The -base loci of factorial Cartier multiples form a descending -sequence of compact analytic subsets and hence stabilize. -Sections exist because \(H_Y\) is big and -Equation~\eqref{rank:big-pullback} descends their pullbacks. -Choose a divisible degree \(v\) whose base locus is -\(\mathbf B(L)\), and let \(\mathfrak b\) be its base ideal. -If \(\mathbf B(L)\ne\varnothing\), the ideal is a unit near -\(D_Z\), while \((Z,\Delta_Z)\) is klt outside \(D_Z\). -Its log canonical threshold -\[ - t=\operatorname{lct}_{(Z,\Delta_Z)}(\mathfrak b) -\] -is therefore a positive rational number, attained by a divisor -whose center is contained in \(\mathbf B(L)\). -This follows directly on a simultaneous log resolution of the -pair and the ideal: the threshold is the minimum of the positive -rational discrepancies divided by the positive integral ideal -orders. - -Choose an integer \(s>t\) and general divisors -\(D_1,\ldots,D_s\in|vL|\). On the same resolution their fixed -part is the divisor of \(\mathfrak b\); their free transforms -are jointly transverse to the SNC data. Hence -\[ - \left(Z,\ \Delta_Z+\frac{t}{s}\sum_{i=1}^s D_i\right) -\] -is log canonical: the fixed part is at its log canonical -threshold and the free components have coefficients \(t/s<1\). -There is a new lc place centered in \(\mathbf B(L)\), and the -new adjoint satisfies the actual identity -\[ - L_{\mathrm{new}}\sim_{\Q}(1+tv)L. -\] -Apply \cite[Theorem~3.3]{HLX} to this compact Kähler lc rational -pair. It gives a projective strongly \(\Q\)-factorial dlt -modification \(d:Z'\to Z\). Since the pair is lc, all extracted -divisors have log discrepancy zero in our convention, so the -modification is crepant. The source is again compact Kähler. -Choose its defining dlt log resolution, with exceptional crepant -coefficients strictly below one and an isomorphism at the general -point of every lc stratum; resolving the remaining data away from -those points gives the resolution required by -Definition~\ref{def:lc-strata-resolution}. The floor on \(Z'\) has a point over the center -of this lc place, hence over \(\mathbf B(L)\). -The new nef adjoint is \(d^*L_{\mathrm{new}}\). On a common -resolution with \(U\), it satisfies -Equation~\eqref{rank:big-pullback} with the big nef line -\((1+tv)H_Y\). The previously proved assertion -\eqref{rank:floor-avoidance} therefore makes its stable base locus -disjoint from its floor. On the -other hand, normality and projection formula identify all -divisible sections under \(d\), and give -\[ - \mathbf B(d^*L_{\mathrm{new}}) - =d^{-1}\mathbf B(L). -\] -This locus contains the stated floor point, a contradiction. -Thus \(\mathbf B(L)=\varnothing\); factorial stabilization -provides an actual globally generated multiple of \(L\). -\end{proof} - -\subsection{Completion of the rank-one case} - -\begin{proof}[Proof of Proposition~\ref{rank:rank-one}] -By Proposition~\ref{bir:resolution-transfer} we can begin on the -resolved source in the statement. We use a secondary induction on -the positive base dimension \(k\). Apply -Lemma~\ref{rank:comparison} and then Lemma~\ref{rank:base-psef} -to obtain the prepared data and the pseudo-effective line \(H\). - -If \(a(W)=0\), Lemma~\ref{rank:zero-base} and the first case of -Lemma~\ref{rank:lift} give -\(J\sim_{\Q}N(J)\), which is already the required decomposition -with trivial positive part. If \(W\) is projective, apply -Lemma~\ref{rank:base-reduction}. Its first alternative has a -strictly smaller positive base dimension, so the secondary induction -and birational transfer finish the proof. In its second alternative, -take a common smooth base resolution in -Equation~\eqref{rank:base-endpoint} and prepare the source over it. -Equation~\eqref{rank:higher-base} adds an effective divisor -exceptional over the old base. It is also exceptional over \(W_m\), -because the base program extracted no divisors. Thus the second -case of Lemma~\ref{rank:lift} applies. - -We have obtained -\[ - J\sim_{\Q}P_0+A,\qquad A=N(J)\geq0, -\] -with \(P_0\) nef. A rationally trivial \(P_0\) finishes the proof. -In the remaining case \(P_0=b^*H_m\), where \(b:X\to W_m\) -is proper and \(H_m\) is big and nef on the projective \(W_m\). -Apply Proposition~\ref{prog:contract-negative} to this -nef-plus-negative presentation. It gives a globally strongly -\(\Q\)-factorial dlt compact Kähler model \((Z,\Delta_Z)\) with nef -adjoint \(L_Z\). The ordinary dlt endpoint has the resolution property -of Definition~\ref{def:lc-strata-resolution}. On a common smooth resolution -\(\mu:U\to X\), \(r:U\to Z\) its exact positive-line comparison is -\[ - r^*L_Z\sim_{\Q}\mu^*P_0 - =(b\mu)^*H_m . -\] -Proposition~\ref{rank:big-nef-generation} makes \(L_Z\) semiample. -Hence \(\mu^*P_0\) is semiample. Lemma~\ref{bir:pullup} identifies -\(\mu^*A=N(\mu^*J)\), so this is precisely \(\Ggood_n\) on \(U\). -Finally Proposition~\ref{bir:resolution-transfer} transfers the -decomposition back through all the prepared source modifications. -\end{proof} +\section{Descent along a fibration} +\label{sec:rank-one} + +Throughout this section we assume Assumption~\ref{asm:log-iitaka}. + +We prove Proposition~\ref{rank:fibration-case}. Its geometric reduction +leads to a fibration whose very general fiber has logarithmic Kodaira +dimension zero. In sufficiently divisible degrees the log pluricanonical +systems on that fiber are one-dimensional, and a relative generating form +determines an adjoint line on the base. We prove the resulting rank-one +proposition by a secondary induction on the positive base dimension: after proving +pseudo-effectivity on the base, we either lower its dimension or lift a nef +line and its entire fixed divisor. The last step proves generation of the +remaining nef line. + +\subsection{Reduction to the rank-one case} + +\begin{proposition}[The rank-one case] +\label{rank:rank-one} +Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume +\(\Ggood_d\) for every \(d0\). If \(a(X)=n\), then \(X\) is Moishezon and +Kähler, hence projective, and Proposition~\ref{proj:good-models} applies. +Otherwise resolve the algebraic reduction as a fibration +\(g:X'\to W\) to a smooth projective variety of dimension \(a(X)\). +Put \(J'=K_{X'}+B'\) for the modified log adjoint. Its restriction to +a very general smooth fiber is pseudo-effective. Indeed, one may choose +a sequence of positive currents in +\(c_1(J')+\varepsilon_j[\omega]\), with analytic singularities and +\(\varepsilon_j\downarrow0\), and restrict all of them outside a +countable union of proper analytic subsets of the base. By +\(\Ggood_{\dim F}\), the restricted adjoint has nonnegative Kodaira +dimension. + +That dimension must be zero. If it were positive, generic coherent +base change would give a divisible \(q\) for which +\(\mathcal E=g_*\OO_{X'}(qJ')\) has a fiber system with a +positive-dimensional image on very general fibers. A sufficiently +large ample twist of the coherent sheaf \(\mathcal E\) is generated at +the generic point of the projective \(W\). Evaluation would therefore +give two sections of +\(\OO_{X'}(qJ')\otimes g^*\OO_W(A)\), for one ample \(A\), whose ratio +is nonconstant on a general \(g\)-fiber. This ratio is a meromorphic +function on \(X'\). Every meromorphic function on \(X'\) factors +through its algebraic reduction, a contradiction. +Proposition~\ref{rank:rank-one} now applies. + +Suppose next that \(a(X)=0\), and resolve the fibration in the +hypothesis as \(f:(X',B')\to Y\). The base may be replaced by a smooth +compact Kähler model, and \(a(Y)=0\), since its meromorphic functions +pull back to \(X'\). Write \(d=\dim X'-\dim Y>0\). +The same restriction argument and \(\Ggood_d\) show that +\[ + \kappa(F,K_F+B'_F)\geq0 +\] +on very general smooth fibers. If this integer is \(j0\), then \(K_S+a_0P_S\) is big for some rational +\(a_0>0\); if \(S\) is a point, \(M\sim_{\Q}0\). +There is a rational divisor \(A_*\) on \(X\) giving an actual rational +line identity +\begin{equation} + J\sim_{\Q}g^*H+A_*. +\label{rank:line-comparison} +\end{equation} +These data satisfy the following properties. + +\begin{enumerate} +\item The horizontal part \(A_{*,\mathrm{hor}}\) is effective. +On a very general smooth fiber it is the zero divisor of a relative +generating section of a divisible multiple of \(K_F+B_F\), divided +by that multiple. + +\item For each prime \(D\subset W\), let \(E\) run over the primes of +\(X\) dominating \(D\), and put \(a_E=\ord_E(g^*D)\). +Then +\begin{equation} + (A_*)_E\geq0,\qquad + \min_{E\to D}\frac{(A_*)_E}{a_E}=0. +\label{rank:zero-minimum} +\end{equation} +There is no assertion here about the sign at a prime whose image has +codimension at least two. + +\item For all degrees \(q\) divisible by one positive integer, +multiplication by the relative generating form identifies +\[ + H^0(W,qH)\simeq H^0(X,qJ) +\] +and preserves ratios of sections. In the same degrees, for every +proper holomorphic \(h:W\to V\), there is a natural injection +\begin{equation} + (hg)_*\OO_X(qJ)\ \lhook\joinrel\longrightarrow\ + h_*\OO_W(qH). +\label{rank:relative-injection} +\end{equation} + +\item Let \(\tau:W'\to W\) be a further smooth modification included +in another diagram with the preceding properties over the same reference +\(X_0\), using the same source-boundary convention. Let \(H'\) be the +base line produced by that diagram. Then, as actual rational lines, +\begin{equation} + H'\sim_{\Q}\tau^*H+E_H, + \qquad E_H\geq0\ \text{\(\tau\)-exceptional}. +\label{rank:higher-base} +\end{equation} +\end{enumerate} +\end{lemma} + +\begin{proof} +The flattening construction of \cite[Lemma~2.4]{CI} gives the stated +reference property, and preserves it after any finite sequence of +further base modifications. Its source boundary is the one in +\cite[Lemma~2.1]{CI}. We may resolve the discriminant and the +horizontal divisor of the relative generator so that all relevant +strata are smooth over the dense smooth log locus. Proposition~2.7 +of \cite{CI} then gives \(T\) and a rational parabolic Hodge line +\(M\). A positive multiple of that line is the highest rank-one +Hodge step of a complex summand of a pure real-polarizable variation +with an integral lattice. Theorem~3.1 of \cite{CI} gives the +factorization \(M=p^*P_S\) and the stated positivity, after a further +modification and a positive rational rescaling. These are identities +in \(\Pic\otimes\Q\), not merely identities of classes. + +Here is the order computation that gives +Equations~\eqref{rank:line-comparison}--\eqref{rank:relative-injection}. +Choose a degree \(m\) for which +\(g_*\OO_X(m(K_{X/W}+B))\) has generic rank one and all boundary +denominators are cleared. Its reflexive hull is a line on the smooth +\(W\). Let \(s\) be a meromorphic relative generating form in a +local frame of this line. At the generic point of a prime \(D\), +choose a nonvanishing ordinary base volume \(\omega_W\), and set +\[ + l_E=\frac1m\ord_E(s\wedge g^*\omega_W^m),\qquad + \delta_E=B_E,\qquad + \alpha_D=\min_{E\to D}\frac{l_E+\delta_E}{a_E}. +\] +The relative-times-base identification in this formula is taken in +the \(m\)-th canonical tensor power. The source is resolved over +the generic point of \(D\), so all the primes needed for the +divisorial test occur in this minimum. In the notation of the +proof of \cite[Proposition~2.7]{CI}, the parabolic order is +\[ + \beta_D=\min_{E\to D}\frac{l_E+1-a_E}{a_E}, + \qquad T_D=\alpha_D-\beta_D. +\] +Thus the corresponding local frame of \(T+M\) has order +\(\alpha_D\). Comparing its pullback with the log adjoint form +defines \(A_*\), with +\begin{equation} + (A_*)_E=l_E+\delta_E-a_E\alpha_D . +\label{rank:vertical-order} +\end{equation} +Changing the generating form multiplies it by a meromorphic base +function. This changes both \(\alpha_D\) and \(\beta_D\) by its +normalized base order and leaves the comparison invariant. The local +comparisons therefore give the rational divisor and the line identity +globally. Equation~\eqref{rank:vertical-order} proves +Equation~\eqref{rank:zero-minimum}. Horizontally the comparison is +precisely the zero divisor of the fiberwise log section, so it is +effective and has the asserted restriction. + +The same order computation proves the relative injection. +In a divisible degree \(q\), a meromorphic base coefficient +\(\varphi\) gives a regular base section at \(D\) exactly when +\[ + \ord_D(\varphi)+q\alpha_D\geq0. +\] +Its corresponding upstairs orders at the primes \(E\to D\) are +\[ + a_E\ord_D(\varphi)+q(l_E+\delta_E). +\] +Their simultaneous nonnegativity is equivalent to the preceding +inequality. Conversely, an upstairs section restricts to a multiple +of the generator on general connected fibers, so its ratio with the +generator descends meromorphically to \(W\). The order inequalities +make it regular outside codimension two on \(W\), and normality +extends it. Consequently +\[ + g_*\OO_X(qJ)\ \lhook\joinrel\longrightarrow\ + \OO_W(qH). +\] +Applying the left exact functor \(h_*\) proves +Equation~\eqref{rank:relative-injection}. The reverse global +comparison follows from the reference property: its only initially +untested poles are on \(g\)-contracted primes, all exceptional over +\(X_0\); pushing to \(X_0\), extending in codimension two, and pulling +back with the reduced-exceptional boundary removes them. This is +the global comparison in \cite[Proposition~2.7]{CI}. + +Finally consider \(\tau:W'\to W\). The parabolic rational line pulls +back under SNC modifications, as in the construction preceding +\cite[Theorem~3.1]{CI}. The coefficient \(T_D\) at an old strict +transform is unchanged. Indeed the sheaf of regular relative +adjoint forms inside the common meromorphic generator line is +unchanged over the generic point of \(D\) by the log modification +formula. Equivalently, the old inequalities above already test all +vertical primes there; in SNC coordinates logarithmic pullback +preserves those inequalities, so new source exceptional primes +impose no stronger one. This keeps \(\alpha_D\) unchanged in +compatible frames, while parabolic pullback keeps \(\beta_D\) +unchanged. Over a new \(\tau\)-exceptional prime every source prime +is exceptional over \(X_0\): a pre-existing one had image of +codimension at least two on \(W\), and a new one is exceptional +by construction. All therefore have boundary coefficient one. +The formula for \(T_D=\alpha_D-\beta_D\) then gives coefficient one +on that new base prime. Thus +\[ + T'=\tau_*^{-1}T+\operatorname{Exc}(\tau)_{\mathrm{red}}. +\] +The usual log discrepancy formula for the SNC pair \((W,T)\) +now gives \(K_{W'}+T'=\tau^*(K_W+T)+E_H\), with \(E_H\) +effective and exceptional. Adding the pulled-back parabolic line +proves Equation~\eqref{rank:higher-base}. +\end{proof} + +\begin{lemma}[Pseudo-effectivity of the base line] +\label{rank:base-psef} +In the setting of Lemma~\ref{rank:comparison}, suppose +\(\dim X=n\), \(0<\dim W0\), Equation~\eqref{rank:first-nef-endpoint} preserves +the divisible section spaces. For \(\kappa(H)=k\) the nef +\(H_{\mathrm{nef}}\) is big, and the second alternative holds with +\(W_m=W_{\mathrm{nef}}\) and \(H_m=H_{\mathrm{nef}}\). If +\(0<\kappa(H)0\). For a rational \(c\in(0,1)\), put +\(H_c=K_W+T+cM\). We claim +\begin{equation} + H_c\ \text{is not pseudo-effective} + \quad(00\) for which \(H-\delta p^*U\) has an +effective representative, forcing \(\kappa(H)\geq\dim S>0\). +Choose a rational \(a>\max\{1,a_0\}\) and a small +rational \(\lambda>0\) such that \(K_S+aP_S-\lambda U\) is big. +This is possible because \(P_S\) is nef and +\(K_S+a_0P_S\) is big. Choose +\[ + 0<\eta<\lambda\frac{1-c}{a-1} + \quad\text{rational}. +\] +The rational line \(cP_S+\eta U\) is ample. A sufficiently high +general member divided by its degree pulls back to a rational +boundary transverse to \(T\) with subunit coefficient. Applying +the nonvanishing part of Proposition~\ref{proj:good-models} to +this ordinary projective log pair gives +\[ + E_c\sim_{\Q}H_c+\eta p^*U,\qquad E_c\geq0. +\] +A nonzero section defining \(E_c\) restricts nontrivially to very +general \(p\)-fibers in a fixed divisible degree. The hypotheses +of the weak-effectivity statement \cite[Lemma~3.3]{CI} are now +satisfied: the source is smooth in class \(\mathcal C\), \(T\) +is rational SNC, the base \(S\) is smooth projective, and a +relative log system is nonzero. Applied to the big line +\(K_S+aP_S-\lambda U\), it gives +\[ + E_a\sim_{\Q}K_W+T+aM-\lambda p^*U,\qquad E_a\geq0. +\] +For +\[ + \theta=\frac{1-c}{a-c},\qquad + \delta=\theta\lambda-(1-\theta)\eta>0, +\] +the rational interpolation is +\[ + (1-\theta)E_c+\theta E_a + \sim_{\Q}H-\delta p^*U. +\] +Multiplying pullbacks of sections of \(U\) by a section of the +effective left-hand side yields +\(\kappa(H)\geq\kappa(p^*U)=\dim S>0\). +This contradicts Equation~\eqref{rank:nontrivial-nef-data} and +proves Equation~\eqref{rank:lowered-nonpsef}. + +We next run a program to fiber type while descending the actual nef line +\(H_{\mathrm{nef}}\) through every contraction. Choose the common +resolution in Equation~\eqref{rank:first-nef-endpoint} to determine the nef +data. The negativity lemma gives +\[ + F_M:=\sigma^*M_{\mathrm{nef}}-\mu^*M\geq0 + \quad\text{\(\sigma\)-exceptional}. +\] +Indeed its negative is \(\sigma\)-nef and exceptional. For every +rational \(00\) with +\(rH_{\mathrm{nef}}\) Cartier, and choose an integer \(d_0>2kr\). +This coefficient will force every ray of the second program to have zero +degree for the descended nef line. Decreasing the nef data to +\((1-\varepsilon)\boldsymbol M\) preserves generalized dlt +singularities: on the displayed resolution the crepant boundary +changes by \(-\varepsilon F_M\), so discrepancies do not decrease. +Add \(d_0H_{\mathrm{nef}}\), determined as nef data on +\(W_{\mathrm{nef}}\). It changes no discrepancies, and the resulting +adjoint is +\[ + (1+d_0)H_{\mathrm{nef}}-\varepsilon M_{\mathrm{nef}} + =(1+d_0)\left( + H_{\mathrm{nef}}-\frac{\varepsilon}{1+d_0}M_{\mathrm{nef}}\right). +\] +It is not pseudo-effective by the preceding paragraph. The +non-pseudo-effective case of Proposition~\ref{proj:generalized-mmp} +therefore gives a chosen terminating program to a Mori fiber +contraction in the \(\Q\)-factorial generalized dlt category. + +At any stage of this second program, let \(H_i\) be the descended rational +line, assuming inductively that it is nef and \(rH_i\) is Cartier, and write +\[ + G_i=K_{W_i}+T_i+(1-\varepsilon)M_i. +\] +An extremal ray negative for \(G_i+d_0H_i\) is negative for \(G_i\). +The pair with adjoint \(G_i\) is still generalized dlt: the added +nef b-divisor \(d_0H_i\) descends to the current model, so removing +it changes no discrepancy. The generalized length bound +\cite[Proposition~3.13]{HanLi} gives a curve \(C\) +on that ray with \(0<-G_i\cdot C\leq2k\). If \(H_i\cdot C>0\), +Cartier integrality would give +\[ + \frac{d_0}{r}\leq d_0H_i\cdot C + <-G_i\cdot C\leq2k, +\] +a contradiction. Thus \(H_i\cdot C=0\). +The exact contraction statement \cite[Theorem~1.5]{Xie2022} +descends the Cartier line \(rH_i\) along this contraction as an +actual Cartier line. In a flip we pull that same line to the +flipped side. The descended line is nef: every curve on the +contraction base has a curve above it mapping with positive degree, +so its degree is nonnegative; pullback preserves nefness. +Consequently \(r\) and nefness persist, and the argument applies +inductively to all steps and to the final contraction. Each step +is crepant for \(H_i\). + +Let \(W_m\) be the last birational model of this second program and let +\(H_m\) be the descended line on it. Crepancy for the \(H_i\), together +with the absence of extraction, turns +Equation~\eqref{rank:first-nef-endpoint} into +Equation~\eqref{rank:base-endpoint} on a common resolution, with an +effective divisor exceptional over \(W_m\). +The final contraction \(h:W_m\to V\) has connected fibers and, +by the same exact descent, +\[ + H_m\sim_{\Q}h^*H_V +\] +for a rational line \(H_V\) on the normal projective \(V\). +If \(V\) is a point, the second alternative holds with +\(H_m\sim_{\Q}0\). + +Assume \(\dim V>0\). To obtain the first alternative, it remains to show +that the log adjoint on the new very general fibers has Kodaira dimension +zero. Resolve the base program by +\(\mu:W''\to W\) and \(\nu:W''\to W_m\), and prepare the source +over \(W''\), calling it \(g'':(X'',B'')\to W''\). The diagram +we use is +\[ +\begin{tikzcd} + X'' \arrow[r,"g''"] & W'' \arrow[r,"\nu"] \arrow[d,"\mu"'] & + W_m \arrow[r,"h"] & V\\ + & W +\end{tikzcd} +\] +with composite \(f= h\nu g''\). +Equation~\eqref{rank:higher-base} and +Equation~\eqref{rank:base-endpoint} give +\[ + H''\sim_{\Q}\nu^*H_m+E'', + \qquad E''\geq0\ \text{\(\nu\)-exceptional}. +\] +To see the exceptionality, the error in +Equation~\eqref{rank:base-endpoint} is already exceptional over +\(W_m\), and every \(\mu\)-exceptional prime is exceptional +over \(W_m\) as well: otherwise \(W_m\) would extract a divisor +over \(W\). +For every degree divisible by one fixed integer, +Equation~\eqref{rank:relative-injection}, exceptional descent, +and projection formula now yield +\begin{align*} + f_*\OO_{X''}(q(K_{X''}+B'')) + &\lhook\joinrel\longrightarrow + (h\nu)_*\OO_{W''}(qH'')\\ + &=h_*\OO_{W_m}(qH_m)=\OO_V(qH_V). +\end{align*} +Here \(\nu_*\OO_{W''}(qE'')=\OO_{W_m}\) by normality and +\(h_*\OO_{W_m}=\OO_V\). Generic coherent base change, +simultaneously for the countably many divisible degrees, bounds +the dimension of every such system on a very general \(f\)-fiber +by one. The log adjoint on that fiber is pseudo-effective by +restriction of a countable sequence of perturbed currents with +analytic singularities. Its dimension is less than \(n\), so +the lower-dimensional \(\Ggood\) supplies a nonzero section in +some divisible degree. Taking a common multiple shows that the +fiber log Kodaira dimension is zero. All maps in the composite +have connected fibers; resolving \(V\) and the source keeps this +property and gives a smooth projective base of dimension +\(0<\dim V0\), since at least one component of +\(Q\) has image dimension \(l\). Thus \(Q\) is not in the radical, +and its square is strictly negative, contrary to +Equation~\eqref{rank:nonnegative-square}. + +It remains to treat \(l=k-1\). Terms from primes whose image has +smaller dimension vanish in this form, as do cross terms from +different divisorial images. Fix a prime divisor \(D\subset Y\) +and take all the primes \(E_1,\ldots,E_s\subset X\) dominating it. +Write \(a_i=\ord_{E_i}(b^*D)>0\) and +\(C_{ij}=\langle E_i,E_j\rangle_{k-1}\). +The divisor \(D\) is rational Cartier, also when \(Y=W_m\). +Consequently +\[ + C_{ij}\geq0\ (i\ne j),\qquad + \sum_j C_{ij}a_j=0. +\] +The first assertion is positivity of the proper intersection of +distinct effective primes. The second pairs \(E_i\) with the +pulled-back divisor \(D\); the additional base factor makes the +intersection vanish by dimension. Components of \(b^*D\) +mapping into a proper subset of \(D\) make no contribution. +The graph with edges \(C_{ij}>0\) is connected. Indeed over a +very general point of \(D\) the fiber is connected, all its points +lie in the components \(E_i\), and an intersection contributes a +positive entry precisely when it dominates \(D\). + +For any real vector \(x=(x_i)\), the two displayed properties give +the exact identity +\[ + x^{\mathsf T}Cx + =-\sum_{i1\) for which +\(uL|_{D_Z}\) is generated, and consider the integral line +\begin{equation} + \mathscr N + :=\OO_U(u\ell+E-C) + =\OO_U\bigl(K_U+\Delta_F+(u-1)\ell\bigr). +\label{rank:vanishing-line} +\end{equation} +Although the expression on the right uses rational divisors, its +sum is the integral line on the left. The restriction sequence is +\[ + 0\longrightarrow\mathscr N + \longrightarrow\OO_U(u\ell+E) + \longrightarrow\OO_C(u\ell+E)\longrightarrow0. +\] +The direct image of the last sheaf is supported on the proper analytic +subset \(f(C)\), so it is torsion. Consequently, the two assertions +\begin{equation} + R^1f_*\mathscr N\ \text{torsion-free}, + \qquad H^1(Y,f_*\mathscr N)=0 +\label{rank:vanishing} +\end{equation} +will give surjectivity on global sections: torsion-freeness makes the +connecting map to \(R^1f_*\mathscr N\) zero, and the \(H^1\)-vanishing +then removes the obstruction to lifting a global section. + +We prove Equation~\eqref{rank:vanishing} using analytic injectivity. +Equation~\eqref{rank:big-pullback} identifies +\(\mathscr N\otimes K_U^{-1}\) with +\(\Delta_F+(u-1)f^*H_Y\) in \(\Pic(U)\otimes\Q\); this comparison +need not be an isomorphism of the corresponding integral lines. Fix a +very ample line \(A_Y\) on \(Y\) and its Fubini--Study metric. +A Kodaira decomposition of the big nef \(H_Y\), given arbitrarily +small weight as in the proof of Lemma~\ref{rank:lift}, writes +\((u-1)H_Y\) as a positive rational multiple of \(A_Y\) plus +an effective rational divisor whose fixed singular contribution +is arbitrarily small and whose remaining contribution is a +general divided ample member. Choose the small weight below +the log canonical threshold of its pullback relative to the +klt SNC boundary \(\Delta_F\), and choose the ample member generally. +Then the resulting boundary on \(U\) is klt. Choose a positive +integer \(q_0\) clearing all its denominators and realizing both the +chosen Kodaira decomposition and the preceding rational-line comparison +as line isomorphisms, in particular +\[ + (\mathscr N\otimes K_U^{-1})^{\otimes q_0} + \simeq \OO_U(q_0\Delta_F)\otimes + f^*\OO_Y\bigl(q_0(u-1)H_Y\bigr). +\] +The divisor metrics and the Fubini--Study metric +define a singular Hermitian metric on +\((\mathscr N\otimes K_U^{-1})^{\otimes q_0}\). Its \(q_0\)-th +root is a metric on the actual line \(\mathscr N\otimes K_U^{-1}\), +with multiplier ideal \(\OO_U\) and curvature dominating a positive +multiple of the pullback Fubini--Study form. The same statements hold after every +nonnegative twist by \(f^*A_Y\). + +The injectivity theorem \cite[Theorem~A]{FujinoMatsumura} applies +on the compact Kähler \(U\): multiplication by the pullback of +any nonzero section of a positive power of \(A_Y\) is injective +on +\[ + H^1(U,\mathscr N\otimes f^*A_Y^j) + \quad(j\geq0) +\] +into the correspondingly higher twist. Its assumptions are +exactly the semipositive smooth metric on the multiplying line, +the preceding positive lower curvature bound, and the trivial +multiplier ideal. +This injectivity implies Equation~\eqref{rank:vanishing}. +For the second assertion, the Leray edge map injects +\(H^1(Y,f_*\mathscr N)\) into \(H^1(U,\mathscr N)\). +Multiplication by a section of a sufficiently high \(A_Y\)-power +sends this subspace to zero, since Serre vanishing kills +\(H^1(Y,f_*\mathscr N\otimes A_Y^j)\) for large \(j\). +Injectivity upstairs forces the original subspace to be zero. +For the first assertion, suppose \(R^1f_*\mathscr N\) has a +nonzero torsion subsheaf. A section of a sufficiently high +power of \(A_Y\) annihilates a nonzero such subsheaf. +After a further sufficiently large twist it has a nonzero +global section, and Serre vanishing and Leray identify that +section with a nonzero class in +\(H^1(U,\mathscr N\otimes f^*A_Y^j)\). +Multiplication annihilates this class, again contradicting +injectivity. This proves both assertions. + +The resulting surjectivity extends the floor sections as follows. +Pull any section of \(uL|_{D_Z}\) +to \(C\), multiply by the canonical section of \(E|_C\), and +extend by this surjectivity. The extended section descends to +\(Z\), since \(r_*\OO_U(E)=\OO_Z\) by exceptionality and +normality. On the strict transforms of the components of +\(D_Z\), division by the canonical section of \(E\) recovers +the prescribed section. The descended restriction therefore +equals it on the whole reduced \(D_Z\), since equality holds +generically on every component. Since \(uL|_{D_Z}\) is generated, +this proves Equation~\eqref{rank:floor-avoidance}. + +We finish by showing that this stable base locus is empty. The +base loci of factorial Cartier multiples form a descending +sequence of compact analytic subsets and hence stabilize. +Sections exist because \(H_Y\) is big and +Equation~\eqref{rank:big-pullback} descends their pullbacks. +Choose a divisible degree \(v\) whose base locus is +\(\mathbf B(L)\), and let \(\mathfrak b\) be its base ideal. +If \(\mathbf B(L)\ne\varnothing\), the ideal is a unit near +\(D_Z\), while \((Z,\Delta_Z)\) is klt outside \(D_Z\). +Its log canonical threshold +\[ + t=\operatorname{lct}_{(Z,\Delta_Z)}(\mathfrak b) +\] +is therefore a positive rational number, attained by a divisor +whose center is contained in \(\mathbf B(L)\). +This follows directly on a simultaneous log resolution of the +pair and the ideal: the threshold is the minimum of the positive +rational discrepancies divided by the positive integral ideal +orders. + +Choose an integer \(s>t\) and general divisors +\(D_1,\ldots,D_s\in|vL|\). On the same resolution their fixed +part is the divisor of \(\mathfrak b\); their free transforms +are jointly transverse to the SNC data. Hence +\[ + \left(Z,\ \Delta_Z+\frac{t}{s}\sum_{i=1}^s D_i\right) +\] +is log canonical: the fixed part is at its log canonical +threshold and the free components have coefficients \(t/s<1\). +There is a new lc place centered in \(\mathbf B(L)\), and the +new adjoint satisfies the actual identity +\[ + L_{\mathrm{new}}\sim_{\Q}(1+tv)L. +\] +Apply \cite[Theorem~3.3]{HLX} to this compact Kähler lc rational +pair. It gives a projective strongly \(\Q\)-factorial dlt +modification \(d:Z'\to Z\). Since the pair is lc, all extracted +divisors have log discrepancy zero in our convention, so the +modification is crepant. The source is again compact Kähler. +Choose its defining dlt log resolution, with exceptional crepant +coefficients strictly below one and an isomorphism at the general +point of every lc stratum; resolving the remaining data away from +those points gives the resolution required by +Definition~\ref{def:lc-strata-resolution}. The floor on \(Z'\) has a point over the center +of this lc place, hence over \(\mathbf B(L)\). +The new nef adjoint is \(d^*L_{\mathrm{new}}\). On a common +resolution with \(U\), it satisfies +Equation~\eqref{rank:big-pullback} with the big nef line +\((1+tv)H_Y\). The previously proved assertion +\eqref{rank:floor-avoidance} therefore makes its stable base locus +disjoint from its floor. On the +other hand, normality and projection formula identify all +divisible sections under \(d\), and give +\[ + \mathbf B(d^*L_{\mathrm{new}}) + =d^{-1}\mathbf B(L). +\] +This locus contains the stated floor point, a contradiction. +Thus \(\mathbf B(L)=\varnothing\); factorial stabilization +provides an actual globally generated multiple of \(L\). +\end{proof} + +\subsection{Completion of the rank-one case} + +\begin{proof}[Proof of Proposition~\ref{rank:rank-one}] +By Proposition~\ref{bir:resolution-transfer} we can begin on the +resolved source in the statement. We use a secondary induction on +the positive base dimension \(k\). Apply +Lemma~\ref{rank:comparison} and then Lemma~\ref{rank:base-psef} +to obtain the prepared data and the pseudo-effective line \(H\). + +If \(a(W)=0\), Lemma~\ref{rank:zero-base} and the first case of +Lemma~\ref{rank:lift} give +\(J\sim_{\Q}N(J)\), which is already the required decomposition +with trivial positive part. If \(W\) is projective, apply +Lemma~\ref{rank:base-reduction}. Its first alternative has a +strictly smaller positive base dimension, so the secondary induction +and birational transfer finish the proof. In its second alternative, +take a common smooth base resolution in +Equation~\eqref{rank:base-endpoint} and prepare the source over it. +Equation~\eqref{rank:higher-base} adds an effective divisor +exceptional over the old base. It is also exceptional over \(W_m\), +because the base program extracted no divisors. Thus the second +case of Lemma~\ref{rank:lift} applies. + +We have obtained +\[ + J\sim_{\Q}P_0+A,\qquad A=N(J)\geq0, +\] +with \(P_0\) nef. A rationally trivial \(P_0\) finishes the proof. +In the remaining case \(P_0=b^*H_m\), where \(b:X\to W_m\) +is proper and \(H_m\) is big and nef on the projective \(W_m\). +Apply Proposition~\ref{prog:contract-negative} to this +nef-plus-negative presentation. It gives a globally strongly +\(\Q\)-factorial dlt compact Kähler model \((Z,\Delta_Z)\) with nef +adjoint \(L_Z\). The ordinary dlt endpoint has the resolution property +of Definition~\ref{def:lc-strata-resolution}. On a common smooth resolution +\(\mu:U\to X\), \(r:U\to Z\) its exact positive-line comparison is +\[ + r^*L_Z\sim_{\Q}\mu^*P_0 + =(b\mu)^*H_m . +\] +Proposition~\ref{rank:big-nef-generation} makes \(L_Z\) semiample. +Hence \(\mu^*P_0\) is semiample. Lemma~\ref{bir:pullup} identifies +\(\mu^*A=N(\mu^*J)\), so this is precisely \(\Ggood_n\) on \(U\). +Finally Proposition~\ref{bir:resolution-transfer} transfers the +decomposition back through all the prepared source modifications. +\end{proof} diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/06-simple-nonvanishing.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/06-simple-nonvanishing.tex index 663be46ff6a9b1f6a2a388066a158669f6447932..4a42d1a150fec8bfce4e73f5e0cf2c7ba1064197 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/06-simple-nonvanishing.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/06-simple-nonvanishing.tex @@ -1,1427 +1,1427 @@ -\section{Meromorphic nonvanishing on simple spaces}\label{sec:nonvanishing} - -The simple case of the induction needs a divisor representing a multiple of -the canonical bundle; its coefficients need not be nonnegative. The -following result supplies precisely this starting point. - -\begin{theorem}\label{nv:meromorphic} -Let \(X\) be a smooth connected compact Kähler manifold. Suppose that -\(a(X)=0\) and that no positive-dimensional proper compact analytic -subvariety passes through a very general point of \(X\). Then -\(K_X^{\otimes m}\) has a nonzero meromorphic section for some integer -\(m>0\). -\end{theorem} - -The proof compares two ways of measuring poles. After normalizing a big -class on \(X\) to have volume one, its pole order at a very general point is -bounded. We construct a projective bundle over \(X^2\) whose volume grows -linearly with a parameter \(q\). Restriction to the exceptional divisor of -the diagonal in the square of this bundle then produces a high-rank -subsheaf of a symmetric cotangent power. A second diagonal forces -substantial vanishing of its determinant; descending that determinant -produces a point pole of order comparable to -\(q^{1/(2\dim X+1)}\), contradicting the point bound. -The argument uses only meromorphic sections of line bundles; it does not -require nonconstant meromorphic functions on \(X\). - -For the proof, suppose that the conclusion fails. Write -\[ - n=\dim X,\qquad \mathscr L=K_X,\qquad L=c_1(\mathscr L) - \in H^{1,1}_{\BC}(X,\R). -\] -We may assume \(n\ge2\): a compact complex curve is projective, and the -assertion for a point is immediate. Throughout this Section, an inequality -between real \((1,1)\)-classes is in pseudo-effective order: -\(\alpha\le\beta\) means that \(\beta-\alpha\) is pseudo-effective. For a -divisor \(D\), \(\{D\}\) denotes \(c_1(\OO(D))\). Pullbacks of classes will -occasionally be suppressed when the map is evident. - -\subsection{Line subsheaves and point poles} - -We first record the consequence of simplicity that controls all the -determinant lines used below. - -\begin{lemma}\label{nv:slope} -Under the contrary hypothesis above, \(H^1(X,\OO_X)=0\), the group -\(\Pic(X)\) is countable, and \(L\ge0\). If \(\mathscr H\) is a holomorphic -line bundle and -\(\mathscr H\longrightarrow\Omega_X^{\otimes k}\) is nonzero, where -\(k\ge0\), then -\begin{equation}\label{nv:slope-X} - c_1(\mathscr H)\le kL . -\end{equation} -There is a complement of a countable union of proper analytic subsets of -\(X\) with the following further property. For every point \(x\) in this -complement, let -\[ - a:Y=\Bl_xX\longrightarrow X,\qquad F=a^{-1}(x). -\] -Every nonzero line map -\(\mathscr H_Y\longrightarrow (a^*\Omega_X)^{\otimes k}\) satisfies -\begin{equation}\label{nv:slope-blowup} - c_1(\mathscr H_Y)\le k a^*L . -\end{equation} -\end{lemma} - -\begin{proof} -If the Albanese map of \(X\) is nonconstant, simplicity makes its image -\(n\)-dimensional: a positive-dimensional general fiber of smaller -dimension would be a forbidden subvariety. At a point where the Albanese -map has rank \(n\), some \(n\) invariant one-forms have nonzero wedge after -pullback. This gives a nonzero holomorphic section of \(K_X\), contrary to -our assumption. Thus \(H^1(X,\OO_X)=0\). The exponential sequence embeds -\(\Pic(X)\) into the countable group \(H^2(X,\Z)\). - -The manifold \(X\) is not uniruled, by simplicity. Ou's -non-pseudo-effectivity criterion for the canonical class therefore gives -\(L\ge0\) \cite[Theorem~1.1]{Ou2025}. We recall how the slope and foliation -results in the same paper give the more precise inequality in -Equation~\eqref{nv:slope-X}. Let \(\gamma\) be any class in the full dual of the -pseudo-effective cone, and use the slope -\[ - \mu_\gamma(\mathscr E) - =\frac{c_1(\mathscr E)\cdot\gamma}{\rk\mathscr E} -\] -for torsion-free sheaves. If -\(\mu_{\gamma,\min}(\Omega_X)<0\), the first Harder--Narasimhan piece -\(\mathscr T\subset T_X\) has strictly positive slope. It is saturated and -semistable. The tensor slope inequality shows that the bracket -\(\bigwedge^2\mathscr T\to T_X/\mathscr T\) is zero: the minimum slope of -its source is at least \(2\mu_\gamma(\mathscr T)\), whereas the maximum -slope of its target is smaller than \(\mu_\gamma(\mathscr T)\). -Consequently \(\mathscr T\) is a foliation. Its dual has strictly negative -maximum slope and is not pseudo-effective by -\cite[Proposition~4.5]{Ou2025}. Ou's foliation theorem -\cite[Theorem~1.4]{Ou2025} makes this foliation the tangent foliation of a -meromorphic fibration with compact general leaf closures. Its rank is strictly -between zero and \(n\): rank \(n\) would give -\(\mu_\gamma(T_X)=-L\cdot\gamma/n\le0\). The general leaf closure is -therefore a positive-dimensional proper subvariety through a general -point, again contradicting simplicity. - -We have proved \(\mu_{\gamma,\min}(\Omega_X)\ge0\). In its -Harder--Narasimhan filtration all quotient slopes are now nonnegative, so -\(\mu_{\gamma,\max}(\Omega_X)\le L\cdot\gamma\). The tensor slope -inequality \cite[Lemma~4.4]{Ou2025} gives -\[ - c_1(\mathscr H)\cdot\gamma - \le \mu_{\gamma,\max}(\Omega_X^{\otimes k}) - \le kL\cdot\gamma . -\] -Separation by the dual cone proves Equation~\eqref{nv:slope-X}. For \(k=0\) this is -also immediate from the effective zero divisor of a nonzero map -\(\mathscr H\to\OO_X\). - -It remains to choose \(x\) uniformly. For any vector bundle -\(\mathscr V\), independent global sections are generically pointwise -independent. Indeed, choose a maximal pointwise independent subfamily on -a dense open set. Expressing any other section in that family by minors -gives meromorphic function coefficients. They are constant because -\(a(X)=0\), so maximality among a linearly independent family forces all -its members to be pointwise independent. Thus evaluation on -\(H^0(X,\mathscr V)\) is injective away from a proper analytic subset -(unless that vector space is zero, in which case there is no restriction). -Apply this to -\(\mathscr V=\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) for every -\(\mathscr H\in\Pic(X)\) and \(k\ge0\). There are countably many such -bundles. - -For \(x\) outside the resulting exceptional set, write any line bundle on -\(Y=\Bl_xX\) uniquely as -\(a^*\mathscr H\otimes\OO_Y(mF)\), \(m\in\Z\). A line map from this bundle -to \((a^*\Omega_X)^{\otimes k}\), restricted away from \(F\), extends over -\(x\) to a section of -\(\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) by Hartogs' theorem. If -\(m>0\), this section vanishes at \(x\), since -\(a_*\OO_Y(-mF)=\mathcal I_x^m\). The choice of \(x\) rules this out. -Thus \(m\le0\), and Equation~\eqref{nv:slope-X} yields -\[ - c_1(a^*\mathscr H\otimes\OO_Y(mF)) - \le k a^*L+m\{F\}\le k a^*L. -\] -If a target has a finite filtration whose graded pieces are direct sums -of the indicated cotangent tensors, take the first nonzero associated -graded component of the line map and then a nonzero direct-sum component. -The preceding argument applies with the tensor order of that component. -\end{proof} - -We will use several standard facts about volumes of real \((1,1)\)-classes. -Our normalization is \(\vol(\alpha)=\int\alpha^e\) for a nef class on an -\(e\)-fold. Volume is continuous, homogeneous, monotone in -pseudo-effective order, invariant under modification, and its \(e\)-th -root is concave on the big cone. Analytic Fujita approximation computes -it by Kähler parts on smooth projective modifications. Here a -\emph{smooth projective modification} means a projective modification -whose source is smooth; the sources used here are compact Kähler -manifolds obtained by resolving coherent analytic ideals. For a smooth -irreducible divisor \(D\) and a big class \(\alpha\), write -\(\operatorname{vol}_{\,|D}(\alpha)\) for the numerical restricted volume. -It is zero when \(D\) is contained in the non-Kähler locus -\cite[Theorem~1.1]{Vu2023}; otherwise it is -the supremum of the masses on \(D\) of restrictions of Kähler currents -with analytic singularities that are not generically singular on \(D\) -\cite[Lemma~2.7]{CollinsTosatti2018}. The divisorial derivative and its -continuity on the big cone are -\begin{equation}\label{nv:volume-derivative} - \frac{\dd}{\dd u}\vol(\alpha-u\{D\}) - =-e\,\operatorname{vol}_{\,|D}(\alpha-u\{D\}); -\end{equation} -see \cite[Theorem~1.1]{Vu2023}. We apply this formula only on smooth -manifolds while the varying class is big. - -Here is a useful precise form of the approximation in the restricted -volume formula. Resolve the log-ideal singularities of a current -restricted to \(D\). Its pullback is an effective real divisor plus a -positive residual current with locally bounded potentials. The latter -dominates a positive multiple of the pulled-back Kähler form. A current -with locally bounded potentials has zero Lelong numbers, so Demailly -regularization makes its class, after subtracting that multiple, nef -\cite[Theorem~1.1]{DemaillyRegularization1992}. -On a projective modification there is an effective exceptional divisor -whose negative is relatively ample. Subtracting a sufficiently small -multiple of this divisor from the residual class therefore makes that -class Kähler; add the same multiple to the divisor part. Round all -divisor coefficients slightly upwards to rational numbers. Openness of -the Kähler cone preserves the Kähler property. These changes can be -arbitrarily small in top intersections, which compute the original mass -by the bounded-potential product formula. Thus we may approximate a -restricted mass by -\begin{equation}\label{nv:rational-fujita} - h^*(\alpha|_D)=\beta+\{D'\},\qquad - \beta\ \text{Kähler},\quad D'\ge0\ \text{a rational divisor}. -\end{equation} -Moreover, \(D'\) has at least the log-ideal orders of the original -restriction on every further resolution. We will use this last -property to turn poles into vanishing conditions. Only \(D'\) is made -rational; the class \(\beta\) and the horizontal part of \(\alpha\) remain -real. - -\begin{lemma}[A point pole bound]\label{nv:pole-threshold} -Let \(W\) be a smooth compact Kähler manifold of dimension \(e\ge2\), -\(\alpha\) a big real \((1,1)\)-class, and \(z\in W\). Suppose a smooth -projective modification \(\mu:W'\to W\), which is an isomorphism near -\(z\), admits a decomposition -\[ - \mu^*\alpha=K+\{D\},\qquad K\ \text{Kähler},\quad D\ge0, -\] -where \(D\) misses the point \(z'\) over \(z\). If the ordinary analytic -Seshadri constant \(\epsilon(K,z')\) is at least \(\eta>0\), then, on the -blowup \(b:\widehat W=\Bl_zW\to W\) with exceptional divisor \(G\), -\begin{equation}\label{nv:pole-formula} - \sup\{u\ge0:b^*\alpha-u\{G\}\ge0\} - \le \frac{\eta}{2} - +2^{e-1}\vol(\alpha)\eta^{-(e-1)} . -\end{equation} -\end{lemma} - -\begin{proof} -Set \(u_0=\eta/2\). Blowing up \(z'\), the class -\(K-u_0\{G'\}\) is Kähler. The Fujita decomposition is unchanged near -\(G'\), so it supplies a Kähler current for -\(\alpha_{u_0}=b^*\alpha-u_0\{G\}\) that is smooth near \(G\). -Its restriction there has class \(u_0c_1(\OO_{\PP^{e-1}}(1))\). -The restricted volume is consequently \(u_0^{e-1}\): the current gives -this lower bound, and the volume of the restricted class gives the -opposite bound. - -Let \(\tau\) denote the left side of Equation~\eqref{nv:pole-formula}. The classes -\(\alpha_u=b^*\alpha-u\{G\}\) are big for \(0\le u<\tau\), since -\(\alpha_0\) is big and the pseudo-effective cone is convex. The -concave function \(f(u)=\vol(\alpha_u)^{1/e}\) satisfies, by -Equation~\eqref{nv:volume-derivative}, -\[ - f'(u_0)=-\frac{u_0^{e-1}}{f(u_0)^{e-1}}. -\] -Its tangent line at \(u_0\) must remain positive up to \(\tau\). -Therefore -\[ - \tau\le u_0+\frac{f(u_0)^e}{u_0^{e-1}} - \le u_0+\frac{\vol(\alpha)}{u_0^{e-1}}, -\] -which is Equation~\eqref{nv:pole-formula}. -\end{proof} - -Fix a Kähler class \(\omega\) on \(X\). The class \(L\) is not big: -a big holomorphic line bundle would make \(X\) Moishezon, contrary to -\(a(X)=0\). For \(t>0\) define -\begin{equation}\label{nv:normalization} - P=r(L+t\omega),\qquad - r=\vol(L+t\omega)^{-1/n}. -\end{equation} -Then \(P\) is a big real class, \(\vol(P)=1\), \(L\le P/r\), and -\(r\to\infty\) as \(t\downarrow0\). Choose a smooth Fujita model -\(\mu:Y_P\to X\) with Kähler part \(P'\) satisfying -\begin{equation}\label{nv:P-prime} - \mu^*P=P'+\{D_P\},\qquad v:=\int_{Y_P}(P')^n>\frac12. -\end{equation} -At a very general point \(y\) of this model there is no -positive-dimensional proper subvariety: its image would be one through a -very general point of \(X\), and \(y\) avoids the exceptional locus. -The ordinary Seshadri formula for a Kähler class, -\[ - \epsilon(K,y) - =\inf_{\substack{W\ni y\\ \dim W>0}} - \left(\frac{\int_W K^{\dim W}} - {\operatorname{mult}_y W}\right)^{1/\dim W}, -\] -therefore gives \(\epsilon(P',y)=v^{1/n}\ge2^{-1/n}\). -This is the ordinary nef/Kähler formula -\cite[Theorem~2.8]{Tosatti2016}; see also -\cite[Equation~(1.5)]{CollinsTosatti2018} and the -Kähler cone criterion of \cite{DemaillyPaun2004}. We use this ordinary -formula in both applications below. Lemma~\ref{nv:pole-threshold}, with -\(\vol(P)=1\), now yields a constant \(C_n\) depending only on \(n\) -such that, for a very general \(x\), -\begin{equation}\label{nv:point-bound} - \tau(P,x):=\sup\{u\ge0:a^*P-u\{F\}\ge0\}\le C_n, - \qquad a:\Bl_xX\to X. -\end{equation} -Here and below positive constants denoted \(c_n,C_n\) may be decreased or -increased from one occurrence to the next. They are independent of all -parameters and choices of currents and modifications. - -\subsection{A projective bundle with controlled volume} - -We now produce a class whose volume grows, while its point pole threshold -grows much more slowly. On \(X^2\), let \(\mathscr L_i\) and \(L_i\) -denote the pullbacks of \(\mathscr L\) and \(L\) from the \(i\)-th factor. -Use the quotient convention and put -\[ - \pi:Z=\PP_{X^2}(\mathscr L_1\oplus\mathscr L_2)\longrightarrow X^2, - \qquad \xi=c_1(\OO_Z(1)),\qquad d=\dim Z=2n+1 . -\] -Thus \(\pi_*\OO_Z(m)=\Sym^m(\mathscr L_1\oplus\mathscr L_2)\) for -\(m\ge0\). The zero divisor \(A_i\) of -\(\pi^*\mathscr L_i\to\OO_Z(1)\) has class \(\xi-L_i\); it is the -section on which \(\xi\) restricts to \(L_{3-i}\). In particular -\(\xi=\{A_i\}+L_i\ge0\). - -\begin{lemma}\label{nv:subvarieties} -For \(X\) and \(Z\) above, assume that \(K_X^{\otimes m}\) has no nonzero -meromorphic section for every integer \(m>0\). -Through a very general point of \(X^2\), the only positive-dimensional -proper compact irreducible analytic subvarieties are the two factor -slices. Through a very general point of \(Z\), the only -positive-dimensional compact irreducible analytic subvarieties are a -fiber of \(\pi\), the full inverse images of the two factor slices, and -\(Z\) itself. -\end{lemma} - -\begin{proof} -Let \(C\subset X^2\) be irreducible through a pair whose coordinates are -very general in \(X\). Each projection image is a point or all of \(X\), -by properness and simplicity. If exactly one is a point, \(C\) is the -corresponding full slice. If both projections are surjective, test their -fibers at a general point of \(C\) whose coordinates are still very -general. A positive-dimensional fiber must be the full other factor. -Thus either \(C=X^2\), or \(\dim C=n\) and both projections are -generically finite. - -We show that subvarieties of the latter kind cannot sweep \(X^2\). -The space of compact \(n\)-cycles on a compact Kähler manifold has -countably many irreducible components, each compact -\cite[Theorem~1.1]{LiebermanCycles}. On a component containing integral cycles -generically finite over both factors, integrality holds on a dense open -set and the two projection degrees remain positive, as can also be -tested by intersection with pulled-back Kähler forms. Suppose the -incidence over one such component dominates \(X^2\). Resolve the -parameter space and the dominating incidence component, obtaining maps -\[ - h:\mathcal Y\longrightarrow\mathcal S,\qquad - F_i:\mathcal Y\longrightarrow X -\] -between smooth spaces. For general \(t\in\mathcal S\), the fiber -\(Y_t=h^{-1}(t)\) is smooth, compact, and bimeromorphic to the integral -cycle, hence irreducible. Its maps \(f_i=F_i|_{Y_t}:Y_t\to X\) are -generically finite. Choose \(t\) also so that \(Y_t\) meets the dense -open set where \(d(F_1,h)\) is invertible and \(d(F_1,F_2)\) has rank -\(2n\). The first assertion follows from generic finiteness of \(f_1\), -and the second from the assumed dominance in \(X^2\). - -Our immediate aim is to turn this dominance into a meromorphic frame of -\(f_2^*T_X\). Its determinant would meromorphically trivialize -\(f_2^*K_X^{-1}\), and hence also \(f_2^*K_X\). -On \(Y_t\), the equal-rank bundle map -\[ - d(F_1,h)|_{Y_t}:T_{\mathcal Y}|_{Y_t} - \longrightarrow f_1^*T_X\oplus - (\OO_{Y_t}\otimes T_t\mathcal S) -\] -has a meromorphic inverse, given by its adjugate and determinant. -For fixed \(v\in T_t\mathcal S\), apply this inverse to \((0,v)\) -and then apply \(dF_2\). The result is a meromorphic section of -\(f_2^*T_X\). At a general point where \(d(F_1,F_2)\) has rank -\(2n\), the resulting map \(T_t\mathcal S\to f_2^*T_X\) is -surjective: in local coordinates \((F_1,h)\), it is the derivative -of \(F_2\) in the parameter directions with \(F_1\) fixed. Choose -\(n\) fixed vectors \(v_i\) whose values there are independent. -Their wedge is a nonzero meromorphic section of \(f_2^*K_X^{-1}\) -on the irreducible \(Y_t\); its inverse trivializes \(f_2^*K_X\) -meromorphically. - -Factor the proper generically finite map \(f_2\) as -\[ - Y_t\longrightarrow \overline Y_t - \xrightarrow{\nu}X -\] -by Stein factorization. Here \(\overline Y_t\) is normal and irreducible, the first -map is a modification, and \(\nu\) is finite of degree \(N>0\). -Meromorphic sections descend across a modification of a normal -space, so the section descends to a nonzero meromorphic section of -\(\nu^*K_X\). Lemma~\ref{bir:finite-norm} gives a nonzero meromorphic -section of \(K_X^{\otimes N}\). This norm is defined in local finite -analytic fraction algebras, so it remains available even though the -global meromorphic function field of \(X\) is just \(\C\). It -contradicts our assumption. - -Consequently the incidence image of each such cycle component is a -proper compact analytic subset of \(X^2\). There are only countably -many components, so their union misses a very general pair. This -proves the assertion for \(X^2\). - -Now let \(W\subset Z\) be irreducible through a very general point. -Its image is a point, a full factor slice, or \(X^2\). If its generic -relative dimension is one, it is the full inverse image of that -image. Otherwise it is a multisection of the restricted -\(\PP^1\)-bundle. Over a slice or over \(X^2\), a multisection is a -divisor with line bundle -\(\OO(k)\otimes\pi^*\mathscr H\), \(k>0\). Its homogeneous equation -has coefficients in -\[ - H^0\bigl(S,\mathscr H\otimes - \mathscr L_1^{\otimes i} - \otimes\mathscr L_2^{\otimes(k-i)}\bigr), - \qquad 0\le i\le k, -\] -where \(S\) is the relevant slice or \(X^2\), and a fixed-factor line is -understood as constant. A single nonzero monomial cuts out only the -axis sections and vertical divisors. A multisection not on an axis -therefore has two nonzero coefficients. Their quotient is a -meromorphic section of a nonzero power of -\(\mathscr L_1\otimes\mathscr L_2^{-1}\). Over a slice this is, up to -inversion and a constant line, a positive power of \(K_X\). Over -\(X^2\), restriction to a general factor slice gives the same -conclusion. This is impossible. The axes themselves miss a very -general point of \(Z\), proving the claim. -\end{proof} - -Let \(q\) tend to infinity through positive integers. For each sufficiently -large \(q\), choose \(t=t(q)\) in Equation~\eqref{nv:normalization} so that -\(r=r(q)\ge q^2\), and use the resulting normalized class \(P=P(q)\). -Define the real class and the scale -\begin{equation}\label{nv:M-definition} - M=P_1+P_2+q\xi,\qquad A_q=q^{1/d}. -\end{equation} - -\begin{lemma}\label{nv:two-slot-volume} -For these choices, \(M\) is big and -\begin{equation}\label{nv:M-bounds} - c_nq\le\vol(M)\le C_nq,\qquad - \tau(M,z):=\sup\{u\ge0:b_z^*M-u\{F_z\}\ge0\}\le C_nA_q -\end{equation} -at a very general point \(z\in Z\), where -\(b_z:\Bl_zZ\to Z\) has exceptional divisor \(F_z\). -\end{lemma} - -\begin{proof} -Pull \(Z\) to the Fujita model \(Y_P^2\) from -Equation~\eqref{nv:P-prime}, and put \(H=P'_1+P'_2\) on this pulled-back bundle. -There is a further smooth projective modification -\(\nu:\widehat Z\to\PP_{Y_P^2}(\mu^*\mathscr L_1\oplus -\mu^*\mathscr L_2)\) and a Fujita decomposition of the pullback of \(M\) -whose Kähler part is exactly -\begin{equation}\label{nv:exact-K} - K=\frac12\nu^*H+\Theta , -\end{equation} -where \(\Theta\) is Kähler and has degree \(q\) on a general vertical -line. The modification and the divisor part miss that line. - -Here is the construction. It suffices to decompose \(H/2+q\xi\) as -\(\Theta\) plus an effective divisor, clean on a whole general vertical -line. Since \(\OO(1)\) is relatively ample, a small positive class of -the form \(\delta(\xi+cH)\) is Kähler for some \(c>0\). Choose -\(\delta>0\) so small that \(\delta0 -\] -on a general fiber. For sufficiently large divisible integers \(j\), -let \(\mathcal F_j=f_*\OO_U(j\Lambda)\) and \(R_j=\rk\mathcal F_j\). -Here \(j\Lambda\) is an actual line bundle and -\(c_1(\mathcal F_j)\) means \(c_1(\det\mathcal F_j)\). Then -\begin{align} - R_j&=\frac{w}{e!}j^e+O(j^{e-1}),\label{nv:GRR-rank}\\ - B_0:=G+\lim_j\frac{c_1(\mathcal F_j)}{jR_j} - &=\frac{f_*\beta^{e+1}}{(e+1)w}\ge0,\label{nv:GRR-base}\\ - \int_U\beta^{b_0+e} - &\le (e+1)^{b_0}w\,\vol(B_0).\label{nv:mixed-bound} -\end{align} -The limit in Equation~\eqref{nv:GRR-base} is a limit of real Bott--Chern classes. -\end{lemma} - -\begin{proof} -The restriction of \(c_1(\Lambda)\) to each fiber is represented by the -restriction of the Kähler form \(\beta\). The fiberwise criterion for -relative ampleness makes \(\Lambda\) relatively ample after clearing -denominators. Relative Serre vanishing then kills the higher direct -images for all sufficiently large divisible \(j\). Analytic -Grothendieck--Riemann--Roch \cite{LevyGRR}, in degrees zero and two, gives -\[ - R_j=\frac{f_*c_1(\Lambda)^e}{e!}j^e+O(j^{e-1}),\qquad - c_1(\mathcal F_j) - =\frac{f_*c_1(\Lambda)^{e+1}}{(e+1)!}j^{e+1}+O(j^e). -\] -The degree-zero pushforward is \(w\). Expanding -\(\beta=f^*G+c_1(\Lambda)\) shows that -\[ - f_*\beta^{e+1} - =f_*c_1(\Lambda)^{e+1}+(e+1)wG, -\] -because terms with at least two horizontal factors have negative -fiber degree after pushforward. This proves the equality in -Equation~\eqref{nv:GRR-base}; the cohomological GRR equality is an equality in -Bott--Chern cohomology by the \(\partial\bar\partial\)-lemma on compact -Kähler manifolds. Pushforward of the positive form -\(\beta^{e+1}\) is a positive closed \((1,1)\)-current, so \(B_0\) is -pseudo-effective. - -To prove Equation~\eqref{nv:mixed-bound}, we need a nef class on the base. -The class \(B_0\) is presently only known to be pseudo-effective. We -flatten \(f\) to obtain a nef class above the base and compare it -with the pullback of \(B_0\). Take a smooth projective flattening -modification \(p:Z'_0\to Z_0\), the equidimensional main transform -\(\overline U\) of \(U\times_{Z_0}Z'_0\), and a resolution -\(U'\to\overline U\). Write \(f':U'\to Z'_0\) and \(q:U'\to U\) for -the maps and \(\beta'=q^*\beta\). The class -\[ - B'_0=\frac{f'_*(\beta')^{e+1}}{(e+1)w} -\] -is nef. It is enough to show that the positive pushforward current -representing this class has zero Lelong numbers. This current can -be computed by integration on the cycle \(\overline U\) of the smooth -form pulled back from \(U\). At a base point, cover the compact fiber -by finitely many coordinate neighborhoods, each embedded in a product -of base coordinates and ambient coordinates. In the mass over a base -ball of radius \(\delta\), after wedging with a base Euclidean form to -power \(b_0-1\), the integrand is bounded by a finite sum of projection -volume forms using \(b_0-1\) base coordinates and \(e+1\) generic -linear combinations of all coordinates of the ambient product, including -the remaining base direction. These projections can be chosen finite on the -neighborhoods: fixing the \(b_0-1\) base coordinates leaves local -dimension at most \(e+1\), by equidimensionality, and generic ambient -coordinates finish a finite projection. After shrinking to compact -subneighborhoods their degrees are bounded. Change of variables -therefore bounds each integral by -\(O(\delta^{2(b_0-1)})\) times the measure of the remaining coordinate -range. On each compact subneighborhood these ranges, taken over closed -base balls, are nested compact sets whose intersection is the image of -the central fiber. That image has measure zero in the \(e+1\) coordinates, -because the fiber has dimension \(e\). Continuity of finite measure from -above shows that the range measures tend to zero. Thus the -mass is -\[ - o\bigl(\delta^{2(b_0-1)}\bigr). -\] -This also covers \(b_0=1\), when it asserts absence of an atom. The -pushforward current has zero Lelong numbers at every point, and -Demailly regularization \cite[Theorem~1.1]{DemaillyRegularization1992} -proves that its class \(B'_0\) is nef. - -Pushforward under \(p\) gives \(p_*B'_0=B_0\). For a modification between -smooth compact Kähler manifolds, -the kernel of pushforward on real Bott--Chern \((1,1)\)-classes is generated -by the classes of its exceptional prime divisors. Since -\(p_*p^*B_0=B_0\), it follows that \(B'_0-p^*B_0\) is an exceptional real -divisor class. It is -\(p\)-nef because \(B'_0\) is nef. Apply relative negativity to this -exceptional real divisor. Locally over the base, the usual proof for a -projective modification cuts by general hyperplanes to a surface and -uses the negative definite intersection matrix of exceptional curves; -it forces every coefficient of a relatively nef exceptional divisor to -be nonpositive. Thus -\[ - B'_0=p^*B_0-\{D_0\},\qquad D_0\ge0\ \text{\(p\)-exceptional}. -\] -In particular -\begin{equation}\label{nv:flattened-volume} - \int_{Z'_0}(B'_0)^{b_0} - =\vol(B'_0)\le\vol(p^*B_0)=\vol(B_0). -\end{equation} -Choose a Kähler class \(\omega'\) on \(Z'_0\) and put -\(C=B'_0+\varepsilon\omega'\). For \(0\le k\le b_0\), set -\[ - I_k=\int_{U'}(\beta')^{e+k}(f'^*C)^{b_0-k}. -\] -These are mixed intersections of nef classes. The first two satisfy -\[ - I_0=w\int_{Z'_0}C^{b_0},\qquad - I_1=(e+1)w\int_{Z'_0}B'_0C^{b_0-1} - \le(e+1)w\int_{Z'_0}C^{b_0}. -\] -The mixed nef inequalities make the sequence \(I_k\) log-concave. -Every \(I_k\) is positive: on the dense open where \(q\) is a local -biholomorphism and \(f'\) is a submersion, \(\beta'\) is positive -definite and \(f'^*C\) has rank \(b_0\), so the defining top form -is strictly positive. The successive ratios are therefore at most -\(I_1/I_0\le e+1\). Thus -\[ - \int_U\beta^{b_0+e}=I_{b_0} - \le(e+1)^{b_0}w\int_{Z'_0}C^{b_0}. -\] -Letting \(\varepsilon\downarrow0\) and using -Equation~\eqref{nv:flattened-volume} proves Equation~\eqref{nv:mixed-bound}. -\end{proof} - -We will also use the determinant formula without its volume bound. - -\begin{corollary}\label{nv:slope-descent} -Let \(Y\) be a smooth compact Kähler manifold and \(\mathscr E\) a -holomorphic vector bundle of rank at least two. Write -\(\pi_{\mathscr E}:\PP_Y(\mathscr E)\to Y\) and -\(\zeta_{\mathscr E}=c_1(\OO_{\PP(\mathscr E)}(1))\). -Suppose that a real class \(D\) on \(Y\) satisfies -\[ - c_1(\mathscr H)\le kD - \quad\text{whenever}\quad - 0\ne\bigl(\mathscr H\longrightarrow\mathscr E^{\otimes k}\bigr) -\] -for a line bundle \(\mathscr H\) and integer \(k\ge0\). -For a real class \(A\) on \(Y\) and a real number \(b\ge0\), -\begin{equation}\label{nv:slope-descent-equation} - \pi_{\mathscr E}^*A+b\zeta_{\mathscr E}\ge0 - \quad\Longrightarrow\quad A+bD\ge0 . -\end{equation} -\end{corollary} - -\begin{proof} -Choose a real class \(H_0\) on \(Y\) so that -\(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0\) is Kähler; relative -ampleness of \(\OO(1)\) permits such a choice. Take numbers -\(\delta\downarrow0\) with \(b+\delta>0\) rational. Adding -\(\delta(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0)\) to the -pseudo-effective class in Equation~\eqref{nv:slope-descent-equation} makes it -big. A Fujita decomposition on a smooth projective modification -\(h:U\to\PP_Y(\mathscr E)\), with the divisor coefficients rounded -upwards as in Equation~\eqref{nv:rational-fujita}, has the form -\[ - \beta=f^*(A+\delta H_0)+c_1(\Lambda),\qquad - \Lambda=(b+\delta)h^*\OO(1)-\OO_U(D') - \ \text{in }\Pic(U)\otimes\Q, -\] -where \(f=\pi_{\mathscr E}h\), \(\beta\) is Kähler, and \(D'\ge0\) -is rational. For divisible \(j\), -\[ - f_*\OO_U(j\Lambda)\ \subseteq\ - \Sym^{j(b+\delta)}\mathscr E . -\] -Its determinant of rank \(R_j\) consequently maps into -\(\mathscr E^{\otimes j(b+\delta)R_j}\). The assumed line inequality -gives -\[ - \frac{c_1(\mathcal F_j)}{jR_j}\le(b+\delta)D. -\] -Lemma~\ref{nv:direct-image} makes -\(A+\delta H_0+\lim c_1(\mathcal F_j)/(jR_j)\) pseudo-effective. -Adding the preceding pseudo-effective difference shows that -\(A+\delta H_0+(b+\delta)D\) is pseudo-effective. Let -\(\delta\downarrow0\). This proves the corollary. In this argument -only \(b+\delta\) and the coefficients of \(D'\) are rational; \(A\), -\(H_0\), and \(D\) remain real classes. -\end{proof} - -\subsection{Restriction to the diagonal} - -We now use the volume of \(M\) to obtain a high-rank subsheaf of a -symmetric cotangent power on \(Z\). Distinguish the two copies of \(Z\) -by bracketed indices and -blow up their diagonal: -\[ - b:B=\Bl_{\Delta_Z}(Z\times Z)\longrightarrow Z\times Z,\qquad - E=b^{-1}(\Delta_Z). -\] -Thus \(E=\PP_Z(\Omega_Z)\); its points are normal lines in \(T_Z\). -The dimensions are \(\dim B=2d\) and \(\dim E=2d-1\), and -\(E\to Z\) has relative dimension \(d-1\). -Put \(\zeta=c_1(\OO_E(1))\), so that \(\{E\}|_E=-\zeta\). -For \(s\ge0\) define -\[ - C_s=b^*(M_{[1]}+M_{[2]})-s\{E\},\qquad - s_{\max}=\sup\{s\ge0:C_s\ge0\}. -\] -The class \(C_0\) is big, so \(s_{\max}>0\) and \(C_s\) is big for -\(0\le s0\) such that, for every sufficiently -large \(q\), one can choose a rational number -\[ - c_nA_q\le s\le C_nA_q -\] -and a Kähler current \(T\) in \(C_s\) with analytic singularities, -not generically singular on \(E\), with the following property. On a -smooth projective modification \(h:U\to E\), its restricted mass has -a rational-divisor approximation -\begin{equation}\label{nv:selected-decomposition} - h^*(2M+s\zeta)=\beta+\{D'\}, - \qquad \beta\ \text{Kähler},\quad D'\ge0\ \text{rational}, -\end{equation} -which retains all log-ideal orders of \(T|_E\). If -\(f:U\to Z\) is the natural map and -\[ - w=\int_{U_z}\beta^{d-1} -\] -on a general fiber, then -\begin{equation}\label{nv:selected-w} - w\ge c_ns^{d-1}. -\end{equation} -\end{lemma} - -\begin{proof} -First let \(s\) be any rational number in \((0,s_{\max})\) with -\(v_E(s)>0\), and use the approximation in -Equation~\eqref{nv:rational-fujita} for any current used -to compute this restricted volume. In -Lemma~\ref{nv:direct-image}, the data for Equation~\eqref{nv:restriction-class} -are -\[ - b_0=d,\quad e=d-1,\quad G=2M,\quad - \Lambda=s h^*\OO_E(1)-\OO_U(D') - \ \text{in }\Pic(U)\otimes\Q. -\] -For large divisible \(j\), the associated sheaves satisfy -\begin{equation}\label{nv:symmetric-inclusion} - \mathcal F_j=f_*\OO_U(j\Lambda) - \subseteq\Sym^{js}\Omega_Z,\qquad - R_j=\rk\mathcal F_j,\qquad 00\). The contribution of -\(0(1-\delta)m\). Summing instead over their internal -degree gives -\[ - H_m(\delta) - \le \binom{m+n}{n}\binom{\lceil\delta m\rceil+n}{n},\qquad - \limsup_{m\to\infty}\frac{H_m(\delta)}{N_m} - \le\binom{2n}{n}\delta^n. -\] -On the other hand, \(w^+\ge w/2\) and -Equation~\eqref{nv:plus-rank} show that the approximation on \(E^+\) -retains at least half the rank fraction in -Equation~\eqref{nv:selected-rank-fraction}: -\[ - \lim_{j\to\infty}\frac{R_j^+}{N_{js}} - =\frac{w^+}{s^{d-1}} - \ge\frac{w}{2s^{d-1}}\ge c_n>0. -\] -Fix \(\delta>0\), depending only on \(n\), so small that, for each fixed -\(q\) and its selected data, fewer than \(R_j^+/2\) monomials have transverse -degree greater than \((1-\delta)js\) for all sufficiently large divisible -\(j\). For each -remaining monomial, Equation~\eqref{nv:coefficient-order} and -Lemma~\ref{nv:incidence-pole} give -\[ - \ord_u(c_{\alpha,\gamma}) - \ge j(s-C_n)-(1-\delta)js - =j(\delta s-C_n)\ge\frac{\delta js}{2}. -\] -The last inequality holds once \(q\) is large, since -\(s\ge c_nA_q\to\infty\). - -Over the same discrete valuation ring \(R\), the torsion-free sheaf -\(\mathcal F_j^+\) becomes a free module of rank \(R_j^+\). Write its -inclusion into the symmetric power as a matrix with the monomials -as rows. Every maximal minor uses at least \(R_j^+/2\) of the rows -whose coefficients have order at least \(\delta js/2\). All maximal -minors therefore have order at least -\(\delta jR_j^+s/4\). By the definition of \(h_j\), this proves -Equation~\eqref{nv:determinant-order-equation}. Dividing the determinant -map by \(\sigma_{J_+}^{h_j}\) gives a holomorphic map at every point of -codimension one: the coefficients have the required order along \(J_+\), -and its local equation is a unit away from \(J_+\). Hartogs' theorem -extends the divided map across the remaining set of codimension at least -two, giving the stated factorization through -\(\det\mathcal F_j^+(h_jJ_+)\). -\end{proof} - -\begin{proof}[Completion of the proof of Theorem~\ref{nv:meromorphic}] -Fix \(q\) sufficiently large for the preceding lemmas, with its chosen -\(P,s,T\) and modifications. Choose a very general point \(x\) in the first -factor of \(X^2\). The slice of \(Z^+\) above \(x\) is -\[ - S=\PP_Y(\OO_Y\oplus a^*\mathscr L),\qquad - a:Y=\Bl_xX\longrightarrow X, -\] -and \(J_+|_S\) is the pullback of \(F\). We choose \(x\) so that -Equation~\eqref{nv:slope-blowup}, Equation~\eqref{nv:point-bound}, the restrictions of -the currents below, and all determinant maps for divisible \(j\) -can be tested on this slice. These exclude at most countably many -proper analytic sets and the measure-zero exceptional sets for -current restrictions. - -Write the restricted determinant line as -\[ - (\det\mathcal F_j^+)|_S - =\pi_S^*\mathscr Q_j^+\otimes\OO_S(\ell_j^+),\qquad - Q_j^+=c_1(\mathscr Q_j^+). -\] -Factoring the common zero of order \(h_j\) from the determinant -map gives a nonzero map whose source on \(S\) is -\[ - \pi_S^*(\mathscr Q_j^+\otimes\OO_Y(h_jF)) - \otimes\OO_S(\ell_j^+). -\] -The plus sign of \(h_jF\) follows because division of the map by -the local equation of \(J_+^{h_j}\) enlarges its source from -\(\det\mathcal F_j^+\) to -\(\det\mathcal F_j^+\otimes\OO(h_jJ_+)\). -Its target embeds into the \(jsR_j^+\)-fold tensor of the -restriction of \(p_+^*\Omega_Z\). In particular we use the -cotangent bundle of the original \(Z\), with its restricted -filtration -\[ - 0\longrightarrow - \pi_S^*(\OO_Y^{\oplus n}\oplus a^*\Omega_X) - \longrightarrow (p_+^*\Omega_Z)|_S - \longrightarrow - \OO_S(-2)\otimes\pi_S^*a^*\mathscr L - \longrightarrow0. -\] -The same homogeneous-monomial calculation as in -Equation~\eqref{nv:determinant-bound-Z}, now applying -Equation~\eqref{nv:slope-blowup} to the second factor of \(X^2\), gives -\begin{equation}\label{nv:determinant-bound-slice} - \ell_j^+\le0,\qquad - Q_j^++h_j\{F\} - \le(jsR_j^+-\ell_j^+)a^*L. -\end{equation} -More explicitly, if the chosen graded term has \(i\) relative -factors, the pushed polynomial has degree -\(-2i-\ell_j^+\ge0\). Its exponent in \(a^*\mathscr L\), together -with the \(i\) relative factors and the cotangent order from the second -factor of \(X^2\), is at most \(jsR_j^+-\ell_j^+\). The cotangent -factors from the first factor of \(X^2\) are constant. This proves the -displayed inequality using -the tensors of \(a^*\Omega_X\), without introducing \(\Omega_Y\) -or \(\Omega_{Z^+}\). - -Apply Lemma~\ref{nv:direct-image} to the decomposition in -Equation~\eqref{nv:plus-decomposition}. Equation~\eqref{nv:GRR-base} -shows that the limits of the restricted determinant components divided by -\(jR_j^+\) exist; denote them by \(Q^+\) and \(\ell^+\). Divide -Equation~\eqref{nv:determinant-bound-slice} by \(jR_j^+\), use -Equation~\eqref{nv:determinant-order-equation}, and pass to the closed -pseudo-effective cone. We obtain -\begin{equation}\label{nv:slice-upper} - \ell^+\le0,\qquad - Q^++c_ns\{F\}\le(s-\ell^+)a^*L. -\end{equation} -The pseudo-effective class in Equation~\eqref{nv:GRR-base}, restricted to the -very general slice \(S\), is -\begin{equation}\label{nv:slice-psef} - \pi_S^*(Q^++2a^*P)+(\ell^++2q)\xi\ge0. -\end{equation} -Testing on a general vertical line gives \(\ell^++2q\ge0\). -Let \(A\subset S\) be the axis of -\(\PP_Y(\OO_Y\oplus a^*\mathscr L)\) with divisor class \(\{A\}=\xi\). -On \(A\simeq Y\), its normal class is \(\{A\}|_A=\xi|_A=a^*L\ge0\). -Add an arbitrarily small Kähler class to the class in -Equation~\eqref{nv:slice-psef}, and choose a Kähler current with analytic -singularities in the resulting big class. Remove its generic divisorial -pole along \(A\) and restrict -the residual current to \(A\). Adding back the removed nonnegative multiple -of \(a^*L\) preserves pseudo-effectivity. Passing to the limit gives -\begin{equation}\label{nv:slice-lower} - Q^++2a^*P+(\ell^++2q)a^*L\ge0. -\end{equation} - -The slope bound in Equation~\eqref{nv:slice-upper} supplies the -pseudo-effective class \((s-\ell^+)a^*L-Q^+-c_ns\{F\}\). Adding it to -Equation~\eqref{nv:slice-lower} cancels the entire normalized determinant -class restricted to the axis, \(Q^++\ell^+a^*L\), leaving -\[ - 2a^*P+(s+2q)a^*L-c_ns\{F\}\ge0. -\] -Since \(L\le P/r\), the point bound in Equation~\eqref{nv:point-bound} implies -\begin{equation}\label{nv:contradiction} - \frac{c_ns}{\,2+(s+2q)/r\,}\le C_n. -\end{equation} -The denominator is bounded independently of \(q\), because -\(s\le C_nq^{1/d}\) and \(r\ge q^2\). The numerator tends to -infinity, because \(s\ge c_nq^{1/d}\). Equation~\eqref{nv:contradiction} -is impossible for arbitrarily large \(q\). - -For each \(q\), the choice of \(t(q)\), the rational \(s\), the current -\(T\), and the modifications precedes the limit in \(j\). The monomial -cutoff \(\delta\) depends only on \(n\), while the required lower bound for -divisible \(j\) may depend on the fixed data. We choose the very general -points after those data are fixed. -Thus every simultaneous general-point test above involves at most -countably many conditions, and no bound depends on the point. -The contradiction disproves the contrary hypothesis and proves -Theorem~\ref{nv:meromorphic}. -\end{proof} +\section{Meromorphic nonvanishing on simple spaces}\label{sec:nonvanishing} + +The simple case of the induction needs a divisor representing a multiple of +the canonical bundle; its coefficients need not be nonnegative. The +following result supplies precisely this starting point. + +\begin{theorem}\label{nv:meromorphic} +Let \(X\) be a smooth connected compact Kähler manifold. Suppose that +\(a(X)=0\) and that no positive-dimensional proper compact analytic +subvariety passes through a very general point of \(X\). Then +\(K_X^{\otimes m}\) has a nonzero meromorphic section for some integer +\(m>0\). +\end{theorem} + +The proof compares two ways of measuring poles. After normalizing a big +class on \(X\) to have volume one, its pole order at a very general point is +bounded. We construct a projective bundle over \(X^2\) whose volume grows +linearly with a parameter \(q\). Restriction to the exceptional divisor of +the diagonal in the square of this bundle then produces a high-rank +subsheaf of a symmetric cotangent power. A second diagonal forces +substantial vanishing of its determinant; descending that determinant +produces a point pole of order comparable to +\(q^{1/(2\dim X+1)}\), contradicting the point bound. +The argument uses only meromorphic sections of line bundles; it does not +require nonconstant meromorphic functions on \(X\). + +For the proof, suppose that the conclusion fails. Write +\[ + n=\dim X,\qquad \mathscr L=K_X,\qquad L=c_1(\mathscr L) + \in H^{1,1}_{\BC}(X,\R). +\] +We may assume \(n\ge2\): a compact complex curve is projective, and the +assertion for a point is immediate. Throughout this Section, an inequality +between real \((1,1)\)-classes is in pseudo-effective order: +\(\alpha\le\beta\) means that \(\beta-\alpha\) is pseudo-effective. For a +divisor \(D\), \(\{D\}\) denotes \(c_1(\OO(D))\). Pullbacks of classes will +occasionally be suppressed when the map is evident. + +\subsection{Line subsheaves and point poles} + +We first record the consequence of simplicity that controls all the +determinant lines used below. + +\begin{lemma}\label{nv:slope} +Under the contrary hypothesis above, \(H^1(X,\OO_X)=0\), the group +\(\Pic(X)\) is countable, and \(L\ge0\). If \(\mathscr H\) is a holomorphic +line bundle and +\(\mathscr H\longrightarrow\Omega_X^{\otimes k}\) is nonzero, where +\(k\ge0\), then +\begin{equation}\label{nv:slope-X} + c_1(\mathscr H)\le kL . +\end{equation} +There is a complement of a countable union of proper analytic subsets of +\(X\) with the following further property. For every point \(x\) in this +complement, let +\[ + a:Y=\Bl_xX\longrightarrow X,\qquad F=a^{-1}(x). +\] +Every nonzero line map +\(\mathscr H_Y\longrightarrow (a^*\Omega_X)^{\otimes k}\) satisfies +\begin{equation}\label{nv:slope-blowup} + c_1(\mathscr H_Y)\le k a^*L . +\end{equation} +\end{lemma} + +\begin{proof} +If the Albanese map of \(X\) is nonconstant, simplicity makes its image +\(n\)-dimensional: a positive-dimensional general fiber of smaller +dimension would be a forbidden subvariety. At a point where the Albanese +map has rank \(n\), some \(n\) invariant one-forms have nonzero wedge after +pullback. This gives a nonzero holomorphic section of \(K_X\), contrary to +our assumption. Thus \(H^1(X,\OO_X)=0\). The exponential sequence embeds +\(\Pic(X)\) into the countable group \(H^2(X,\Z)\). + +The manifold \(X\) is not uniruled, by simplicity. Ou's +non-pseudo-effectivity criterion for the canonical class therefore gives +\(L\ge0\) \cite[Theorem~1.1]{Ou2025}. We recall how the slope and foliation +results in the same paper give the more precise inequality in +Equation~\eqref{nv:slope-X}. Let \(\gamma\) be any class in the full dual of the +pseudo-effective cone, and use the slope +\[ + \mu_\gamma(\mathscr E) + =\frac{c_1(\mathscr E)\cdot\gamma}{\rk\mathscr E} +\] +for torsion-free sheaves. If +\(\mu_{\gamma,\min}(\Omega_X)<0\), the first Harder--Narasimhan piece +\(\mathscr T\subset T_X\) has strictly positive slope. It is saturated and +semistable. The tensor slope inequality shows that the bracket +\(\bigwedge^2\mathscr T\to T_X/\mathscr T\) is zero: the minimum slope of +its source is at least \(2\mu_\gamma(\mathscr T)\), whereas the maximum +slope of its target is smaller than \(\mu_\gamma(\mathscr T)\). +Consequently \(\mathscr T\) is a foliation. Its dual has strictly negative +maximum slope and is not pseudo-effective by +\cite[Proposition~4.5]{Ou2025}. Ou's foliation theorem +\cite[Theorem~1.4]{Ou2025} makes this foliation the tangent foliation of a +meromorphic fibration with compact general leaf closures. Its rank is strictly +between zero and \(n\): rank \(n\) would give +\(\mu_\gamma(T_X)=-L\cdot\gamma/n\le0\). The general leaf closure is +therefore a positive-dimensional proper subvariety through a general +point, again contradicting simplicity. + +We have proved \(\mu_{\gamma,\min}(\Omega_X)\ge0\). In its +Harder--Narasimhan filtration all quotient slopes are now nonnegative, so +\(\mu_{\gamma,\max}(\Omega_X)\le L\cdot\gamma\). The tensor slope +inequality \cite[Lemma~4.4]{Ou2025} gives +\[ + c_1(\mathscr H)\cdot\gamma + \le \mu_{\gamma,\max}(\Omega_X^{\otimes k}) + \le kL\cdot\gamma . +\] +Separation by the dual cone proves Equation~\eqref{nv:slope-X}. For \(k=0\) this is +also immediate from the effective zero divisor of a nonzero map +\(\mathscr H\to\OO_X\). + +It remains to choose \(x\) uniformly. For any vector bundle +\(\mathscr V\), independent global sections are generically pointwise +independent. Indeed, choose a maximal pointwise independent subfamily on +a dense open set. Expressing any other section in that family by minors +gives meromorphic function coefficients. They are constant because +\(a(X)=0\), so maximality among a linearly independent family forces all +its members to be pointwise independent. Thus evaluation on +\(H^0(X,\mathscr V)\) is injective away from a proper analytic subset +(unless that vector space is zero, in which case there is no restriction). +Apply this to +\(\mathscr V=\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) for every +\(\mathscr H\in\Pic(X)\) and \(k\ge0\). There are countably many such +bundles. + +For \(x\) outside the resulting exceptional set, write any line bundle on +\(Y=\Bl_xX\) uniquely as +\(a^*\mathscr H\otimes\OO_Y(mF)\), \(m\in\Z\). A line map from this bundle +to \((a^*\Omega_X)^{\otimes k}\), restricted away from \(F\), extends over +\(x\) to a section of +\(\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) by Hartogs' theorem. If +\(m>0\), this section vanishes at \(x\), since +\(a_*\OO_Y(-mF)=\mathcal I_x^m\). The choice of \(x\) rules this out. +Thus \(m\le0\), and Equation~\eqref{nv:slope-X} yields +\[ + c_1(a^*\mathscr H\otimes\OO_Y(mF)) + \le k a^*L+m\{F\}\le k a^*L. +\] +If a target has a finite filtration whose graded pieces are direct sums +of the indicated cotangent tensors, take the first nonzero associated +graded component of the line map and then a nonzero direct-sum component. +The preceding argument applies with the tensor order of that component. +\end{proof} + +We will use several standard facts about volumes of real \((1,1)\)-classes. +Our normalization is \(\vol(\alpha)=\int\alpha^e\) for a nef class on an +\(e\)-fold. Volume is continuous, homogeneous, monotone in +pseudo-effective order, invariant under modification, and its \(e\)-th +root is concave on the big cone. Analytic Fujita approximation computes +it by Kähler parts on smooth projective modifications. Here a +\emph{smooth projective modification} means a projective modification +whose source is smooth; the sources used here are compact Kähler +manifolds obtained by resolving coherent analytic ideals. For a smooth +irreducible divisor \(D\) and a big class \(\alpha\), write +\(\operatorname{vol}_{\,|D}(\alpha)\) for the numerical restricted volume. +It is zero when \(D\) is contained in the non-Kähler locus +\cite[Theorem~1.1]{Vu2023}; otherwise it is +the supremum of the masses on \(D\) of restrictions of Kähler currents +with analytic singularities that are not generically singular on \(D\) +\cite[Lemma~2.7]{CollinsTosatti2018}. The divisorial derivative and its +continuity on the big cone are +\begin{equation}\label{nv:volume-derivative} + \frac{\dd}{\dd u}\vol(\alpha-u\{D\}) + =-e\,\operatorname{vol}_{\,|D}(\alpha-u\{D\}); +\end{equation} +see \cite[Theorem~1.1]{Vu2023}. We apply this formula only on smooth +manifolds while the varying class is big. + +Here is a useful precise form of the approximation in the restricted +volume formula. Resolve the log-ideal singularities of a current +restricted to \(D\). Its pullback is an effective real divisor plus a +positive residual current with locally bounded potentials. The latter +dominates a positive multiple of the pulled-back Kähler form. A current +with locally bounded potentials has zero Lelong numbers, so Demailly +regularization makes its class, after subtracting that multiple, nef +\cite[Theorem~1.1]{DemaillyRegularization1992}. +On a projective modification there is an effective exceptional divisor +whose negative is relatively ample. Subtracting a sufficiently small +multiple of this divisor from the residual class therefore makes that +class Kähler; add the same multiple to the divisor part. Round all +divisor coefficients slightly upwards to rational numbers. Openness of +the Kähler cone preserves the Kähler property. These changes can be +arbitrarily small in top intersections, which compute the original mass +by the bounded-potential product formula. Thus we may approximate a +restricted mass by +\begin{equation}\label{nv:rational-fujita} + h^*(\alpha|_D)=\beta+\{D'\},\qquad + \beta\ \text{Kähler},\quad D'\ge0\ \text{a rational divisor}. +\end{equation} +Moreover, \(D'\) has at least the log-ideal orders of the original +restriction on every further resolution. We will use this last +property to turn poles into vanishing conditions. Only \(D'\) is made +rational; the class \(\beta\) and the horizontal part of \(\alpha\) remain +real. + +\begin{lemma}[A point pole bound]\label{nv:pole-threshold} +Let \(W\) be a smooth compact Kähler manifold of dimension \(e\ge2\), +\(\alpha\) a big real \((1,1)\)-class, and \(z\in W\). Suppose a smooth +projective modification \(\mu:W'\to W\), which is an isomorphism near +\(z\), admits a decomposition +\[ + \mu^*\alpha=K+\{D\},\qquad K\ \text{Kähler},\quad D\ge0, +\] +where \(D\) misses the point \(z'\) over \(z\). If the ordinary analytic +Seshadri constant \(\epsilon(K,z')\) is at least \(\eta>0\), then, on the +blowup \(b:\widehat W=\Bl_zW\to W\) with exceptional divisor \(G\), +\begin{equation}\label{nv:pole-formula} + \sup\{u\ge0:b^*\alpha-u\{G\}\ge0\} + \le \frac{\eta}{2} + +2^{e-1}\vol(\alpha)\eta^{-(e-1)} . +\end{equation} +\end{lemma} + +\begin{proof} +Set \(u_0=\eta/2\). Blowing up \(z'\), the class +\(K-u_0\{G'\}\) is Kähler. The Fujita decomposition is unchanged near +\(G'\), so it supplies a Kähler current for +\(\alpha_{u_0}=b^*\alpha-u_0\{G\}\) that is smooth near \(G\). +Its restriction there has class \(u_0c_1(\OO_{\PP^{e-1}}(1))\). +The restricted volume is consequently \(u_0^{e-1}\): the current gives +this lower bound, and the volume of the restricted class gives the +opposite bound. + +Let \(\tau\) denote the left side of Equation~\eqref{nv:pole-formula}. The classes +\(\alpha_u=b^*\alpha-u\{G\}\) are big for \(0\le u<\tau\), since +\(\alpha_0\) is big and the pseudo-effective cone is convex. The +concave function \(f(u)=\vol(\alpha_u)^{1/e}\) satisfies, by +Equation~\eqref{nv:volume-derivative}, +\[ + f'(u_0)=-\frac{u_0^{e-1}}{f(u_0)^{e-1}}. +\] +Its tangent line at \(u_0\) must remain positive up to \(\tau\). +Therefore +\[ + \tau\le u_0+\frac{f(u_0)^e}{u_0^{e-1}} + \le u_0+\frac{\vol(\alpha)}{u_0^{e-1}}, +\] +which is Equation~\eqref{nv:pole-formula}. +\end{proof} + +Fix a Kähler class \(\omega\) on \(X\). The class \(L\) is not big: +a big holomorphic line bundle would make \(X\) Moishezon, contrary to +\(a(X)=0\). For \(t>0\) define +\begin{equation}\label{nv:normalization} + P=r(L+t\omega),\qquad + r=\vol(L+t\omega)^{-1/n}. +\end{equation} +Then \(P\) is a big real class, \(\vol(P)=1\), \(L\le P/r\), and +\(r\to\infty\) as \(t\downarrow0\). Choose a smooth Fujita model +\(\mu:Y_P\to X\) with Kähler part \(P'\) satisfying +\begin{equation}\label{nv:P-prime} + \mu^*P=P'+\{D_P\},\qquad v:=\int_{Y_P}(P')^n>\frac12. +\end{equation} +At a very general point \(y\) of this model there is no +positive-dimensional proper subvariety: its image would be one through a +very general point of \(X\), and \(y\) avoids the exceptional locus. +The ordinary Seshadri formula for a Kähler class, +\[ + \epsilon(K,y) + =\inf_{\substack{W\ni y\\ \dim W>0}} + \left(\frac{\int_W K^{\dim W}} + {\operatorname{mult}_y W}\right)^{1/\dim W}, +\] +therefore gives \(\epsilon(P',y)=v^{1/n}\ge2^{-1/n}\). +This is the ordinary nef/Kähler formula +\cite[Theorem~2.8]{Tosatti2016}; see also +\cite[Equation~(1.5)]{CollinsTosatti2018} and the +Kähler cone criterion of \cite{DemaillyPaun2004}. We use this ordinary +formula in both applications below. Lemma~\ref{nv:pole-threshold}, with +\(\vol(P)=1\), now yields a constant \(C_n\) depending only on \(n\) +such that, for a very general \(x\), +\begin{equation}\label{nv:point-bound} + \tau(P,x):=\sup\{u\ge0:a^*P-u\{F\}\ge0\}\le C_n, + \qquad a:\Bl_xX\to X. +\end{equation} +Here and below positive constants denoted \(c_n,C_n\) may be decreased or +increased from one occurrence to the next. They are independent of all +parameters and choices of currents and modifications. + +\subsection{A projective bundle with controlled volume} + +We now produce a class whose volume grows, while its point pole threshold +grows much more slowly. On \(X^2\), let \(\mathscr L_i\) and \(L_i\) +denote the pullbacks of \(\mathscr L\) and \(L\) from the \(i\)-th factor. +Use the quotient convention and put +\[ + \pi:Z=\PP_{X^2}(\mathscr L_1\oplus\mathscr L_2)\longrightarrow X^2, + \qquad \xi=c_1(\OO_Z(1)),\qquad d=\dim Z=2n+1 . +\] +Thus \(\pi_*\OO_Z(m)=\Sym^m(\mathscr L_1\oplus\mathscr L_2)\) for +\(m\ge0\). The zero divisor \(A_i\) of +\(\pi^*\mathscr L_i\to\OO_Z(1)\) has class \(\xi-L_i\); it is the +section on which \(\xi\) restricts to \(L_{3-i}\). In particular +\(\xi=\{A_i\}+L_i\ge0\). + +\begin{lemma}\label{nv:subvarieties} +For \(X\) and \(Z\) above, assume that \(K_X^{\otimes m}\) has no nonzero +meromorphic section for every integer \(m>0\). +Through a very general point of \(X^2\), the only positive-dimensional +proper compact irreducible analytic subvarieties are the two factor +slices. Through a very general point of \(Z\), the only +positive-dimensional compact irreducible analytic subvarieties are a +fiber of \(\pi\), the full inverse images of the two factor slices, and +\(Z\) itself. +\end{lemma} + +\begin{proof} +Let \(C\subset X^2\) be irreducible through a pair whose coordinates are +very general in \(X\). Each projection image is a point or all of \(X\), +by properness and simplicity. If exactly one is a point, \(C\) is the +corresponding full slice. If both projections are surjective, test their +fibers at a general point of \(C\) whose coordinates are still very +general. A positive-dimensional fiber must be the full other factor. +Thus either \(C=X^2\), or \(\dim C=n\) and both projections are +generically finite. + +We show that subvarieties of the latter kind cannot sweep \(X^2\). +The space of compact \(n\)-cycles on a compact Kähler manifold has +countably many irreducible components, each compact +\cite[Theorem~1.1]{LiebermanCycles}. On a component containing integral cycles +generically finite over both factors, integrality holds on a dense open +set and the two projection degrees remain positive, as can also be +tested by intersection with pulled-back Kähler forms. Suppose the +incidence over one such component dominates \(X^2\). Resolve the +parameter space and the dominating incidence component, obtaining maps +\[ + h:\mathcal Y\longrightarrow\mathcal S,\qquad + F_i:\mathcal Y\longrightarrow X +\] +between smooth spaces. For general \(t\in\mathcal S\), the fiber +\(Y_t=h^{-1}(t)\) is smooth, compact, and bimeromorphic to the integral +cycle, hence irreducible. Its maps \(f_i=F_i|_{Y_t}:Y_t\to X\) are +generically finite. Choose \(t\) also so that \(Y_t\) meets the dense +open set where \(d(F_1,h)\) is invertible and \(d(F_1,F_2)\) has rank +\(2n\). The first assertion follows from generic finiteness of \(f_1\), +and the second from the assumed dominance in \(X^2\). + +Our immediate aim is to turn this dominance into a meromorphic frame of +\(f_2^*T_X\). Its determinant would meromorphically trivialize +\(f_2^*K_X^{-1}\), and hence also \(f_2^*K_X\). +On \(Y_t\), the equal-rank bundle map +\[ + d(F_1,h)|_{Y_t}:T_{\mathcal Y}|_{Y_t} + \longrightarrow f_1^*T_X\oplus + (\OO_{Y_t}\otimes T_t\mathcal S) +\] +has a meromorphic inverse, given by its adjugate and determinant. +For fixed \(v\in T_t\mathcal S\), apply this inverse to \((0,v)\) +and then apply \(dF_2\). The result is a meromorphic section of +\(f_2^*T_X\). At a general point where \(d(F_1,F_2)\) has rank +\(2n\), the resulting map \(T_t\mathcal S\to f_2^*T_X\) is +surjective: in local coordinates \((F_1,h)\), it is the derivative +of \(F_2\) in the parameter directions with \(F_1\) fixed. Choose +\(n\) fixed vectors \(v_i\) whose values there are independent. +Their wedge is a nonzero meromorphic section of \(f_2^*K_X^{-1}\) +on the irreducible \(Y_t\); its inverse trivializes \(f_2^*K_X\) +meromorphically. + +Factor the proper generically finite map \(f_2\) as +\[ + Y_t\longrightarrow \overline Y_t + \xrightarrow{\nu}X +\] +by Stein factorization. Here \(\overline Y_t\) is normal and irreducible, the first +map is a modification, and \(\nu\) is finite of degree \(N>0\). +Meromorphic sections descend across a modification of a normal +space, so the section descends to a nonzero meromorphic section of +\(\nu^*K_X\). Lemma~\ref{bir:finite-norm} gives a nonzero meromorphic +section of \(K_X^{\otimes N}\). This norm is defined in local finite +analytic fraction algebras, so it remains available even though the +global meromorphic function field of \(X\) is just \(\C\). It +contradicts our assumption. + +Consequently the incidence image of each such cycle component is a +proper compact analytic subset of \(X^2\). There are only countably +many components, so their union misses a very general pair. This +proves the assertion for \(X^2\). + +Now let \(W\subset Z\) be irreducible through a very general point. +Its image is a point, a full factor slice, or \(X^2\). If its generic +relative dimension is one, it is the full inverse image of that +image. Otherwise it is a multisection of the restricted +\(\PP^1\)-bundle. Over a slice or over \(X^2\), a multisection is a +divisor with line bundle +\(\OO(k)\otimes\pi^*\mathscr H\), \(k>0\). Its homogeneous equation +has coefficients in +\[ + H^0\bigl(S,\mathscr H\otimes + \mathscr L_1^{\otimes i} + \otimes\mathscr L_2^{\otimes(k-i)}\bigr), + \qquad 0\le i\le k, +\] +where \(S\) is the relevant slice or \(X^2\), and a fixed-factor line is +understood as constant. A single nonzero monomial cuts out only the +axis sections and vertical divisors. A multisection not on an axis +therefore has two nonzero coefficients. Their quotient is a +meromorphic section of a nonzero power of +\(\mathscr L_1\otimes\mathscr L_2^{-1}\). Over a slice this is, up to +inversion and a constant line, a positive power of \(K_X\). Over +\(X^2\), restriction to a general factor slice gives the same +conclusion. This is impossible. The axes themselves miss a very +general point of \(Z\), proving the claim. +\end{proof} + +Let \(q\) tend to infinity through positive integers. For each sufficiently +large \(q\), choose \(t=t(q)\) in Equation~\eqref{nv:normalization} so that +\(r=r(q)\ge q^2\), and use the resulting normalized class \(P=P(q)\). +Define the real class and the scale +\begin{equation}\label{nv:M-definition} + M=P_1+P_2+q\xi,\qquad A_q=q^{1/d}. +\end{equation} + +\begin{lemma}\label{nv:two-slot-volume} +For these choices, \(M\) is big and +\begin{equation}\label{nv:M-bounds} + c_nq\le\vol(M)\le C_nq,\qquad + \tau(M,z):=\sup\{u\ge0:b_z^*M-u\{F_z\}\ge0\}\le C_nA_q +\end{equation} +at a very general point \(z\in Z\), where +\(b_z:\Bl_zZ\to Z\) has exceptional divisor \(F_z\). +\end{lemma} + +\begin{proof} +Pull \(Z\) to the Fujita model \(Y_P^2\) from +Equation~\eqref{nv:P-prime}, and put \(H=P'_1+P'_2\) on this pulled-back bundle. +There is a further smooth projective modification +\(\nu:\widehat Z\to\PP_{Y_P^2}(\mu^*\mathscr L_1\oplus +\mu^*\mathscr L_2)\) and a Fujita decomposition of the pullback of \(M\) +whose Kähler part is exactly +\begin{equation}\label{nv:exact-K} + K=\frac12\nu^*H+\Theta , +\end{equation} +where \(\Theta\) is Kähler and has degree \(q\) on a general vertical +line. The modification and the divisor part miss that line. + +Here is the construction. It suffices to decompose \(H/2+q\xi\) as +\(\Theta\) plus an effective divisor, clean on a whole general vertical +line. Since \(\OO(1)\) is relatively ample, a small positive class of +the form \(\delta(\xi+cH)\) is Kähler for some \(c>0\). Choose +\(\delta>0\) so small that \(\delta0 +\] +on a general fiber. For sufficiently large divisible integers \(j\), +let \(\mathcal F_j=f_*\OO_U(j\Lambda)\) and \(R_j=\rk\mathcal F_j\). +Here \(j\Lambda\) is an actual line bundle and +\(c_1(\mathcal F_j)\) means \(c_1(\det\mathcal F_j)\). Then +\begin{align} + R_j&=\frac{w}{e!}j^e+O(j^{e-1}),\label{nv:GRR-rank}\\ + B_0:=G+\lim_j\frac{c_1(\mathcal F_j)}{jR_j} + &=\frac{f_*\beta^{e+1}}{(e+1)w}\ge0,\label{nv:GRR-base}\\ + \int_U\beta^{b_0+e} + &\le (e+1)^{b_0}w\,\vol(B_0).\label{nv:mixed-bound} +\end{align} +The limit in Equation~\eqref{nv:GRR-base} is a limit of real Bott--Chern classes. +\end{lemma} + +\begin{proof} +The restriction of \(c_1(\Lambda)\) to each fiber is represented by the +restriction of the Kähler form \(\beta\). The fiberwise criterion for +relative ampleness makes \(\Lambda\) relatively ample after clearing +denominators. Relative Serre vanishing then kills the higher direct +images for all sufficiently large divisible \(j\). Analytic +Grothendieck--Riemann--Roch \cite{LevyGRR}, in degrees zero and two, gives +\[ + R_j=\frac{f_*c_1(\Lambda)^e}{e!}j^e+O(j^{e-1}),\qquad + c_1(\mathcal F_j) + =\frac{f_*c_1(\Lambda)^{e+1}}{(e+1)!}j^{e+1}+O(j^e). +\] +The degree-zero pushforward is \(w\). Expanding +\(\beta=f^*G+c_1(\Lambda)\) shows that +\[ + f_*\beta^{e+1} + =f_*c_1(\Lambda)^{e+1}+(e+1)wG, +\] +because terms with at least two horizontal factors have negative +fiber degree after pushforward. This proves the equality in +Equation~\eqref{nv:GRR-base}; the cohomological GRR equality is an equality in +Bott--Chern cohomology by the \(\partial\bar\partial\)-lemma on compact +Kähler manifolds. Pushforward of the positive form +\(\beta^{e+1}\) is a positive closed \((1,1)\)-current, so \(B_0\) is +pseudo-effective. + +To prove Equation~\eqref{nv:mixed-bound}, we need a nef class on the base. +The class \(B_0\) is presently only known to be pseudo-effective. We +flatten \(f\) to obtain a nef class above the base and compare it +with the pullback of \(B_0\). Take a smooth projective flattening +modification \(p:Z'_0\to Z_0\), the equidimensional main transform +\(\overline U\) of \(U\times_{Z_0}Z'_0\), and a resolution +\(U'\to\overline U\). Write \(f':U'\to Z'_0\) and \(q:U'\to U\) for +the maps and \(\beta'=q^*\beta\). The class +\[ + B'_0=\frac{f'_*(\beta')^{e+1}}{(e+1)w} +\] +is nef. It is enough to show that the positive pushforward current +representing this class has zero Lelong numbers. This current can +be computed by integration on the cycle \(\overline U\) of the smooth +form pulled back from \(U\). At a base point, cover the compact fiber +by finitely many coordinate neighborhoods, each embedded in a product +of base coordinates and ambient coordinates. In the mass over a base +ball of radius \(\delta\), after wedging with a base Euclidean form to +power \(b_0-1\), the integrand is bounded by a finite sum of projection +volume forms using \(b_0-1\) base coordinates and \(e+1\) generic +linear combinations of all coordinates of the ambient product, including +the remaining base direction. These projections can be chosen finite on the +neighborhoods: fixing the \(b_0-1\) base coordinates leaves local +dimension at most \(e+1\), by equidimensionality, and generic ambient +coordinates finish a finite projection. After shrinking to compact +subneighborhoods their degrees are bounded. Change of variables +therefore bounds each integral by +\(O(\delta^{2(b_0-1)})\) times the measure of the remaining coordinate +range. On each compact subneighborhood these ranges, taken over closed +base balls, are nested compact sets whose intersection is the image of +the central fiber. That image has measure zero in the \(e+1\) coordinates, +because the fiber has dimension \(e\). Continuity of finite measure from +above shows that the range measures tend to zero. Thus the +mass is +\[ + o\bigl(\delta^{2(b_0-1)}\bigr). +\] +This also covers \(b_0=1\), when it asserts absence of an atom. The +pushforward current has zero Lelong numbers at every point, and +Demailly regularization \cite[Theorem~1.1]{DemaillyRegularization1992} +proves that its class \(B'_0\) is nef. + +Pushforward under \(p\) gives \(p_*B'_0=B_0\). For a modification between +smooth compact Kähler manifolds, +the kernel of pushforward on real Bott--Chern \((1,1)\)-classes is generated +by the classes of its exceptional prime divisors. Since +\(p_*p^*B_0=B_0\), it follows that \(B'_0-p^*B_0\) is an exceptional real +divisor class. It is +\(p\)-nef because \(B'_0\) is nef. Apply relative negativity to this +exceptional real divisor. Locally over the base, the usual proof for a +projective modification cuts by general hyperplanes to a surface and +uses the negative definite intersection matrix of exceptional curves; +it forces every coefficient of a relatively nef exceptional divisor to +be nonpositive. Thus +\[ + B'_0=p^*B_0-\{D_0\},\qquad D_0\ge0\ \text{\(p\)-exceptional}. +\] +In particular +\begin{equation}\label{nv:flattened-volume} + \int_{Z'_0}(B'_0)^{b_0} + =\vol(B'_0)\le\vol(p^*B_0)=\vol(B_0). +\end{equation} +Choose a Kähler class \(\omega'\) on \(Z'_0\) and put +\(C=B'_0+\varepsilon\omega'\). For \(0\le k\le b_0\), set +\[ + I_k=\int_{U'}(\beta')^{e+k}(f'^*C)^{b_0-k}. +\] +These are mixed intersections of nef classes. The first two satisfy +\[ + I_0=w\int_{Z'_0}C^{b_0},\qquad + I_1=(e+1)w\int_{Z'_0}B'_0C^{b_0-1} + \le(e+1)w\int_{Z'_0}C^{b_0}. +\] +The mixed nef inequalities make the sequence \(I_k\) log-concave. +Every \(I_k\) is positive: on the dense open where \(q\) is a local +biholomorphism and \(f'\) is a submersion, \(\beta'\) is positive +definite and \(f'^*C\) has rank \(b_0\), so the defining top form +is strictly positive. The successive ratios are therefore at most +\(I_1/I_0\le e+1\). Thus +\[ + \int_U\beta^{b_0+e}=I_{b_0} + \le(e+1)^{b_0}w\int_{Z'_0}C^{b_0}. +\] +Letting \(\varepsilon\downarrow0\) and using +Equation~\eqref{nv:flattened-volume} proves Equation~\eqref{nv:mixed-bound}. +\end{proof} + +We will also use the determinant formula without its volume bound. + +\begin{corollary}\label{nv:slope-descent} +Let \(Y\) be a smooth compact Kähler manifold and \(\mathscr E\) a +holomorphic vector bundle of rank at least two. Write +\(\pi_{\mathscr E}:\PP_Y(\mathscr E)\to Y\) and +\(\zeta_{\mathscr E}=c_1(\OO_{\PP(\mathscr E)}(1))\). +Suppose that a real class \(D\) on \(Y\) satisfies +\[ + c_1(\mathscr H)\le kD + \quad\text{whenever}\quad + 0\ne\bigl(\mathscr H\longrightarrow\mathscr E^{\otimes k}\bigr) +\] +for a line bundle \(\mathscr H\) and integer \(k\ge0\). +For a real class \(A\) on \(Y\) and a real number \(b\ge0\), +\begin{equation}\label{nv:slope-descent-equation} + \pi_{\mathscr E}^*A+b\zeta_{\mathscr E}\ge0 + \quad\Longrightarrow\quad A+bD\ge0 . +\end{equation} +\end{corollary} + +\begin{proof} +Choose a real class \(H_0\) on \(Y\) so that +\(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0\) is Kähler; relative +ampleness of \(\OO(1)\) permits such a choice. Take numbers +\(\delta\downarrow0\) with \(b+\delta>0\) rational. Adding +\(\delta(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0)\) to the +pseudo-effective class in Equation~\eqref{nv:slope-descent-equation} makes it +big. A Fujita decomposition on a smooth projective modification +\(h:U\to\PP_Y(\mathscr E)\), with the divisor coefficients rounded +upwards as in Equation~\eqref{nv:rational-fujita}, has the form +\[ + \beta=f^*(A+\delta H_0)+c_1(\Lambda),\qquad + \Lambda=(b+\delta)h^*\OO(1)-\OO_U(D') + \ \text{in }\Pic(U)\otimes\Q, +\] +where \(f=\pi_{\mathscr E}h\), \(\beta\) is Kähler, and \(D'\ge0\) +is rational. For divisible \(j\), +\[ + f_*\OO_U(j\Lambda)\ \subseteq\ + \Sym^{j(b+\delta)}\mathscr E . +\] +Its determinant of rank \(R_j\) consequently maps into +\(\mathscr E^{\otimes j(b+\delta)R_j}\). The assumed line inequality +gives +\[ + \frac{c_1(\mathcal F_j)}{jR_j}\le(b+\delta)D. +\] +Lemma~\ref{nv:direct-image} makes +\(A+\delta H_0+\lim c_1(\mathcal F_j)/(jR_j)\) pseudo-effective. +Adding the preceding pseudo-effective difference shows that +\(A+\delta H_0+(b+\delta)D\) is pseudo-effective. Let +\(\delta\downarrow0\). This proves the corollary. In this argument +only \(b+\delta\) and the coefficients of \(D'\) are rational; \(A\), +\(H_0\), and \(D\) remain real classes. +\end{proof} + +\subsection{Restriction to the diagonal} + +We now use the volume of \(M\) to obtain a high-rank subsheaf of a +symmetric cotangent power on \(Z\). Distinguish the two copies of \(Z\) +by bracketed indices and +blow up their diagonal: +\[ + b:B=\Bl_{\Delta_Z}(Z\times Z)\longrightarrow Z\times Z,\qquad + E=b^{-1}(\Delta_Z). +\] +Thus \(E=\PP_Z(\Omega_Z)\); its points are normal lines in \(T_Z\). +The dimensions are \(\dim B=2d\) and \(\dim E=2d-1\), and +\(E\to Z\) has relative dimension \(d-1\). +Put \(\zeta=c_1(\OO_E(1))\), so that \(\{E\}|_E=-\zeta\). +For \(s\ge0\) define +\[ + C_s=b^*(M_{[1]}+M_{[2]})-s\{E\},\qquad + s_{\max}=\sup\{s\ge0:C_s\ge0\}. +\] +The class \(C_0\) is big, so \(s_{\max}>0\) and \(C_s\) is big for +\(0\le s0\) such that, for every sufficiently +large \(q\), one can choose a rational number +\[ + c_nA_q\le s\le C_nA_q +\] +and a Kähler current \(T\) in \(C_s\) with analytic singularities, +not generically singular on \(E\), with the following property. On a +smooth projective modification \(h:U\to E\), its restricted mass has +a rational-divisor approximation +\begin{equation}\label{nv:selected-decomposition} + h^*(2M+s\zeta)=\beta+\{D'\}, + \qquad \beta\ \text{Kähler},\quad D'\ge0\ \text{rational}, +\end{equation} +which retains all log-ideal orders of \(T|_E\). If +\(f:U\to Z\) is the natural map and +\[ + w=\int_{U_z}\beta^{d-1} +\] +on a general fiber, then +\begin{equation}\label{nv:selected-w} + w\ge c_ns^{d-1}. +\end{equation} +\end{lemma} + +\begin{proof} +First let \(s\) be any rational number in \((0,s_{\max})\) with +\(v_E(s)>0\), and use the approximation in +Equation~\eqref{nv:rational-fujita} for any current used +to compute this restricted volume. In +Lemma~\ref{nv:direct-image}, the data for Equation~\eqref{nv:restriction-class} +are +\[ + b_0=d,\quad e=d-1,\quad G=2M,\quad + \Lambda=s h^*\OO_E(1)-\OO_U(D') + \ \text{in }\Pic(U)\otimes\Q. +\] +For large divisible \(j\), the associated sheaves satisfy +\begin{equation}\label{nv:symmetric-inclusion} + \mathcal F_j=f_*\OO_U(j\Lambda) + \subseteq\Sym^{js}\Omega_Z,\qquad + R_j=\rk\mathcal F_j,\qquad 00\). The contribution of +\(0(1-\delta)m\). Summing instead over their internal +degree gives +\[ + H_m(\delta) + \le \binom{m+n}{n}\binom{\lceil\delta m\rceil+n}{n},\qquad + \limsup_{m\to\infty}\frac{H_m(\delta)}{N_m} + \le\binom{2n}{n}\delta^n. +\] +On the other hand, \(w^+\ge w/2\) and +Equation~\eqref{nv:plus-rank} show that the approximation on \(E^+\) +retains at least half the rank fraction in +Equation~\eqref{nv:selected-rank-fraction}: +\[ + \lim_{j\to\infty}\frac{R_j^+}{N_{js}} + =\frac{w^+}{s^{d-1}} + \ge\frac{w}{2s^{d-1}}\ge c_n>0. +\] +Fix \(\delta>0\), depending only on \(n\), so small that, for each fixed +\(q\) and its selected data, fewer than \(R_j^+/2\) monomials have transverse +degree greater than \((1-\delta)js\) for all sufficiently large divisible +\(j\). For each +remaining monomial, Equation~\eqref{nv:coefficient-order} and +Lemma~\ref{nv:incidence-pole} give +\[ + \ord_u(c_{\alpha,\gamma}) + \ge j(s-C_n)-(1-\delta)js + =j(\delta s-C_n)\ge\frac{\delta js}{2}. +\] +The last inequality holds once \(q\) is large, since +\(s\ge c_nA_q\to\infty\). + +Over the same discrete valuation ring \(R\), the torsion-free sheaf +\(\mathcal F_j^+\) becomes a free module of rank \(R_j^+\). Write its +inclusion into the symmetric power as a matrix with the monomials +as rows. Every maximal minor uses at least \(R_j^+/2\) of the rows +whose coefficients have order at least \(\delta js/2\). All maximal +minors therefore have order at least +\(\delta jR_j^+s/4\). By the definition of \(h_j\), this proves +Equation~\eqref{nv:determinant-order-equation}. Dividing the determinant +map by \(\sigma_{J_+}^{h_j}\) gives a holomorphic map at every point of +codimension one: the coefficients have the required order along \(J_+\), +and its local equation is a unit away from \(J_+\). Hartogs' theorem +extends the divided map across the remaining set of codimension at least +two, giving the stated factorization through +\(\det\mathcal F_j^+(h_jJ_+)\). +\end{proof} + +\begin{proof}[Completion of the proof of Theorem~\ref{nv:meromorphic}] +Fix \(q\) sufficiently large for the preceding lemmas, with its chosen +\(P,s,T\) and modifications. Choose a very general point \(x\) in the first +factor of \(X^2\). The slice of \(Z^+\) above \(x\) is +\[ + S=\PP_Y(\OO_Y\oplus a^*\mathscr L),\qquad + a:Y=\Bl_xX\longrightarrow X, +\] +and \(J_+|_S\) is the pullback of \(F\). We choose \(x\) so that +Equation~\eqref{nv:slope-blowup}, Equation~\eqref{nv:point-bound}, the restrictions of +the currents below, and all determinant maps for divisible \(j\) +can be tested on this slice. These exclude at most countably many +proper analytic sets and the measure-zero exceptional sets for +current restrictions. + +Write the restricted determinant line as +\[ + (\det\mathcal F_j^+)|_S + =\pi_S^*\mathscr Q_j^+\otimes\OO_S(\ell_j^+),\qquad + Q_j^+=c_1(\mathscr Q_j^+). +\] +Factoring the common zero of order \(h_j\) from the determinant +map gives a nonzero map whose source on \(S\) is +\[ + \pi_S^*(\mathscr Q_j^+\otimes\OO_Y(h_jF)) + \otimes\OO_S(\ell_j^+). +\] +The plus sign of \(h_jF\) follows because division of the map by +the local equation of \(J_+^{h_j}\) enlarges its source from +\(\det\mathcal F_j^+\) to +\(\det\mathcal F_j^+\otimes\OO(h_jJ_+)\). +Its target embeds into the \(jsR_j^+\)-fold tensor of the +restriction of \(p_+^*\Omega_Z\). In particular we use the +cotangent bundle of the original \(Z\), with its restricted +filtration +\[ + 0\longrightarrow + \pi_S^*(\OO_Y^{\oplus n}\oplus a^*\Omega_X) + \longrightarrow (p_+^*\Omega_Z)|_S + \longrightarrow + \OO_S(-2)\otimes\pi_S^*a^*\mathscr L + \longrightarrow0. +\] +The same homogeneous-monomial calculation as in +Equation~\eqref{nv:determinant-bound-Z}, now applying +Equation~\eqref{nv:slope-blowup} to the second factor of \(X^2\), gives +\begin{equation}\label{nv:determinant-bound-slice} + \ell_j^+\le0,\qquad + Q_j^++h_j\{F\} + \le(jsR_j^+-\ell_j^+)a^*L. +\end{equation} +More explicitly, if the chosen graded term has \(i\) relative +factors, the pushed polynomial has degree +\(-2i-\ell_j^+\ge0\). Its exponent in \(a^*\mathscr L\), together +with the \(i\) relative factors and the cotangent order from the second +factor of \(X^2\), is at most \(jsR_j^+-\ell_j^+\). The cotangent +factors from the first factor of \(X^2\) are constant. This proves the +displayed inequality using +the tensors of \(a^*\Omega_X\), without introducing \(\Omega_Y\) +or \(\Omega_{Z^+}\). + +Apply Lemma~\ref{nv:direct-image} to the decomposition in +Equation~\eqref{nv:plus-decomposition}. Equation~\eqref{nv:GRR-base} +shows that the limits of the restricted determinant components divided by +\(jR_j^+\) exist; denote them by \(Q^+\) and \(\ell^+\). Divide +Equation~\eqref{nv:determinant-bound-slice} by \(jR_j^+\), use +Equation~\eqref{nv:determinant-order-equation}, and pass to the closed +pseudo-effective cone. We obtain +\begin{equation}\label{nv:slice-upper} + \ell^+\le0,\qquad + Q^++c_ns\{F\}\le(s-\ell^+)a^*L. +\end{equation} +The pseudo-effective class in Equation~\eqref{nv:GRR-base}, restricted to the +very general slice \(S\), is +\begin{equation}\label{nv:slice-psef} + \pi_S^*(Q^++2a^*P)+(\ell^++2q)\xi\ge0. +\end{equation} +Testing on a general vertical line gives \(\ell^++2q\ge0\). +Let \(A\subset S\) be the axis of +\(\PP_Y(\OO_Y\oplus a^*\mathscr L)\) with divisor class \(\{A\}=\xi\). +On \(A\simeq Y\), its normal class is \(\{A\}|_A=\xi|_A=a^*L\ge0\). +Add an arbitrarily small Kähler class to the class in +Equation~\eqref{nv:slice-psef}, and choose a Kähler current with analytic +singularities in the resulting big class. Remove its generic divisorial +pole along \(A\) and restrict +the residual current to \(A\). Adding back the removed nonnegative multiple +of \(a^*L\) preserves pseudo-effectivity. Passing to the limit gives +\begin{equation}\label{nv:slice-lower} + Q^++2a^*P+(\ell^++2q)a^*L\ge0. +\end{equation} + +The slope bound in Equation~\eqref{nv:slice-upper} supplies the +pseudo-effective class \((s-\ell^+)a^*L-Q^+-c_ns\{F\}\). Adding it to +Equation~\eqref{nv:slice-lower} cancels the entire normalized determinant +class restricted to the axis, \(Q^++\ell^+a^*L\), leaving +\[ + 2a^*P+(s+2q)a^*L-c_ns\{F\}\ge0. +\] +Since \(L\le P/r\), the point bound in Equation~\eqref{nv:point-bound} implies +\begin{equation}\label{nv:contradiction} + \frac{c_ns}{\,2+(s+2q)/r\,}\le C_n. +\end{equation} +The denominator is bounded independently of \(q\), because +\(s\le C_nq^{1/d}\) and \(r\ge q^2\). The numerator tends to +infinity, because \(s\ge c_nq^{1/d}\). Equation~\eqref{nv:contradiction} +is impossible for arbitrarily large \(q\). + +For each \(q\), the choice of \(t(q)\), the rational \(s\), the current +\(T\), and the modifications precedes the limit in \(j\). The monomial +cutoff \(\delta\) depends only on \(n\), while the required lower bound for +divisible \(j\) may depend on the fixed data. We choose the very general +points after those data are fixed. +Thus every simultaneous general-point test above involves at most +countably many conditions, and no bound depends on the point. +The contradiction disproves the contrary hypothesis and proves +Theorem~\ref{nv:meromorphic}. +\end{proof} diff --git a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/07-signed-rigidity.tex b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/07-signed-rigidity.tex index f3f5665f4946de8f895287f9a38c9700aa9be2d2..0cfc5c96c711a4fd848746a4d1df3607c2771f2b 100644 --- a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/07-signed-rigidity.tex +++ b/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/07-signed-rigidity.tex @@ -1,1644 +1,1644 @@ -\section{Signed rigidity on simple spaces} -\label{sec:signed} - -The meromorphic section furnished by Theorem~\ref{nv:meromorphic} need -not have an effective divisor. We therefore need a boundary argument -that retains both its zeroes and its poles. The result below supplies -that argument. - -\begin{theorem}[Signed rigidity]\label{signed:torsion} -Let \((X,D)\) be a dlt pair on a normal irreducible compact Kähler -space, and suppose that the pair has the resolution property of -Definition~\ref{def:lc-strata-resolution}. Suppose also that \(X\) is -globally \(\Q\)-factorial, that \(a(X)=0\), that \(X\) is simple, -and that \(D=\sum_{i=1}^sD_i\) is reduced. Put \(L=K_X+D\), and -assume the following. -\begin{enumerate} -\item The rational line \(L\) is analytically nef and has an actual -rational linear equivalence -\[ - L\sim_{\Q}G=\sum_{i=1}^s a_iD_i,\qquad a_i\in\Q. -\] -\item For some real \(c>0\), the pullback of the class -\(\{L-cD\}\) to a projective resolution with smooth compact -Kähler source is pseudo-effective. -\item The rational holomorphic line \(L|_D\) is semiample on the -whole reduced analytic space \(D\): some Cartier multiple is -generated at every point of \(D\). -\end{enumerate} -Then \(L\) is torsion. More precisely, for some positive integer -\(q\), the holomorphic line \(\OO_X(qL)\) is isomorphic to -\(\OO_X\). -\end{theorem} - -To apply this theorem in the proof of Proposition~\ref{simple:case}, -Theorem~\ref{nv:meromorphic} supplies a signed representative of the -canonical bundle. After resolving, we enlarge the support of the -transformed boundary and this signed canonical divisor to a reduced SNC -divisor; the resulting adjoint remains pseudo-effective. -Corollary~\ref{prog:signed-nef} gives an ordinary dlt nef model -\((X_{\mathrm{nef}},D_{\mathrm{nef}})\) whose adjoint -\(L_{\mathrm{nef}}=K_{X_{\mathrm{nef}}}+D_{\mathrm{nef}}\) has a -signed representative supported on \(D_{\mathrm{nef}}\), supplying -the first hypothesis. Theorem~\ref{floor:generation} supplies -semiampleness on the whole reduced \(D_{\mathrm{nef}}\). On a common -smooth resolution, simplicity excludes uniruledness, so Ou's criterion -\cite[Theorem~1.1]{Ou2025} and exceptional translation for the canonical -comparison give pseudo-effectivity of the pullback of -\(K_{X_{\mathrm{nef}}}=L_{\mathrm{nef}}-D_{\mathrm{nef}}\). Thus the -second hypothesis holds with \(c=1\). Once signed rigidity makes the -actual line \(L_{\mathrm{nef}}\) torsion, the program comparison and -Lemma~\ref{bir:unique-current} let us subtract the added boundary and give -the required decomposition for the original adjoint. The final subsection -verifies these steps. - -The signed construction in \cite[Proposition~3.3]{LI} is the source -of the root and separation operations used below. The lifting -method is that of \cite[Sections~10--12]{BL}. -We will establish the analytic form of the signed construction and -the change to the lifting argument caused by its residual poles. -The numerical argument first singles out positive-dimensional fibers in -the positive boundary. We then lift those fibers through every finite -boundary neighborhood and deform them to compact subspaces outside the -boundary. An argument using the cycle space and Baire's theorem turns these -deformations into a covering family, which simplicity excludes. - -\subsection{The boundary detected by the nef class} - -If \(D=0\), the signed equivalence already gives -\(L\sim_{\Q}0\). We may therefore assume in the constructions -below that \(D\ne0\); in particular \(\dim X>0\). - -Fix the data of Theorem~\ref{signed:torsion}, and write -\[ - G=G_+-G_-,\qquad D_0=\sum_{a_i=0}D_i, -\] -where \(G_+\) and \(G_-\) are effective with disjoint prime supports. -Choose a positive integer \(m\) so that \(mG_+\), \(mG_-\), -\(mD_0\), and \(mL\) are Cartier, the given rational equivalence -induces a fixed isomorphism -\[ - N_0:=\OO_X(mG)\simeq\OO_X(mL), -\] -and \(N_0|_D\) is generated. Its sections give a morphism and an -actual line isomorphism -\begin{equation}\label{signed:boundary-map} - \phi:D\longrightarrow P=\PP^b,\qquad - N_0|_D\simeq\phi^*\OO_P(1). -\end{equation} -Let \(n=\dim X\). Choose a projective resolution -\(\mu:\widetilde X\to X\) as in the second hypothesis of the theorem, -and a Kähler class \(\omega\) on \(\widetilde X\). The numerical -dimension of the nef pullback is -\[ - v=\nu(L):=\max\bigl\{k\in\{0,\ldots,n\}: - (\mu^*\{L\})^k\omega^{n-k}>0\bigr\}. -\] - -The next argument is the Kähler form of -\cite[Lemma~3.2]{LI}. - -\begin{lemma}\label{signed:boundary-geometry} -If \(v=0\), then \(D=0\) and \(L\sim_{\Q}0\). If \(v=n>0\), -then \(a(X)=n\). If \(00}\dim\phi(D_i)=r, -\] -and -\begin{equation}\label{signed:small-intersections} - \dim\phi\bigl(\Supp G_+\cap(\Supp G_-\cup D_0)\bigr)0\) such that -\(C\omega-h\) is nef and \(h-\epsilon\omega\) is -pseudo-effective. Pairing these two inequalities successively with -products of the nef classes \(\mu^*\{L\}\), \(h\), and -\(\omega\) shows that -\[ - (\mu^*\{L\})^k h^{n-k}>0 - \quad\Longleftrightarrow\quad - (\mu^*\{L\})^k\omega^{n-k}>0. -\] -We may therefore use \(\theta\) in all the tests defining \(v\). - -Suppose first that \(v0\) attains dimension \(r\). - -It remains to separate its intersections from a general fiber. -On \(H^{1,1}_{\BC}(\widetilde X,\R)\) consider the form -\[ - q(\alpha,\beta)= - \alpha\beta(\mu^*\{L\})^r h^{n-r-2},\qquad - Q_{ij}=q(\mu^*\{D_i\},\mu^*\{D_j\}). -\] -The mixed Kähler Hodge index theorem, followed by approximation of -the nef factors by Kähler classes, says that \(q\) has at most one -positive direction; see \cite[Theorems~A and C]{DinhNguyen}. Put -\(l=\mu^*\{L\}\). The definition of \(v\) and -Equation~\eqref{signed:boundary-zero} give -\[ - q(l,l)=0,\qquad q(l,\mu^*\{D_i\})=0, - \qquad q(l,h)=\{L\}^{v}\theta^{n-v}>0. -\] -Linear algebra now makes \(q\) negative semidefinite on -\(l^\perp\): a positive vector in \(l^\perp\), together with a -suitable vector in the span of \(l,h\), would give two positive -directions. In particular \(Q\) is negative semidefinite. - -For \(i\ne j\), one has \(Q_{ij}\geq0\). This assertion uses -the effective intersection cycle of the distinct \(\Q\)-Cartier -divisors \(D_i,D_j\) on \(X\), and projection formula. It does -not require their total transforms to intersect effectively. The -two Cartier equations have a proper intersection: the dlt ambient -space is klt, hence Cohen--Macaulay, after dropping the reduced -boundary, and the distinct divisor equations form a regular -sequence. The intersection cycle is consequently pure of -codimension two, and each of its mixed nef degrees is nonnegative. - -Write the coefficient vector as -\(\mathbf a=\mathbf a^+-\mathbf a^-\), with nonnegative vectors -of disjoint supports. The signed equivalence and orthogonality to -\(l\) give \(Q\mathbf a=0\). Thus -\[ - Q(\mathbf a^+,\mathbf a^+) - =Q(\mathbf a^+,\mathbf a^-)\geq0. -\] -Negative semidefiniteness forces equality and -\(Q\mathbf a^+=0\). For every \(j\) with \(a_j\leq0\), its -row is a sum of nonnegative off-diagonal terms: -\[ - 0=\sum_{a_i>0}a_iQ_{ji}. -\] -Each term is zero. Applying the generated-line degree test to the -effective intersection cycles proves -Equation~\eqref{signed:small-intersections}. If \(r=0\), any -nonempty intersection would have positive \(\theta^{n-2}\)-degree, -so these intersections are empty. - -Finally, if \(v=n>0\), the nef volume criterion makes the rational -line on \(\widetilde X\) big -\cite[Theorem~0.5]{DemaillyPaun2004}. A big holomorphic line on a -compact Kähler manifold has maximal Iitaka dimension. Hence -\(a(\widetilde X)=n\), and birational invariance gives \(a(X)=n\). -\end{proof} - -Thus, when \(00\). -Equation~\eqref{signed:small-intersections} makes those fibers disjoint -from \(\Supp G_-\cup D_0\). The next construction prepares their normal -directions and adjunction for infinitesimal lifting. - -\subsection{Local roots and residue with residual poles} - -We work for the rest of the signed argument in the range -\(01\) divisible by \(m\) -and by every nonzero integer \(|ma_i|\), and put -\(a=\ell/m\), a positive integer. - -\begin{lemma}[Analytic signed charts]\label{signed:root-charts} -For each \(t\in P\), after shrinking an open neighborhood -\(U\subset P\), choose a line \(R_U\) together with an isomorphism -\(R_U^\ell\simeq\OO_P(1)|_U\). A comparison of two such roots on an -overlap is power-compatible if its \(\ell\)-th power commutes with these -isomorphisms. If -\(t\notin\phi(\Supp G_+)\), one may take \(U\) disjoint from this -closed image and set \(Z_U=S=T=\varnothing\); the assertions below are -then vacuous for the unique maps from the empty spaces. If -\(t\in\phi(\Supp G_+)\), there are a Hausdorff normal -analytic space \(Z_U\) of pure dimension \(n\), considered near a -reduced Cartier divisor \(S\) of pure dimension \(n-1\), a -reduced Weil divisor \(T\) with no component in common -with \(S\), and maps -\[ - \pi:Z_U\longrightarrow X,\qquad g:S\longrightarrow U -\] -with the following properties. -\begin{enumerate} -\item The map \(g\) is proper, and \(\pi|_S\) is finite over -\(D_U=\phi^{-1}(U)\), with \(g=\phi\pi|_S\). Its image covers -the positive components over \(U\), and -\[ - \OO_S(S)\simeq g^*R_U. -\] -The image of \(S\cap T\) lies in -\(\phi(\Supp G_+\cap(\Supp G_-\cup D_0))\). -Near \(S\), the underlying sets satisfy -\[ - \pi^{-1}(|D|)=|S|\cup|T|. -\] -\item The pair \((Z_U,S+T)\) is log canonical, and there is a fixed -actual isomorphism of divisorial sheaves -\begin{equation}\label{signed:adjoint-chart} - \tau:\OO_{Z_U}(aS)\xrightarrow{\ \simeq\ }\omega_{Z_U}(S+T). -\end{equation} -The spaces \(Z_U\) and \(S\) are Cohen--Macaulay. The support -of \(T\) is locally set-theoretically principal. -Off \(T\), the space \(Z_U\) is klt and its canonical sheaf is -invertible. The map \(\pi\) is quasi-finite near \(S\setminus T\). -\item Projective log resolutions of the total divisor data can be -chosen with Kähler forms on neighborhoods of the compact sets over -each compact fiber of \(g\). Power-compatible changes of the -boundary root extend to isomorphisms of ambient germs that preserve -the full ideal \(\OO_{Z_U}(-S)\), the reduced divisor \(T\), and -Equation~\eqref{signed:adjoint-chart}. Resolutions may be chosen -functorially for these germ isomorphisms. -\end{enumerate} -Here the nonempty space \(Z_U\) is a neighborhood of the whole compact -fiber under consideration; no extension of \(g\) to \(Z_U\) is asserted. -\end{lemma} - -\begin{proof} -The empty branch follows by shrinking away from the closed image -\(\phi(\Supp G_+)\). We construct the nonempty charts in four stages: -take roots and trivialize the remaining adjoint torsion, separate the -positive and negative root divisors, retain the root line that controls the -normal direction, and pass to an ordinary neighborhood of the compact fiber. - -First consider an analytic open set on which the Cartier lines of -\(mG_+\), \(mG_-\), and \(mD_0\) are trivial, and let -\(f_+,f_-,f_0\) be the functions representing their sections. Normalize -the entire finite space -\[ - \{y_+^\ell=f_+,\qquad y_-^\ell=f_-,\qquad y_0^\ell=f_0\}, -\] -retaining all its components and the lifted action of \(\mu_\ell^3\) -that multiplies the three root coordinates. On an overlap, the functions -\(f_+,f_-,f_0\) change by holomorphic units. Local \(\ell\)-th roots of -those units give comparisons of the finite spaces that lift uniquely to -their normalizations; different choices differ by \(\mu_\ell^3\). -The quotient stacks of these normalized charts by \(\mu_\ell^3\) glue to -the simultaneous normalized root stack over \(X\), which we denote by -\(\mathcal X_{\mathrm{rt}}\). The zero divisors -of the three root sections are denoted by \(S_+,S_-,S_0\). At a -generic prime of multiplicity \(b\), a normalized chart of -\(y^\ell=x^b\) has ramification index \(\ell/b\) and -\(\ord(y)=1\). Here \(b=|ma_i|\) or \(b=m\), so it divides -\(\ell\). Thus each root divisor is generically reduced and -Cartier. A Cartier divisor on a normal space satisfies Serre's -condition \(S_1\), so it is reduced. Log ramification with the -full reduced boundary gives a log canonical pair and the rational -equivalence -\[ - K_{\mathcal X_{\mathrm{rt}}}+S_++S_-+S_0 - \sim_{\Q}a(S_+-S_-). -\] -The ambient charts of \(\mathcal X_{\mathrm{rt}}\) are klt. Off the -boundary this follows from -the klt complement downstairs and the unramified charts. For a -place centered in the Cartier boundary, dropping that boundary -increases its log discrepancy by its strictly positive order; -this also makes every zero-discrepancy place positive. - -To turn the rational adjoint equivalence into an actual isomorphism, -consider the integral reflexive sheaf on \(\mathcal X_{\mathrm{rt}}\) -\[ - \mathcal F= - \bigl(\omega_{\mathcal X_{\mathrm{rt}}}(S_++S_-+S_0)\otimes - \OO_{\mathcal X_{\mathrm{rt}}}(-a(S_+-S_-))\bigr)^{**}. -\] -It is torsion. Choose a periodicity -\(\mathcal F^{[q]}\simeq\OO\), take the relative spectrum of its -reflexive-power algebra, and normalize; write \(\rho_{\mathrm{ind}}\) -for this finite map, and retain \(S_+,S_-,S_0\) for the pulled-back -root divisors on the cover. At codimension one this -is a cover \(w^q=u\) for a unit \(u\), hence is unramified. -Log canonicity, the klt ambient property, and reduced Cartier -root divisors persist. The tautological evaluation of this -algebra trivializes the reflexive pullback -\((\rho_{\mathrm{ind}}^*\mathcal F)^{**}\): it does so in -codimension one, and the identity then extends reflexively. -The evaluation is a sheaf morphism on the cover stack itself, -so the resulting actual divisorial isomorphism -\[ - \omega(S_++S_-+S_0)\simeq\OO\bigl(a(S_+-S_-)\bigr) -\] -on the cover is equivariant on every atlas. - -Next separate the positive and negative root divisors. Let -\(\mathcal X_{\mathrm{sep}}\) be the normalization of the blowup of the -ideal generated by their equations on the cover, and write \(p_1\) for its -map. This blowup embeds in a relative -\(\PP^1\); its normalization still has fibers of dimension at -most one. Each exceptional prime therefore lies over a -codimension-two positive--negative intersection. At its generic -point the original dlt pair is a two-branch SNC pair. After -extracting roots of units the normalized Kummer chart there has -smooth coordinates -\(x=u^{\ell/b_+}\), \(z=w^{\ell/b_-}\), where \(b_+\) and -\(b_-\) are the two corresponding multiplicities. On this chart -\(\mathcal F\) is a line and its periodicity cover is unramified. -The two-coordinate blowup calculation consequently applies at -every exceptional generic point. If \(E\) denotes its -exceptional divisor, it gives reduced Cartier divisors -\[ - E,\qquad S'_+=p_1^*S_+-E,\qquad S'_-=p_1^*S_--E, -\] -with \(S'_+\cap S'_-=\varnothing\). No such two-branch stratum -is contained in \(S_0\), so \(p_1^*S_0\) is reduced and has no -exceptional component. The full boundary -\(S'_++S'_-+E+p_1^*S_0\) is crepant and reduced. Its -codimension-one equality with the pullback adjoint gives the -discrepancy equality for every further valuation, hence log -canonicity. The ambient charts after this blowup are still -klt. Off the full boundary the blowup is an isomorphism to -the old klt complement. A place of zero pair discrepancy -centered in that boundary gains its strictly positive order -when the effective Cartier boundary is dropped, while a -place of positive pair discrepancy remains positive. The -strict divisor \(S'_+\) is finite over \(S_+\): -before normalization it lies in one section of the projective -ratio coordinate, so its proper map has finite fibers, and -normalization is finite. - -On \(\mathcal X_{\mathrm{sep}}\), put \(S=S'_+\) and, temporarily, -\(T=S'_-+E+p_1^*S_0\). Write -\(\pi_{\mathrm{sep}}:\mathcal X_{\mathrm{sep}}\to X\) for the composite -map, and define the retained root line -\[ - \mathcal R:=\OO_{\mathcal X_{\mathrm{sep}}}(S'_+-S'_-), - \qquad \mathcal R^\ell\simeq\pi_{\mathrm{sep}}^*N_0. -\] -Near \(S\), the negative strict root section is a unit. The fixed adjoint -isomorphism becomes -\(\omega_{\mathcal X_{\mathrm{sep}}}(S+T) - \simeq\OO_{\mathcal X_{\mathrm{sep}}}(aS)\), and the positive root -section divided by that negative unit section is a section of -\(\mathcal R\) with zero divisor \(S\). This line and section determine -the Cartier divisor \(S\) and its normal line; they must survive passage -to an ordinary chart. - -The pair consisting of \(\mathcal R\) and its displayed power isomorphism -defines a map to the gerbe \(\sqrt[\ell]{N_0}\), which parametrizes -\(\ell\)-th roots of \(N_0\). Take the relative coarse space over this -gerbe. On an atlas this quotients by the finite relative inertia kernel, -the subgroup of each stabilizer acting trivially on \(\mathcal R\). -Thus the quotient retains the root character of \(\mathcal R\). The -deck transformations of the representable periodicity cover -are not this inertia kernel and are not removed. - -Both \(\mathcal R\) and its section descend through the kernel, so the -reduced image \(S\) remains Cartier and its full ideal descends. We retain -\(S,T\) for the reduced images. A finite-group norm of -a local equation of the upstairs \(T\) has precisely its image -as zero set. Thus the reduced image \(T\) is locally -set-theoretically principal; it need not be Cartier. All -divisorial ramification lies on the full boundary. Log -ramification identifies the invariant log dualizing sheaf in -codimension one with the downstairs one. Taking reflexive -hulls and invariants of the equivariant adjoint isomorphism -gives an actual descended isomorphism \(\OO(aS)\simeq\omega(S+T)\). -Its ordinary pullback below is Equation~\eqref{signed:adjoint-chart}. -Outside \(S+T\), the local -finite quotient is quasi-étale, so its klt upstairs complement gives -a klt downstairs complement by finite discrepancy comparison. Away from -\(T\), dropping the Cartier divisor \(S\) from the lc pair -increases the log discrepancy of every place centered in \(S\) by -its strictly positive order. Thus the quotient charts are klt away from -\(T\), and the same isomorphism makes their canonical sheaves invertible -there. Off \(T\) the separating blowup is an -isomorphism, while the other operations are finite; hence -the composite map to \(X\) is quasi-finite near \(S\setminus T\). - -The same quotient preserves Cohen--Macaulayness of both the -ambient space and \(S\). Before the quotient the ambient -space is klt and hence Cohen--Macaulay, and \(S\) is Cartier -there. Locally its finite quotient ring, and the quotient -ring of \(S\), are invariant direct summands of those finite -Cohen--Macaulay rings. A system of parameters downstairs is -one upstairs; the vanishing of lower local cohomology upstairs -and the invariant projection give its vanishing for the -downstairs direct summand. This proves the stated depth. - -Removal of the relative inertia kernel also makes the strict -positive divisor finite and representable over -\[ - D\times_P\sqrt[\ell]{\OO_P(1)}. -\] -Its root line is \(\OO_S(S)\), and its image covers -\(\Supp G_+\). The intersection \(S\cap T\) maps to the -positive--negative or positive--zero intersections: the only -new component is the separating exceptional divisor. Before the quotient, -the inverse image of \(|D|\) is the support of -\(S'_++S'_-+E+p_1^*S_0\). The finite quotient sends this full support -to \(|S|\cup|T|\), and the equality of underlying sets persists under the -ordinary base change below. The image of \(S\cap T\) in \(P\) consequently -has dimension \(0\). This -proves Equation~\eqref{signed:full-residual-comparison}. - -Projection formula gives the same comparison after adding -\(D_S=p^*S\). The quotient sequence therefore identifies -\[ -\begin{aligned} - R(p_D)_*\left( - \frac{\omega_{\widehat Z}(D_S+C)}{\omega_{\widehat Z}(C)} - \right) - &\simeq \omega_Z(S+T)|_S[0]\\ - &\simeq \OO_S(aS)[0], -\end{aligned} -\] -where \(p_D:D_S\to S\) and the quotient is a sheaf on the -possibly nonreduced divisor \(D_S\). To justify the displayed -derived statement, first push it by the exact closed immersion -\(S\hookrightarrow Z\). There it is the quotient of the two -acyclic comparisons. Closed pushforward detects cohomology -sheaves, and canonical truncation yields the statement on \(S\). - -There is a quotient morphism -\begin{equation}\label{signed:quotient-map} -\begin{aligned} - \omega_H(A)&= - \frac{\omega_{\widehat Z}(H+A)}{\omega_{\widehat Z}(A)} - \\ - &\longrightarrow - \frac{\omega_{\widehat Z}(D_S+C)}{\omega_{\widehat Z}(C)}. -\end{aligned} -\end{equation} -Indeed \(H+A\leq D_S+C\) and \(A\leq C\). This morphism -need not be injective when \(C\) and \(H\) share a component. -Compose its derived pushforward with the preceding isomorphism. -The result retracts Equation~\eqref{signed:derived-insertion}: -at each generic point of \(S\), its composite is the identity -under the fixed residue convention. The composite is an -endomorphism of the invertible sheaf \(\OO_S(aS)\) in degree -zero; since \(S\) is reduced, agreement at every generic point -implies agreement everywhere. Applying \(Rg_*\) proves -Equation~\eqref{signed:residue-injection}. All maps use the -fixed adjoint isomorphism, divisor ideals, and residue, so they -commute with the germ comparisons. Relative canonical forms -under products, followed by wedging with the canonical frame -of the new factor, give the last assertion. -\end{proof} - -\subsection{The localized Hodge argument} - -The target in Equation~\eqref{signed:residue-injection} contains -the additional poles \(A\). The following version of the filtered -SNC calculation incorporates them. In its application, we embed \(H\) -by the graph of \(h\) in \(\widehat Z\times U\) and use the projection to -\(U\) as the ambient map. This projection is submersive, while its -restriction to the graph is the proper map \(h\); it requires no extension -of \(h\) to \(\widehat Z\). We will use the same graph construction after -a change of parameter. The parameter is projective only in the last -assertion below; the closed-stratum maps are proper Kähler maps. - -\begin{proposition}\label{signed:localized-hodge} -Let \(q:\mathscr V\to Y\) be a holomorphic map of complex -manifolds, submersive near a reduced closed analytic subspace -\(i:H\hookrightarrow\mathscr V\) of pure dimension \(d\) and -codimension \(c_0>0\). Suppose \(h=q|_H\) is proper. Assume -that locally \(H\) is an SNC divisor in a smooth submanifold of -codimension \(c_0-1\), and that a reduced divisor \(A\) is -simultaneously SNC and transverse to those support equations. -Assume that the components \(H_i\) and \(A_j\) near \(H\) have -finite global smooth indexing, and that every closed stratum -\(H_I\cap A_J\), with \(I\ne\varnothing\), is smooth and proper -over \(Y\) and carries a global relative Kähler form. Empty or -disconnected strata are allowed. -Here \(H_I=\bigcap_{i\in I}H_i\) and -\(A_J=\bigcap_{j\in J}A_j\), with \(A_\varnothing=\mathscr V\). - -Use right \(\mathscr D\)-modules and finite-pole local cohomology, -and put -\[ - \mathcal K_A=\mathcal H^{c_0}_{[H]}(\omega_{\mathscr V})(*A). -\] -Its order filtration is generated from -\(F_0\mathcal K_A=i_*\omega_H(A)\), with \(F_p=0\) for \(p<0\). -For \(M^j=\mathcal H^j q_+\mathcal K_A\), the filtered direct -image is strict at every level, and \(M^j\) has a finite -filtration by submodules, strict for \(F\), whose quotients are -polarizable real pure Hodge modules. In particular, -\begin{equation}\label{signed:lowest-step} - F_0M^j=R^jh_*\omega_H(A)\lhook\joinrel\longrightarrow M^j, - \qquad F_pM^j=0\quad(p<0). -\end{equation} -Define the symbol morphism -\[ - \sigma:F_0M^j\otimes T_Y\longrightarrow\gr^F_1M^j, - \qquad \sigma(m\otimes\xi)=[m\xi]. -\] -If \(Y\) is smooth projective, then for every ample line \(N\) on \(Y\), -\begin{equation}\label{signed:symbol-vanishing} - \Hom_{\OO_Y}(N,\Ker\sigma)=0. -\end{equation} -This includes any coherent torsion in the kernel. -\end{proposition} - -\begin{proof} -We verify the residual localization in the proof of -\cite[Proposition~11.1]{BL}. We first identify the lowest order step and -a filtration by the number of branch poles. Its graded pieces will be -dual constant modules on smooth strata, to which proper Kähler direct -image applies. The resulting strictness gives the injection at the lowest -filtration step, -and the graded de Rham complex on a projective base gives the symbol -vanishing. - -At a point of \(H\), choose -simultaneous coordinates -\[ - (z_1,\ldots,z_{c_0-1},x_1,\ldots,x_t,y_1,\ldots,y_s,\xi), - \quad - \mathscr I_H=(z_1,\ldots,z_{c_0-1},x_1\cdots x_t), - \quad A=(y_1\cdots y_s=0). -\] -Let \(\eta\) be the coordinate volume form. The localization -Čech complex for the displayed regular support equations has -cohomology only in degree \(c_0\). Localizing this cohomology -also in the \(y\)'s gives \(\mathcal K_A\). Successive finite -Taylor divisions give its unique additive polar normal forms: -finite sums of -\begin{equation}\label{signed:polar-form} - f(x_{I^c},y_{J^c},\xi) - \prod_{\nu=1}^{c_0-1}z_\nu^{-u_\nu} - \prod_{i\in I}x_i^{-v_i} - \prod_{j\in J}y_j^{-w_j}\eta, - \quad I\ne\varnothing, -\end{equation} -where every displayed order is positive and \(J\) is arbitrary, -including empty. The coefficient is independent of exactly -the variables occurring negatively. The normal form is an -additive statement, not an \(\OO_{\mathscr V}\)-linear -decomposition. - -The simple fractions, with each displayed order equal to one, -are exactly the image of \(\omega_H(A)\). More explicitly, -their unreduced representatives are -\[ - \frac{f\eta} - {z_1\cdots z_{c_0-1}(x_1\cdots x_t)(y_1\cdots y_s)}. -\] -The regular-equation residue identifies their numerators modulo -\(\mathscr I_H\) with the absolute dualizing sheaf of \(H\), -twisted by \(A\). This map is injective. If the fraction has -no remaining \(x\)-pole, Taylor division says that its -numerator modulo the \(z\)'s is divisible by \(x_1\cdots x_t\). -Localization in the \(y\)'s does not enlarge this kernel, -because each \(y_j\) is a nonzerodivisor modulo -\(\mathscr I_H\). This also proves the identification under -changes of coordinates, including the determinant of the -normal conormal frame. - -Give a term of Equation~\eqref{signed:polar-form} the excess -\[ - e=\sum_\nu(u_\nu-1)+\sum_{i\in I}(v_i-1) - +\sum_{j\in J}(w_j-1). -\] -The right canonical action of a coordinate derivative is minus -differentiation of the coefficient of \(\eta\). A derivative -in a denominator variable increases its order by one with a -nonzero scalar. Because the coefficient in the normal form -is independent of those variables, every term of excess -\(e\) is obtained from a simple term by \(e\) such derivatives. -Conversely, order \(p\) differentiations produce excess at -most \(p\). Thus \(F_p\mathcal K_A\) consists exactly of the -finite sums of excess at most \(p\). This is the good order -filtration generated from \(i_*\omega_H(A)\). - -There is also an increasing filtration \(W\) by the total -number of branch poles. Intrinsically, its step of index -\(-d+k-1\) is the sum of the images of support modules for -subunions of the \(H_i\), localized along sublists of the -\(A_j\), for which the total number of selected branches is -at most \(k\). In the normal form it imposes -\(|I|+|J|\leq k\). The induced double filtration therefore -has -\begin{equation}\label{signed:localized-graded} - \gr^W_{-d+k-1}(\mathcal K_A,F) - \simeq - \bigoplus_{\substack{|I|+|J|=k\\ I\ne\varnothing}} - (i_{I,J})_+(\omega_{H_I\cap A_J},F^{(0)}). -\end{equation} -Here \(F^{(0)}_p\omega_{H_I\cap A_J}=0\) for \(p<0\) and is -\(\omega_{H_I\cap A_J}\) for \(p\geq0\). -Here \(A_\varnothing=\mathscr V\), and empty intersections -contribute zero. To check this formula, retain the polar type -using exactly \(I,J\). Its normal denominators are the -\(c_0-1\) equations \(z_\nu\), the \(|I|\) selected \(x\)'s, -and the \(|J|\) selected \(y\)'s. They are the regular -equations of the smooth stratum \(H_I\cap A_J\), whose -dimension is \(d-k+1\). Right closed transfer has no -codimension shift in \(F\), and the excess counts exactly its -normal derivative orders. Mayer--Vietoris boundary maps for -the \(x\)-subunions and the localization boundary for each -selected \(y\) identify these quotients intrinsically with -the support module of that intersection. For the latter -assertion, localization modulo the nonlocalized module in one -transverse \(y\) direction is its first local cohomology in -that direction. A fixed order of the component labels fixes -the residue signs, so the identifications glue. The smooth- -support graded modules are regular holonomic, and their finite -extension \(\mathcal K_A\) is regular holonomic as well. - -The filtration \(W\) has a real realization. Tensor the -finite-pole de Rham comparisons for the regular support -coordinates with the open localization comparison for the -\(y\) coordinates, whose one-coordinate factor is the -cohomology of a punctured disk. These comparisons are -natural for all sublists. The resulting real support and -localization complexes complexify to the de Rham complexes -just computed. The latter are perverse by regular -holonomicity; exact and faithful complexification gives the -same assertion for the real complexes. Their real perverse -images then give \(W\). -On a graded stratum of dimension \(d-k+1\), the real object -is its dual constant object -\(\R^H[d-k+1](d-k+1)\). Its first right filtration step is -zero and its weight is \(-d+k-1\), in agreement with -Equation~\eqref{signed:localized-graded}. Residue signs and -constant normalizations preserve its real polarization. This -is the real comparison in \cite[Section~11.2]{BL}, with the -punctured-disk factors supplying the additional localization -boundaries. - -The proper-support check also survives localization. If a -holomorphic germ \(f\) vanishes on reduced \(H\), then -\begin{equation}\label{signed:support-condition} - fF_p\mathcal K_A\subset F_{p-1}\mathcal K_A. -\end{equation} -For a generator \(z_\nu\) of \(\mathscr I_H\), multiplication -reduces its pole order or kills the class. Multiplication by -\(x_1\cdots x_t\) also lowers the excess or kills the class, -because the polar type always has \(I\ne\varnothing\). -The \(y\)-poles do not affect this argument. In particular -every graded stratum stays in the proper support \(H\), and -properness is needed only there. - -We have now identified the order filtration, its real pure stratum pieces, -and the proper support of each level. We next pass these data to the base. -Apply Saito's constant-source Kähler direct-image theorem -\cite[Theorem~1 and Remark~1.2]{SaitoKahler} to each smooth -closed stratum in Equation~\eqref{signed:localized-graded}, -using its stipulated relative Kähler form. The resulting -degree-\(j\) direct image of the step indexed by \(\lambda\) -is strict and polarizable real pure of weight \(\lambda+j\). -In the submersion neighborhood, the level-\(F_p\) relative -right de Rham complex has term -\[ - F_{p-e}\mathcal K_A\otimes\bigwedge^e T_{\mathscr V/Y} - \quad\text{in degree }-e. -\] -The simultaneous excess and branch bounds show that passing -to a \(W\)-quotient commutes with taking each such level. - -To prove strictness, use the finite \(W\) spectral sequence -\[ - E_1^{-\lambda,j+\lambda} - =\mathcal H^j q_+\gr^W_\lambda\mathcal K_A - \quad\Longrightarrow\quad \mathcal H^j q_+\mathcal K_A. -\] -Its first-page terms and their level filtrations are strict -pure objects by the preceding application. The differentials -are real and filtration preserving, because the support and -localization comparisons are real and natural. The first -differential preserves weight, hence is a strict morphism of -pure Hodge modules; its kernels and cokernels remain pure. -A higher differential sends \((\lambda,j)\) to -\((\lambda-k,j+1)\) for \(k\geq2\) and strictly lowers -weight. Such a real filtered map is zero. On distinct strict -supports this follows from strict support; on the same dense -smooth support a real map preserves both Hodge filtrations, -and a source vector of type \((p,q)\) would have image in -\(F^p\cap\overline F^q\) of a pure object of smaller weight, -which is zero. Both the unfiltered and each level spectral -sequence therefore degenerate at the second page. Their -comparison is injective there. A least-\(W\)-step argument -on the finite abutment proves injection at every \(F\) level -and strictness of its induced \(W\) filtration. This is the -strict-direct-image argument of \cite[Proposition~11.1]{BL}, -now justified for every graded term with \(A\)-poles. - -For \(p<0\), the relative complex is zero. For \(p=0\), its -only term is \(F_0\mathcal K_A=i_*\omega_H(A)\) in degree -zero. The level injection gives -Equation~\eqref{signed:lowest-step}. - -Suppose finally that \(Y\) is smooth projective. Each strict- -support pure quotient just obtained is of exactly the type in -\cite[Lemma~11.2]{BL}: a polarizable real pure direct-image -piece arising from a dual constant on a smooth Kähler source. -That lemma compares its actual right filtration with the -Sabbah--Schnell pure component extending the same generic -polarized real variation. Its proof uses uniqueness of the -intermediate extension and the reconstruction of \(F\) from -the canonical \(V\)-filtration, so it depends only on this -pure piece, not on the divisor presentation of its source. -It gives the negative-ample vanishing -\[ - \mathbb H^u\bigl(Y,N^{-1}\otimes\gr^F_p\mathrm{DR}_Y Q\bigr)=0 - \quad(u<0) -\] -for each pure quotient \(Q\), by -\cite[Theorem~16.3.10]{SabbahSchnell}. The finite strict -filtration extends this vanishing to \(M^j\). Since its -negative \(F\) levels vanish, its degree-one graded right -de Rham complex is exactly -\[ - \gr^F_1\mathrm{DR}_Y M^j= - \left[F_0M^j\otimes T_Y\xrightarrow{\sigma}\gr^F_1M^j\right] - \quad\text{in degrees }-1,0. -\] -Its negative hypercohomology after tensoring by \(N^{-1}\) -is \(H^0(Y,N^{-1}\otimes\Ker\sigma)\). It vanishes, which -is Equation~\eqref{signed:symbol-vanishing}. This calculation -does not assume that the kernel is locally free. -\end{proof} - -\subsection{Lifting every finite boundary neighborhood} - -Return to the charts and resolution of -Lemma~\ref{signed:root-charts} and -Equation~\eqref{signed:resolved-divisors}. On an empty chart the sheaves -below are zero. On a nonempty chart put \(I=\OO_Z(-S)\), an invertible -ideal. For every integer \(j\) -and positive integer \(k\), put -\[ - \mathcal A_j=I^j/I^{j+1}=\OO_S(-jS),\qquad - \mathcal T_{j,k}=I^j/I^{j+k}. -\] -These are sheaves on the underlying topological space of \(S\). -Pushforward by \(g\) has that meaning for \(\mathcal T_{j,k}\); -it does not require a morphism from a thickening to \(U\). -A local frame of \(R_U^{-1}\) gives a Laurent graded frame -\(u^j\) of \(\mathcal A_j\) for every \(j\). It is a frame -on the associated graded, not an extension of a frame to \(Z\). - -\begin{proposition}[All finite lifting orders]\label{signed:all-orders} -For every root chart, every \(j\in\Z\), and every \(k\geq1\), -the map of sheaves of complex vector spaces on \(U\) -\begin{equation}\label{signed:all-order-surjection} - g_*\mathcal T_{j,k+1}\longrightarrow g_*\mathcal T_{j,k} -\end{equation} -is surjective. The assertion holds on every parameter germ, -including germs supported at special parameters. -\end{proposition} - -\begin{proof} -The assertion is immediate for an empty chart. We induct on \(k\), -simultaneously for all integer \(j\) and all nonempty charts. The layer -algebra and its connecting maps are the -ones in \cite[Lemma~12.1]{BL}; we recall their construction to -specify exactly which obstruction is to be killed. - -Assume all smaller orders. The connecting map for -\[ - 0\longrightarrow\mathcal A_{j+k} - \longrightarrow\mathcal T_{j,k+1} - \longrightarrow\mathcal T_{j,k}\longrightarrow0 -\] -factors uniquely through a map -\begin{equation}\label{signed:obstruction} - \delta_k:g_*\mathcal A_j\longrightarrow R^1g_*\mathcal A_{j+k}. -\end{equation} -Indeed the smaller orders make -\(g_*\mathcal T_{j,k}\to g_*\mathcal A_j\) surjective. For -\(k>1\), the kernel comes from \(g_*\mathcal T_{j+1,k-1}\), -by the leading-layer sequence. The order \(k-1\) in degree -\(j+1\) lifts this kernel into \(g_*\mathcal T_{j,k+1}\), so -the current connecting map kills it. For \(k=1\) the leading -map is the identity. Vanishing of all maps in -Equation~\eqref{signed:obstruction} is equivalent to the -surjectivity at the current order. - -The maps \(\delta_k\) form a degree-\(k\) derivation on the -Laurent graded algebra \(\bigoplus_jg_*\mathcal A_j\), with -values in its first-cohomology module. On a parameter germ, choose a lift -of a leading section \(x\) to \(g_*\mathcal T_{j,k}\), and represent that -lift locally by \(x_\alpha\in I^j\). Then -\(x_\beta-x_\alpha\in I^{j+k}\). -Their differences modulo \(I^{j+k+1}\) represent -\(\delta_k(x)\). For another degree-\(l\) section \(z\), -the product difference is -\[ - x_\beta z_\beta-x_\alpha z_\alpha - =x_\alpha(z_\beta-z_\alpha)+z_\alpha(x_\beta-x_\alpha) - +(x_\beta-x_\alpha)(z_\beta-z_\alpha). -\] -The last term lies in \(I^{j+l+2k}\subset I^{j+l+k+1}\). -This proves the derivation rule. If \(t_1,\ldots,t_b\) are -coordinates on \(U\), choose simultaneous representatives -\(G_{i,\alpha}\) of their lifts to \(g_*\mathcal T_{0,k}\); their -differences lie in \(I^k\). Taylor's -formula modulo their squared differences gives, for any -holomorphic \(F\), -\begin{equation}\label{signed:chain-rule} - \delta_k(F(t))=\sum_i F_{t_i}(t)\delta_k(t_i). -\end{equation} -Both formulas hold on germs, without removing torsion in the -target, and commute with the prescribed germ comparisons. - -Put \(m_k=a+k>0\). An order-\(k\) obstruction raises degree by \(k\), -whereas the residue injection accepts degree \(-a\). We therefore test -leading sections \(x\) of degree \(-m_k=-(a+k)\). Apply the split residue -injection to define -\[ - c(x)=\operatorname{res}_*\delta_k(x),\qquad - e_i(x)=\operatorname{res}_*\bigl(x\delta_k(t_i)\bigr) - \quad\text{in }R^1h_*\omega_H(A). -\] -Both arguments have degree \(-a\), the degree of -\(\OO_S(aS)\) in Equation~\eqref{signed:residue-injection}. -The next construction places these classes in a filtered direct-image -module. For the identity graph, the calculation will give the undivided -relation \(c(x)+\sum_i e_i(x)\partial_{t_i}=0\). Taking its degree-one -symbol gives \(\sigma(\sum_i e_i(x)\otimes\partial_{t_i})=0\). -We will globalize that -symbol on a projective coordinate-power cover, where the boundary root is -an ample line. The adjugate formula expresses the symbol across the -ramification of this cover without dividing by its Jacobian determinant. -The symbol vanishing will then kill the obstructions of the parameter coordinates -on every stalk; the undivided relation will kill \(\delta_k(x)\), and the -derivation rule will recover the other degrees. - -Let \(\psi:U'\to U\) be a holomorphic map between coordinate -opens of dimension \(b\), with coordinates \(w\) on \(U'\), -transverse to every smooth closed stratum \(H_I\cap A_J\). -The identity map is allowed. In \(\widehat Z\times U'\), -the graph support \(H^\flat=H\times_U U'\) is an SNC divisor -in a smooth complete intersection of codimension \(b\). -The residual divisor remains transverse. After shrinking about a -compact fiber, the finite-component Lemma~10.3 of \cite{BL} -gives finite smooth indexing of the graph strata, and their -relative Kähler forms restrict from the product neighborhood. -Complete-intersection adjunction gives the canonical factor -\[ - \omega_{U'}\otimes\psi^*\omega_U^{-1}, -\] -whose coordinate frame we denote by \(dw/dt\). This is a -ratio of canonical frames, with no inverse Jacobian. Pull -the Čech representatives of \(c(x),e_i(x)\) to this graph -and tensor by that frame. Proposition~\ref{signed:localized-hodge} -places the resulting classes \(\widetilde c,\widetilde e_i\) -in \(F_0M^\flat\), where \(M^\flat\) is the degree-one -direct-image module of the localized graph support. - -Write \(J=(\partial\psi_i/\partial w_j)\) and -\(J^\#=\operatorname{adj}(J)\). Because the adjugate is polynomial in -\(J\), the following identity remains meaningful where \(J\) is singular: -\begin{equation}\label{signed:adjugate} - \widetilde c\det J+ - \sum_{i,j}(\widetilde e_i\partial_{w_j})J^\#_{ji}=0 - \quad\text{in }M^\flat. -\end{equation} -We verify the change from \cite[Lemma~12.2]{BL} explicitly. -Choose simultaneous representatives of the lifts to length \(k\) -\[ - x_\alpha\in I^{-a-k},\quad x_\beta-x_\alpha\in I^{-a}, - \qquad G_{i,\beta}-G_{i,\alpha}\in I^k. -\] -After pulling to \(\widehat Z\), put -\(\eta_\alpha=x_\alpha\tau\), using the meromorphic form -defined by Equation~\eqref{signed:adjoint-chart}, and put -\(Q_{i,\alpha}=\psi_i(w)-G_{i,\alpha}\), -\(P_\alpha=\prod_iQ_{i,\alpha}\). On an overlap write -\(\Delta_i=G_{i,\beta}-G_{i,\alpha}\). The insertion and -\(D_S\geq H\) give the precise pole bounds -\begin{align} - \eta_\alpha&\in\omega_{\widehat Z}(H+A+kD_S),\nonumber\\ - \eta_\beta-\eta_\alpha,\quad \eta_\alpha\Delta_i - &\in\omega_{\widehat Z}(H+A),\label{signed:pole-bounds}\\ - \eta_\alpha\Delta_i\Delta_l,\quad - (\eta_\beta-\eta_\alpha)\Delta_i - &\in\omega_{\widehat Z}(H+A-kD_S) - \subset\omega_{\widehat Z}(A).\nonumber -\end{align} -Thus every quadratic or cross difference has no \(H\)-pole, -although it may retain a pole on \(A\). Such a term is zero -in the support module localized along \(A\). - -In that localized graph support module consider the finite -generalized fractions -\[ - S_\alpha=\left[\frac{\eta_\alpha\wedge dw}{P_\alpha}\right]. -\] -The \(H\)-pole is already in the numerator. Each polar class -is killed by a power of a reduced equation of \(H\), and each -\(\Delta_i\) is a multiple of that equation. -Changing \(Q_{i,\alpha}\) to \(Q_{i,\beta}\) therefore has a -finite geometric expansion on each class. The last line of -Equation~\eqref{signed:pole-bounds} kills all terms beyond the -linear terms, and gives the exact equality -\[ - S_\beta-S_\alpha= - \left[\frac{(\eta_\beta-\eta_\alpha)\wedge dw}{P_\alpha}\right] - +\sum_i\left[ - \frac{\eta_\alpha\Delta_i\wedge dw} - {Q_{i,\alpha}P_\alpha}\right]. -\] -Let \(C_\bullet\) be the first cochain. Let \(E_{i,\bullet}\) -be the cochain with the numerator of the \(i\)-th summand and -only denominator \(P_\alpha\), and let \(E_{i,\bullet}^{(l)}\) -have its additional denominator \(Q_{l,\alpha}\). The simple -cochains \(C_\bullet,E_{i,\bullet}\) are cocycles representing -\(\widetilde c,\widetilde e_i\); their possible triple-overlap -errors are precisely the cross terms killed above. The normal -residue frame in this identification is \(dw/dt\). - -The numerator of \(E_{i,\bullet}\), including its \(A\)-poles, -is independent of \(w\). Right differentiation therefore gives -\[ - E_{i,\bullet}\partial_{w_j}=\sum_lJ_{lj}E_{i,\bullet}^{(l)}. -\] -The preceding fraction equality is -\(\check dS_\bullet=C_\bullet+\sum_iE_{i,\bullet}^{(i)}\). -Multiplying by \(\det J\) and using \(JJ^\#=(\det J)\id\) -gives at the cochain level -\[ - \check d(S_\bullet\det J)=C_\bullet\det J+ - \sum_{i,j}(E_{i,\bullet}\partial_{w_j})J^\#_{ji}. -\] -These cochains are in the end term of the relative right de -Rham complex, so a Čech coboundary there is a total -coboundary. Passing to the direct image proves -Equation~\eqref{signed:adjugate}, with \(J^\#_{ji}\) placed -after the right derivative as displayed. Passing to -\(\gr^F_1\) yields -\begin{equation}\label{signed:local-symbol} - \sigma\left(\sum_{j,i}J^\#_{ji}\widetilde e_i - \otimes\partial_{w_j}\right)=0. -\end{equation} -For \(\psi=\id\) the undivided identity is -\begin{equation}\label{signed:undivided} - c(x)+\sum_i e_i(x)\partial_{t_i}=0. -\end{equation} -For \(b=0\) there are no graph equations, and the same -undivided cochain computation gives \(c(x)=0\). - -We now prove that the obstructions of the parameter coordinates vanish on every -stalk, including an obstruction supported only at a special parameter. -Suppose \(b>0\), and fix \(t_*\in P\). We will use one compact graph over -a projective parameter space, obtained by a parameter change unbranched -above \(t_*\). The vanishing of -\(\Hom\) in Equation~\eqref{signed:symbol-vanishing} applies to the entire -coherent symbol kernel, and the local isomorphism -above \(t_*\) will detect vanishing on the full germ there. The -construction of \cite[Proposition~10.5]{BL} gives a coordinate- -power map, here of exponent \(\ell\), -\[ -\begin{gathered} - \psi:P'=\PP^b\longrightarrow P,\qquad - [w_0:\cdots:w_b]\longmapsto[w_0^\ell:\cdots:w_b^\ell],\\ - R=\OO_{P'}(1),\qquad R^\ell\simeq\psi^*\OO_P(1), -\end{gathered} -\] -in suitably chosen coordinates, unbranched over \(t_*\). -We spell out why its geometry also accommodates \(A\). -Choose finitely many relatively compact parameter opens -covering \(\phi(D)\), with the chart resolutions defined on -slightly larger opens. Empty charts contribute no strata. Properness of -the nonempty supports leaves -only finitely many closed smooth strata \(H_I\cap A_J\) meeting -the corresponding compact sets. A general tuple of coordinate -hyperplanes is transverse to the maps of every such stratum, -for every subtuple, and avoids \(t_*\). This follows by Sard's -theorem applied to the incidence with variable hyperplanes; -varying each hyperplane supplies its normal direction. In -these coordinates \(d\psi\) has image the tangent space to -the intersection of its vanishing coordinate hyperplanes. -The transversality condition is consequently -\[ - dh(T_z(H_I\cap A_J))+d\psi(T_wP')=T_{h(z)}P - \quad\text{whenever }h(z)=\psi(w). -\] -It gives the required simultaneous regular graph equations -and SNC transverse residual divisors. - -To obtain a single compact graph support, first form the compact -graph \(\Xi=D\times_P P'\subset X\times P'\). On it the -pullback of \(N_0\) has the specified root \(R\). Lemma~10.1 -of \cite{BL} extends that root to a neighborhood of \(\Xi\). -Perform the full construction of Lemma~\ref{signed:root-charts} -on this neighborhood, including the relative quotient and its ordinary -pullback, and use the canonical sheaf relative to \(P'\) in the periodicity -algebra; the absolute canonical factor is added only when -taking graph residues. Resolve the same ordered total divisor data before imposing -the graph equations. Near a compact graph slice, the root -comparison lemma identifies this construction with the ordinary -product of a local chart with a parameter open, as an ambient -germ preserving the ideal and the adjoint form. Uniqueness -of the comparison as a germ and properness of the graph allow -a base shrink on which the identification holds on the entire -slice. Normalization thus occurs before ramified graph base -change. Smooth-functorial resolution preserves the product -identification. - -The graph cut \(H'\) in this resolved ambient space is compact, -has pure dimension \(n-1\), and is locally an SNC divisor in -a smooth complete intersection of codimension \(b\). Its -ambient projection to \(P'\) is submersive near \(H'\). -Let \(A'\) be the reduced ambient resolved residual divisor, restricted -to a neighborhood of \(H'\). Under each product-germ identification -it is \(A\times U'\); its restriction to the graph complete intersection -is the transverse residual cut. The chosen transversality makes \(A'\) -simultaneously SNC with the graph support equations. -All closed strata \(H'_I\cap A'_J\) are compact and smooth; -splitting their connected components leaves finitely many -indices. The relative metric construction in -Lemma~\ref{signed:root-charts} gives one Kähler form near -\(H'\), and hence the required forms on every stratum. Thus -Proposition~\ref{signed:localized-hodge} applies to this single -global graph. Write \(M'\) for its degree-one module. - -Take \(x=u^{-m_k}\). On a graph chart choose a frame -\(\vartheta\) of \(R\) compatible with the downstairs frame -of \(\OO_P(1)\). The local rule -\begin{equation}\label{signed:global-symbol} - \vartheta^{m_k}\longmapsto - \sum_{j,i}J^\#_{ji}\widetilde e_i(u^{-m_k}) - \otimes\partial_{w_j} -\end{equation} -defines a global morphism -\(R^{m_k}\to F_0M'\otimes T_{P'}\). Here is the transition -check, including the ramification locus. For changes of -coordinates let \(A_t=\partial t'/\partial t\) and -\(B_w=\partial w'/\partial w\), and let the root frame change -by \(\vartheta'=\lambda\vartheta\). Locally \(\lambda\) -descends from a holomorphic unit downstairs: its \(\ell\)-th -power does, and after choosing a local \(\ell\)-th root of -that unit the remaining ratio is a locally constant element -of \(\mu_\ell\). Root germ compatibility and -Equation~\eqref{signed:chain-rule} transform the downstairs -column of \(e_i\)'s by \(\lambda^{m_k}A_t\). No derivative -of \(\lambda\) appears, since \(x\) multiplies outside -\(\delta_k(t_i)\). The absolute canonical factor changes -by \(\det B_w/\det A_t\). Hence -\[ -\begin{aligned} - \widetilde e'&=\frac{\det B_w}{\det A_t}\lambda^{m_k}A_t\widetilde e, - &J'&=A_tJB_w^{-1},\\ - (J')^\#\widetilde e'&=\lambda^{m_k}B_wJ^\#\widetilde e. -\end{aligned} -\] -The adjugate equality is polynomial in \(J\), so holds also -when \(J\) is singular. The last formula, together with the -inverse change of the vector basis, is exactly the transition -of Equation~\eqref{signed:global-symbol}. The graph classes -here are natural Čech pullbacks through ambient germs; no -ramified base-change isomorphism for \(R^1h_*\) is used. - -Equation~\eqref{signed:local-symbol} puts the image of this -morphism in the first-symbol kernel. Since -\(R^{m_k}=\OO_{\PP^b}(a+k)\) is ample, -Equation~\eqref{signed:symbol-vanishing} makes the morphism -zero. At a point above \(t_*\), the map \(\psi\) is locally -biholomorphic and \(J^\#\) is invertible. The ambient germ -comparison identifies the graph classes with the downstairs -classes there. Thus \(e_i(u^{-m_k})=0\) as germs at -\(t_*\), including any class supported there. The point was -arbitrary, so these classes vanish on every chart. - -The split injection in Lemma~\ref{signed:residue} and the -invertible Laurent frame now give \(\delta_k(t_i)=0\). -For every leading section \(x\) of degree \(-m_k\), the -classes \(e_i(x)\) vanish. Equation~\eqref{signed:undivided}, -the injection in Equation~\eqref{signed:lowest-step}, and -then the split residue injection give \(\delta_k(x)=0\). -When \(b=0\), the direct identity \(c(x)=0\) and the same -two injections give this conclusion without a graph test. -In particular, the derivation rule and characteristic zero give -\[ - 0=\delta_k(u^{-m_k})=-m_k u^{-m_k-1}\delta_k(u), - \qquad \delta_k(u)=0. -\] -For any local \(F\in g_*\OO_S\), the degree-\(-m_k\) section -\(Fu^{-m_k}\) then gives -\(0=\delta_k(Fu^{-m_k})=u^{-m_k}\delta_k(F)\). -Every graded section is \(Fu^j\), so \(\delta_k\) vanishes -in every degree. This proves the current order and completes -the induction. -\end{proof} - -\subsection{Compact deformations and the signed theorem} - -We now turn the infinitesimal lifting into actual compact -subspaces. The role of all the integer layers in -Proposition~\ref{signed:all-orders} is that both the equations -of a boundary fiber and a generator of its normal direction -can be lifted compatibly. This is the analytic counterpart of -the compact-family step in \cite[Proposition~5.1]{LI}. - -\begin{lemma}\label{signed:escaping-fibers} -Assume \(00\). -On the selected component the locus where it is singular or -where \(g\) has rank less than \(r\) is proper. Removing its -image, along with the preceding proper bad subsets, under the -finite map leaves a nonempty open \(D_i^\circ\subset D_i\). -Its points have preimages at smooth points of these fiber -components. - -Fix \(x\in D_i^\circ\) and a point above it in the compact -fiber \(F=g^{-1}(t)\), for \(t\in Y\). Work on the single -ordinary chart neighborhood of that fiber from -Lemma~\ref{signed:root-charts}. Take regular coordinates -\(t_1,\ldots,t_r\) on \(Y\) centered at \(t\), extended to -local coordinates of its smooth embedding in \(P\). Their -pullbacks form a regular sequence on \(S\) along \(F\), -by flatness. In addition, \(S\) is Cohen--Macaulay, so -\(F\) is Cohen--Macaulay of pure dimension \(d\). - -Choose once and for all a Hausdorff open neighborhood -\(W\subset Z\) of \(F\), disjoint from \(T\), on which -\(\pi\) is quasi-finite. Set \(I=\OO_Z(-S)\). -Proposition~\ref{signed:all-orders} lets us lift the finitely -many \(t_i\)'s to compatible germs of sections of \(\OO_Z/I^q\), -\(q\geq1\), along \(F\). It also lifts the conormal frame -\(u\in I/I^2\) to compatible germs -\(\widetilde y_q\in I/I^{q+1}\) along \(F\); write \(\widetilde y\) -for this system. At each order choose simultaneous -representatives on a neighborhood of \(F\) contained in -\(W\). These neighborhoods may shrink with the order. -The leading coefficient of \(\widetilde y\) is a unit -frame, so it generates \(I\) on each finite thickening. - -For \(R_q=\C[s]/(s^q)\), cut the lifted \(r\) equations in -the Cartier thickening \(qS\), and map \(s\) to -\(\widetilde y\). Denote the resulting subspace by -\(F_q\). It is flat over the Artin analytic point -\(\Spec R_q\), with closed fiber \(F\). Before cutting, -multiplication by the Cartier generator identifies each -successive \(s\)-layer with \(\OO_S\); this is the -flatness criterion over \(R_q\). Quotienting by lifts of -the closed-fiber regular sequence preserves flatness, by -the local flatness criterion. Compatibility of the lifted -equations gives compatible reductions of all \(F_q\). - -Each of these is an embedded compact deformation in the -same fixed neighborhood \(W\). To see why the shrinking -representative neighborhoods cause no difficulty, the -support of \(F_q\) is the fixed compact set \(F\). Its -ideal on its representative neighborhood glues with the -unit ideal on \(W\setminus F\), since the deformation is -empty on their overlap away from \(F\). This realizes it -as a closed subspace of \(W\times\Spec R_q\). The ideal -is coherent locally on both opens, and the family is proper -over the Artin point because its support is compact. - -The Douady space of compact subspaces of \(W\) represents -proper flat embedded families \cite[Section~9]{Douady1966}. -The compatible \(F_q\)'s thus define a formal arc in its -analytic germ at \([F]\). Embed that germ in a finite -dimensional analytic coordinate space. The arc is a formal -solution of its convergent defining equations. Analytic -Artin approximation \cite[Theorem~1.2]{Artin1968} gives a -convergent arc agreeing modulo \(s^2\). Pull back the -universal family and shrink its disk to obtain -\(\mathscr F\subset W\times\Delta\), proper and flat over -\(\Delta\), with closed fiber \(F\) and the prescribed -first normal displacement. - -The family may be taken to have pure \(d\)-dimensional -fibers throughout the disk. Here is a local justification. -The central fiber is Cohen--Macaulay of dimension \(d\). -Flatness over the regular one-dimensional base makes its -parameter a nonzerodivisor and gives the depth and dimension -of the total local ring as \(d+1\) along that fiber. -Cohen--Macaulayness is open for analytic local rings, and -properness allows a disk shrink for which it holds on the -whole family. Every total component then meets the central -fiber after shrinking, and flatness makes it dominate the -disk; the dimension formula gives it dimension \(d+1\). -Quotient by the nonzerodivisor at each -parameter gives Cohen--Macaulay fibers of dimension \(d\). -The analytic dimension formula and unmixedness of these -local rings give the asserted purity. - -Let \(f\) be a local equation of \(S\). On \(\mathscr F\) -it vanishes on the central fiber, so flatness writes it as -\(f=s h\). Agreement modulo \(s^2\) with the formal -deformation says that \(h|_F\) is a unit: this is precisely -the lifted conormal generator. Finitely many such -neighborhoods cover the compact \(F\). Properness of -\(\mathscr F\to\Delta\) lets us shrink the disk so that -they cover the entire family and all their \(h\)'s are -units; remove the closed image of the complement. Thus a -nonzero fiber misses \(S\). The neighborhood \(W\) already -misses \(T\), and the set equality in Lemma~\ref{signed:root-charts} gives -\(\pi^{-1}(|D|)\cap W=|S|\cap W\). Hence the -nonzero fiber images miss \(D\). - -The induced map \(\Pi:\mathscr F\to X\times\Delta\) is proper: -the source is proper over \(\Delta\) and the target is -Hausdorff over it, so the graph is closed and the projection -is proper. It is quasi-finite by the choice of \(W\), -hence finite. It therefore preserves the dimension of -every fiber component. Every total component through a central -point dominates the disk by flatness, and its punctured part is dense. -Thus every central point, including the point chosen above \(x\), -is a limit of points of nonzero fibers of dimension \(d\). - -Let \(\mathscr Z\subset X\times\Delta\) be the schematic -image of this finite map. Its algebra is the coherent image -of \(\OO_{X\times\Delta}\to\Pi_*\OO_{\mathscr F}\). -Multiplication by a nonzero -germ from the disk is injective on \(\OO_{\mathscr F}\), -by flatness, and remains injective after the exact finite -pushforward. It is therefore injective on the image algebra. -That algebra is torsion-free, hence flat, over each local -discrete valuation ring of the disk. Thus \(\mathscr Z\) -is a proper flat family of subspaces of \(X\). The finite -map to its schematic image is surjective, also on each -fiber as a map of supports. Each \(\mathscr Z_s\) consequently -has pure dimension \(d\). The Douady-to-cycle morphism, in the -form recorded in \cite[Section~3.3]{Fujiki1978}, makes their -fundamental cycles an analytic family in the cycle space -of \(X\). The restriction to a smaller closed disk is -compact. The volumes of these cycles against a Kähler -form are bounded by continuity, proving the last assertion. -\end{proof} - -\begin{proof}[Proof of Theorem~\ref{signed:torsion}] -The assertion is immediate in dimension zero, and the case -\(D=0\) follows directly from the signed equivalence. By -Lemma~\ref{signed:boundary-geometry}, the case \(v=0\) -already gives \(L\sim_{\Q}0\), and \(v=n>0\) contradicts -\(a(X)=0\). Suppose, therefore, that \(00\), the pullback of the class +\(\{L-cD\}\) to a projective resolution with smooth compact +Kähler source is pseudo-effective. +\item The rational holomorphic line \(L|_D\) is semiample on the +whole reduced analytic space \(D\): some Cartier multiple is +generated at every point of \(D\). +\end{enumerate} +Then \(L\) is torsion. More precisely, for some positive integer +\(q\), the holomorphic line \(\OO_X(qL)\) is isomorphic to +\(\OO_X\). +\end{theorem} + +To apply this theorem in the proof of Proposition~\ref{simple:case}, +Theorem~\ref{nv:meromorphic} supplies a signed representative of the +canonical bundle. After resolving, we enlarge the support of the +transformed boundary and this signed canonical divisor to a reduced SNC +divisor; the resulting adjoint remains pseudo-effective. +Corollary~\ref{prog:signed-nef} gives an ordinary dlt nef model +\((X_{\mathrm{nef}},D_{\mathrm{nef}})\) whose adjoint +\(L_{\mathrm{nef}}=K_{X_{\mathrm{nef}}}+D_{\mathrm{nef}}\) has a +signed representative supported on \(D_{\mathrm{nef}}\), supplying +the first hypothesis. Theorem~\ref{floor:generation} supplies +semiampleness on the whole reduced \(D_{\mathrm{nef}}\). On a common +smooth resolution, simplicity excludes uniruledness, so Ou's criterion +\cite[Theorem~1.1]{Ou2025} and exceptional translation for the canonical +comparison give pseudo-effectivity of the pullback of +\(K_{X_{\mathrm{nef}}}=L_{\mathrm{nef}}-D_{\mathrm{nef}}\). Thus the +second hypothesis holds with \(c=1\). Once signed rigidity makes the +actual line \(L_{\mathrm{nef}}\) torsion, the program comparison and +Lemma~\ref{bir:unique-current} let us subtract the added boundary and give +the required decomposition for the original adjoint. The final subsection +verifies these steps. + +The signed construction in \cite[Proposition~3.3]{LI} is the source +of the root and separation operations used below. The lifting +method is that of \cite[Sections~10--12]{BL}. +We will establish the analytic form of the signed construction and +the change to the lifting argument caused by its residual poles. +The numerical argument first singles out positive-dimensional fibers in +the positive boundary. We then lift those fibers through every finite +boundary neighborhood and deform them to compact subspaces outside the +boundary. An argument using the cycle space and Baire's theorem turns these +deformations into a covering family, which simplicity excludes. + +\subsection{The boundary detected by the nef class} + +If \(D=0\), the signed equivalence already gives +\(L\sim_{\Q}0\). We may therefore assume in the constructions +below that \(D\ne0\); in particular \(\dim X>0\). + +Fix the data of Theorem~\ref{signed:torsion}, and write +\[ + G=G_+-G_-,\qquad D_0=\sum_{a_i=0}D_i, +\] +where \(G_+\) and \(G_-\) are effective with disjoint prime supports. +Choose a positive integer \(m\) so that \(mG_+\), \(mG_-\), +\(mD_0\), and \(mL\) are Cartier, the given rational equivalence +induces a fixed isomorphism +\[ + N_0:=\OO_X(mG)\simeq\OO_X(mL), +\] +and \(N_0|_D\) is generated. Its sections give a morphism and an +actual line isomorphism +\begin{equation}\label{signed:boundary-map} + \phi:D\longrightarrow P=\PP^b,\qquad + N_0|_D\simeq\phi^*\OO_P(1). +\end{equation} +Let \(n=\dim X\). Choose a projective resolution +\(\mu:\widetilde X\to X\) as in the second hypothesis of the theorem, +and a Kähler class \(\omega\) on \(\widetilde X\). The numerical +dimension of the nef pullback is +\[ + v=\nu(L):=\max\bigl\{k\in\{0,\ldots,n\}: + (\mu^*\{L\})^k\omega^{n-k}>0\bigr\}. +\] + +The next argument is the Kähler form of +\cite[Lemma~3.2]{LI}. + +\begin{lemma}\label{signed:boundary-geometry} +If \(v=0\), then \(D=0\) and \(L\sim_{\Q}0\). If \(v=n>0\), +then \(a(X)=n\). If \(00}\dim\phi(D_i)=r, +\] +and +\begin{equation}\label{signed:small-intersections} + \dim\phi\bigl(\Supp G_+\cap(\Supp G_-\cup D_0)\bigr)0\) such that +\(C\omega-h\) is nef and \(h-\epsilon\omega\) is +pseudo-effective. Pairing these two inequalities successively with +products of the nef classes \(\mu^*\{L\}\), \(h\), and +\(\omega\) shows that +\[ + (\mu^*\{L\})^k h^{n-k}>0 + \quad\Longleftrightarrow\quad + (\mu^*\{L\})^k\omega^{n-k}>0. +\] +We may therefore use \(\theta\) in all the tests defining \(v\). + +Suppose first that \(v0\) attains dimension \(r\). + +It remains to separate its intersections from a general fiber. +On \(H^{1,1}_{\BC}(\widetilde X,\R)\) consider the form +\[ + q(\alpha,\beta)= + \alpha\beta(\mu^*\{L\})^r h^{n-r-2},\qquad + Q_{ij}=q(\mu^*\{D_i\},\mu^*\{D_j\}). +\] +The mixed Kähler Hodge index theorem, followed by approximation of +the nef factors by Kähler classes, says that \(q\) has at most one +positive direction; see \cite[Theorems~A and C]{DinhNguyen}. Put +\(l=\mu^*\{L\}\). The definition of \(v\) and +Equation~\eqref{signed:boundary-zero} give +\[ + q(l,l)=0,\qquad q(l,\mu^*\{D_i\})=0, + \qquad q(l,h)=\{L\}^{v}\theta^{n-v}>0. +\] +Linear algebra now makes \(q\) negative semidefinite on +\(l^\perp\): a positive vector in \(l^\perp\), together with a +suitable vector in the span of \(l,h\), would give two positive +directions. In particular \(Q\) is negative semidefinite. + +For \(i\ne j\), one has \(Q_{ij}\geq0\). This assertion uses +the effective intersection cycle of the distinct \(\Q\)-Cartier +divisors \(D_i,D_j\) on \(X\), and projection formula. It does +not require their total transforms to intersect effectively. The +two Cartier equations have a proper intersection: the dlt ambient +space is klt, hence Cohen--Macaulay, after dropping the reduced +boundary, and the distinct divisor equations form a regular +sequence. The intersection cycle is consequently pure of +codimension two, and each of its mixed nef degrees is nonnegative. + +Write the coefficient vector as +\(\mathbf a=\mathbf a^+-\mathbf a^-\), with nonnegative vectors +of disjoint supports. The signed equivalence and orthogonality to +\(l\) give \(Q\mathbf a=0\). Thus +\[ + Q(\mathbf a^+,\mathbf a^+) + =Q(\mathbf a^+,\mathbf a^-)\geq0. +\] +Negative semidefiniteness forces equality and +\(Q\mathbf a^+=0\). For every \(j\) with \(a_j\leq0\), its +row is a sum of nonnegative off-diagonal terms: +\[ + 0=\sum_{a_i>0}a_iQ_{ji}. +\] +Each term is zero. Applying the generated-line degree test to the +effective intersection cycles proves +Equation~\eqref{signed:small-intersections}. If \(r=0\), any +nonempty intersection would have positive \(\theta^{n-2}\)-degree, +so these intersections are empty. + +Finally, if \(v=n>0\), the nef volume criterion makes the rational +line on \(\widetilde X\) big +\cite[Theorem~0.5]{DemaillyPaun2004}. A big holomorphic line on a +compact Kähler manifold has maximal Iitaka dimension. Hence +\(a(\widetilde X)=n\), and birational invariance gives \(a(X)=n\). +\end{proof} + +Thus, when \(00\). +Equation~\eqref{signed:small-intersections} makes those fibers disjoint +from \(\Supp G_-\cup D_0\). The next construction prepares their normal +directions and adjunction for infinitesimal lifting. + +\subsection{Local roots and residue with residual poles} + +We work for the rest of the signed argument in the range +\(01\) divisible by \(m\) +and by every nonzero integer \(|ma_i|\), and put +\(a=\ell/m\), a positive integer. + +\begin{lemma}[Analytic signed charts]\label{signed:root-charts} +For each \(t\in P\), after shrinking an open neighborhood +\(U\subset P\), choose a line \(R_U\) together with an isomorphism +\(R_U^\ell\simeq\OO_P(1)|_U\). A comparison of two such roots on an +overlap is power-compatible if its \(\ell\)-th power commutes with these +isomorphisms. If +\(t\notin\phi(\Supp G_+)\), one may take \(U\) disjoint from this +closed image and set \(Z_U=S=T=\varnothing\); the assertions below are +then vacuous for the unique maps from the empty spaces. If +\(t\in\phi(\Supp G_+)\), there are a Hausdorff normal +analytic space \(Z_U\) of pure dimension \(n\), considered near a +reduced Cartier divisor \(S\) of pure dimension \(n-1\), a +reduced Weil divisor \(T\) with no component in common +with \(S\), and maps +\[ + \pi:Z_U\longrightarrow X,\qquad g:S\longrightarrow U +\] +with the following properties. +\begin{enumerate} +\item The map \(g\) is proper, and \(\pi|_S\) is finite over +\(D_U=\phi^{-1}(U)\), with \(g=\phi\pi|_S\). Its image covers +the positive components over \(U\), and +\[ + \OO_S(S)\simeq g^*R_U. +\] +The image of \(S\cap T\) lies in +\(\phi(\Supp G_+\cap(\Supp G_-\cup D_0))\). +Near \(S\), the underlying sets satisfy +\[ + \pi^{-1}(|D|)=|S|\cup|T|. +\] +\item The pair \((Z_U,S+T)\) is log canonical, and there is a fixed +actual isomorphism of divisorial sheaves +\begin{equation}\label{signed:adjoint-chart} + \tau:\OO_{Z_U}(aS)\xrightarrow{\ \simeq\ }\omega_{Z_U}(S+T). +\end{equation} +The spaces \(Z_U\) and \(S\) are Cohen--Macaulay. The support +of \(T\) is locally set-theoretically principal. +Off \(T\), the space \(Z_U\) is klt and its canonical sheaf is +invertible. The map \(\pi\) is quasi-finite near \(S\setminus T\). +\item Projective log resolutions of the total divisor data can be +chosen with Kähler forms on neighborhoods of the compact sets over +each compact fiber of \(g\). Power-compatible changes of the +boundary root extend to isomorphisms of ambient germs that preserve +the full ideal \(\OO_{Z_U}(-S)\), the reduced divisor \(T\), and +Equation~\eqref{signed:adjoint-chart}. Resolutions may be chosen +functorially for these germ isomorphisms. +\end{enumerate} +Here the nonempty space \(Z_U\) is a neighborhood of the whole compact +fiber under consideration; no extension of \(g\) to \(Z_U\) is asserted. +\end{lemma} + +\begin{proof} +The empty branch follows by shrinking away from the closed image +\(\phi(\Supp G_+)\). We construct the nonempty charts in four stages: +take roots and trivialize the remaining adjoint torsion, separate the +positive and negative root divisors, retain the root line that controls the +normal direction, and pass to an ordinary neighborhood of the compact fiber. + +First consider an analytic open set on which the Cartier lines of +\(mG_+\), \(mG_-\), and \(mD_0\) are trivial, and let +\(f_+,f_-,f_0\) be the functions representing their sections. Normalize +the entire finite space +\[ + \{y_+^\ell=f_+,\qquad y_-^\ell=f_-,\qquad y_0^\ell=f_0\}, +\] +retaining all its components and the lifted action of \(\mu_\ell^3\) +that multiplies the three root coordinates. On an overlap, the functions +\(f_+,f_-,f_0\) change by holomorphic units. Local \(\ell\)-th roots of +those units give comparisons of the finite spaces that lift uniquely to +their normalizations; different choices differ by \(\mu_\ell^3\). +The quotient stacks of these normalized charts by \(\mu_\ell^3\) glue to +the simultaneous normalized root stack over \(X\), which we denote by +\(\mathcal X_{\mathrm{rt}}\). The zero divisors +of the three root sections are denoted by \(S_+,S_-,S_0\). At a +generic prime of multiplicity \(b\), a normalized chart of +\(y^\ell=x^b\) has ramification index \(\ell/b\) and +\(\ord(y)=1\). Here \(b=|ma_i|\) or \(b=m\), so it divides +\(\ell\). Thus each root divisor is generically reduced and +Cartier. A Cartier divisor on a normal space satisfies Serre's +condition \(S_1\), so it is reduced. Log ramification with the +full reduced boundary gives a log canonical pair and the rational +equivalence +\[ + K_{\mathcal X_{\mathrm{rt}}}+S_++S_-+S_0 + \sim_{\Q}a(S_+-S_-). +\] +The ambient charts of \(\mathcal X_{\mathrm{rt}}\) are klt. Off the +boundary this follows from +the klt complement downstairs and the unramified charts. For a +place centered in the Cartier boundary, dropping that boundary +increases its log discrepancy by its strictly positive order; +this also makes every zero-discrepancy place positive. + +To turn the rational adjoint equivalence into an actual isomorphism, +consider the integral reflexive sheaf on \(\mathcal X_{\mathrm{rt}}\) +\[ + \mathcal F= + \bigl(\omega_{\mathcal X_{\mathrm{rt}}}(S_++S_-+S_0)\otimes + \OO_{\mathcal X_{\mathrm{rt}}}(-a(S_+-S_-))\bigr)^{**}. +\] +It is torsion. Choose a periodicity +\(\mathcal F^{[q]}\simeq\OO\), take the relative spectrum of its +reflexive-power algebra, and normalize; write \(\rho_{\mathrm{ind}}\) +for this finite map, and retain \(S_+,S_-,S_0\) for the pulled-back +root divisors on the cover. At codimension one this +is a cover \(w^q=u\) for a unit \(u\), hence is unramified. +Log canonicity, the klt ambient property, and reduced Cartier +root divisors persist. The tautological evaluation of this +algebra trivializes the reflexive pullback +\((\rho_{\mathrm{ind}}^*\mathcal F)^{**}\): it does so in +codimension one, and the identity then extends reflexively. +The evaluation is a sheaf morphism on the cover stack itself, +so the resulting actual divisorial isomorphism +\[ + \omega(S_++S_-+S_0)\simeq\OO\bigl(a(S_+-S_-)\bigr) +\] +on the cover is equivariant on every atlas. + +Next separate the positive and negative root divisors. Let +\(\mathcal X_{\mathrm{sep}}\) be the normalization of the blowup of the +ideal generated by their equations on the cover, and write \(p_1\) for its +map. This blowup embeds in a relative +\(\PP^1\); its normalization still has fibers of dimension at +most one. Each exceptional prime therefore lies over a +codimension-two positive--negative intersection. At its generic +point the original dlt pair is a two-branch SNC pair. After +extracting roots of units the normalized Kummer chart there has +smooth coordinates +\(x=u^{\ell/b_+}\), \(z=w^{\ell/b_-}\), where \(b_+\) and +\(b_-\) are the two corresponding multiplicities. On this chart +\(\mathcal F\) is a line and its periodicity cover is unramified. +The two-coordinate blowup calculation consequently applies at +every exceptional generic point. If \(E\) denotes its +exceptional divisor, it gives reduced Cartier divisors +\[ + E,\qquad S'_+=p_1^*S_+-E,\qquad S'_-=p_1^*S_--E, +\] +with \(S'_+\cap S'_-=\varnothing\). No such two-branch stratum +is contained in \(S_0\), so \(p_1^*S_0\) is reduced and has no +exceptional component. The full boundary +\(S'_++S'_-+E+p_1^*S_0\) is crepant and reduced. Its +codimension-one equality with the pullback adjoint gives the +discrepancy equality for every further valuation, hence log +canonicity. The ambient charts after this blowup are still +klt. Off the full boundary the blowup is an isomorphism to +the old klt complement. A place of zero pair discrepancy +centered in that boundary gains its strictly positive order +when the effective Cartier boundary is dropped, while a +place of positive pair discrepancy remains positive. The +strict divisor \(S'_+\) is finite over \(S_+\): +before normalization it lies in one section of the projective +ratio coordinate, so its proper map has finite fibers, and +normalization is finite. + +On \(\mathcal X_{\mathrm{sep}}\), put \(S=S'_+\) and, temporarily, +\(T=S'_-+E+p_1^*S_0\). Write +\(\pi_{\mathrm{sep}}:\mathcal X_{\mathrm{sep}}\to X\) for the composite +map, and define the retained root line +\[ + \mathcal R:=\OO_{\mathcal X_{\mathrm{sep}}}(S'_+-S'_-), + \qquad \mathcal R^\ell\simeq\pi_{\mathrm{sep}}^*N_0. +\] +Near \(S\), the negative strict root section is a unit. The fixed adjoint +isomorphism becomes +\(\omega_{\mathcal X_{\mathrm{sep}}}(S+T) + \simeq\OO_{\mathcal X_{\mathrm{sep}}}(aS)\), and the positive root +section divided by that negative unit section is a section of +\(\mathcal R\) with zero divisor \(S\). This line and section determine +the Cartier divisor \(S\) and its normal line; they must survive passage +to an ordinary chart. + +The pair consisting of \(\mathcal R\) and its displayed power isomorphism +defines a map to the gerbe \(\sqrt[\ell]{N_0}\), which parametrizes +\(\ell\)-th roots of \(N_0\). Take the relative coarse space over this +gerbe. On an atlas this quotients by the finite relative inertia kernel, +the subgroup of each stabilizer acting trivially on \(\mathcal R\). +Thus the quotient retains the root character of \(\mathcal R\). The +deck transformations of the representable periodicity cover +are not this inertia kernel and are not removed. + +Both \(\mathcal R\) and its section descend through the kernel, so the +reduced image \(S\) remains Cartier and its full ideal descends. We retain +\(S,T\) for the reduced images. A finite-group norm of +a local equation of the upstairs \(T\) has precisely its image +as zero set. Thus the reduced image \(T\) is locally +set-theoretically principal; it need not be Cartier. All +divisorial ramification lies on the full boundary. Log +ramification identifies the invariant log dualizing sheaf in +codimension one with the downstairs one. Taking reflexive +hulls and invariants of the equivariant adjoint isomorphism +gives an actual descended isomorphism \(\OO(aS)\simeq\omega(S+T)\). +Its ordinary pullback below is Equation~\eqref{signed:adjoint-chart}. +Outside \(S+T\), the local +finite quotient is quasi-étale, so its klt upstairs complement gives +a klt downstairs complement by finite discrepancy comparison. Away from +\(T\), dropping the Cartier divisor \(S\) from the lc pair +increases the log discrepancy of every place centered in \(S\) by +its strictly positive order. Thus the quotient charts are klt away from +\(T\), and the same isomorphism makes their canonical sheaves invertible +there. Off \(T\) the separating blowup is an +isomorphism, while the other operations are finite; hence +the composite map to \(X\) is quasi-finite near \(S\setminus T\). + +The same quotient preserves Cohen--Macaulayness of both the +ambient space and \(S\). Before the quotient the ambient +space is klt and hence Cohen--Macaulay, and \(S\) is Cartier +there. Locally its finite quotient ring, and the quotient +ring of \(S\), are invariant direct summands of those finite +Cohen--Macaulay rings. A system of parameters downstairs is +one upstairs; the vanishing of lower local cohomology upstairs +and the invariant projection give its vanishing for the +downstairs direct summand. This proves the stated depth. + +Removal of the relative inertia kernel also makes the strict +positive divisor finite and representable over +\[ + D\times_P\sqrt[\ell]{\OO_P(1)}. +\] +Its root line is \(\OO_S(S)\), and its image covers +\(\Supp G_+\). The intersection \(S\cap T\) maps to the +positive--negative or positive--zero intersections: the only +new component is the separating exceptional divisor. Before the quotient, +the inverse image of \(|D|\) is the support of +\(S'_++S'_-+E+p_1^*S_0\). The finite quotient sends this full support +to \(|S|\cup|T|\), and the equality of underlying sets persists under the +ordinary base change below. The image of \(S\cap T\) in \(P\) consequently +has dimension \(0\). This +proves Equation~\eqref{signed:full-residual-comparison}. + +Projection formula gives the same comparison after adding +\(D_S=p^*S\). The quotient sequence therefore identifies +\[ +\begin{aligned} + R(p_D)_*\left( + \frac{\omega_{\widehat Z}(D_S+C)}{\omega_{\widehat Z}(C)} + \right) + &\simeq \omega_Z(S+T)|_S[0]\\ + &\simeq \OO_S(aS)[0], +\end{aligned} +\] +where \(p_D:D_S\to S\) and the quotient is a sheaf on the +possibly nonreduced divisor \(D_S\). To justify the displayed +derived statement, first push it by the exact closed immersion +\(S\hookrightarrow Z\). There it is the quotient of the two +acyclic comparisons. Closed pushforward detects cohomology +sheaves, and canonical truncation yields the statement on \(S\). + +There is a quotient morphism +\begin{equation}\label{signed:quotient-map} +\begin{aligned} + \omega_H(A)&= + \frac{\omega_{\widehat Z}(H+A)}{\omega_{\widehat Z}(A)} + \\ + &\longrightarrow + \frac{\omega_{\widehat Z}(D_S+C)}{\omega_{\widehat Z}(C)}. +\end{aligned} +\end{equation} +Indeed \(H+A\leq D_S+C\) and \(A\leq C\). This morphism +need not be injective when \(C\) and \(H\) share a component. +Compose its derived pushforward with the preceding isomorphism. +The result retracts Equation~\eqref{signed:derived-insertion}: +at each generic point of \(S\), its composite is the identity +under the fixed residue convention. The composite is an +endomorphism of the invertible sheaf \(\OO_S(aS)\) in degree +zero; since \(S\) is reduced, agreement at every generic point +implies agreement everywhere. Applying \(Rg_*\) proves +Equation~\eqref{signed:residue-injection}. All maps use the +fixed adjoint isomorphism, divisor ideals, and residue, so they +commute with the germ comparisons. Relative canonical forms +under products, followed by wedging with the canonical frame +of the new factor, give the last assertion. +\end{proof} + +\subsection{The localized Hodge argument} + +The target in Equation~\eqref{signed:residue-injection} contains +the additional poles \(A\). The following version of the filtered +SNC calculation incorporates them. In its application, we embed \(H\) +by the graph of \(h\) in \(\widehat Z\times U\) and use the projection to +\(U\) as the ambient map. This projection is submersive, while its +restriction to the graph is the proper map \(h\); it requires no extension +of \(h\) to \(\widehat Z\). We will use the same graph construction after +a change of parameter. The parameter is projective only in the last +assertion below; the closed-stratum maps are proper Kähler maps. + +\begin{proposition}\label{signed:localized-hodge} +Let \(q:\mathscr V\to Y\) be a holomorphic map of complex +manifolds, submersive near a reduced closed analytic subspace +\(i:H\hookrightarrow\mathscr V\) of pure dimension \(d\) and +codimension \(c_0>0\). Suppose \(h=q|_H\) is proper. Assume +that locally \(H\) is an SNC divisor in a smooth submanifold of +codimension \(c_0-1\), and that a reduced divisor \(A\) is +simultaneously SNC and transverse to those support equations. +Assume that the components \(H_i\) and \(A_j\) near \(H\) have +finite global smooth indexing, and that every closed stratum +\(H_I\cap A_J\), with \(I\ne\varnothing\), is smooth and proper +over \(Y\) and carries a global relative Kähler form. Empty or +disconnected strata are allowed. +Here \(H_I=\bigcap_{i\in I}H_i\) and +\(A_J=\bigcap_{j\in J}A_j\), with \(A_\varnothing=\mathscr V\). + +Use right \(\mathscr D\)-modules and finite-pole local cohomology, +and put +\[ + \mathcal K_A=\mathcal H^{c_0}_{[H]}(\omega_{\mathscr V})(*A). +\] +Its order filtration is generated from +\(F_0\mathcal K_A=i_*\omega_H(A)\), with \(F_p=0\) for \(p<0\). +For \(M^j=\mathcal H^j q_+\mathcal K_A\), the filtered direct +image is strict at every level, and \(M^j\) has a finite +filtration by submodules, strict for \(F\), whose quotients are +polarizable real pure Hodge modules. In particular, +\begin{equation}\label{signed:lowest-step} + F_0M^j=R^jh_*\omega_H(A)\lhook\joinrel\longrightarrow M^j, + \qquad F_pM^j=0\quad(p<0). +\end{equation} +Define the symbol morphism +\[ + \sigma:F_0M^j\otimes T_Y\longrightarrow\gr^F_1M^j, + \qquad \sigma(m\otimes\xi)=[m\xi]. +\] +If \(Y\) is smooth projective, then for every ample line \(N\) on \(Y\), +\begin{equation}\label{signed:symbol-vanishing} + \Hom_{\OO_Y}(N,\Ker\sigma)=0. +\end{equation} +This includes any coherent torsion in the kernel. +\end{proposition} + +\begin{proof} +We verify the residual localization in the proof of +\cite[Proposition~11.1]{BL}. We first identify the lowest order step and +a filtration by the number of branch poles. Its graded pieces will be +dual constant modules on smooth strata, to which proper Kähler direct +image applies. The resulting strictness gives the injection at the lowest +filtration step, +and the graded de Rham complex on a projective base gives the symbol +vanishing. + +At a point of \(H\), choose +simultaneous coordinates +\[ + (z_1,\ldots,z_{c_0-1},x_1,\ldots,x_t,y_1,\ldots,y_s,\xi), + \quad + \mathscr I_H=(z_1,\ldots,z_{c_0-1},x_1\cdots x_t), + \quad A=(y_1\cdots y_s=0). +\] +Let \(\eta\) be the coordinate volume form. The localization +Čech complex for the displayed regular support equations has +cohomology only in degree \(c_0\). Localizing this cohomology +also in the \(y\)'s gives \(\mathcal K_A\). Successive finite +Taylor divisions give its unique additive polar normal forms: +finite sums of +\begin{equation}\label{signed:polar-form} + f(x_{I^c},y_{J^c},\xi) + \prod_{\nu=1}^{c_0-1}z_\nu^{-u_\nu} + \prod_{i\in I}x_i^{-v_i} + \prod_{j\in J}y_j^{-w_j}\eta, + \quad I\ne\varnothing, +\end{equation} +where every displayed order is positive and \(J\) is arbitrary, +including empty. The coefficient is independent of exactly +the variables occurring negatively. The normal form is an +additive statement, not an \(\OO_{\mathscr V}\)-linear +decomposition. + +The simple fractions, with each displayed order equal to one, +are exactly the image of \(\omega_H(A)\). More explicitly, +their unreduced representatives are +\[ + \frac{f\eta} + {z_1\cdots z_{c_0-1}(x_1\cdots x_t)(y_1\cdots y_s)}. +\] +The regular-equation residue identifies their numerators modulo +\(\mathscr I_H\) with the absolute dualizing sheaf of \(H\), +twisted by \(A\). This map is injective. If the fraction has +no remaining \(x\)-pole, Taylor division says that its +numerator modulo the \(z\)'s is divisible by \(x_1\cdots x_t\). +Localization in the \(y\)'s does not enlarge this kernel, +because each \(y_j\) is a nonzerodivisor modulo +\(\mathscr I_H\). This also proves the identification under +changes of coordinates, including the determinant of the +normal conormal frame. + +Give a term of Equation~\eqref{signed:polar-form} the excess +\[ + e=\sum_\nu(u_\nu-1)+\sum_{i\in I}(v_i-1) + +\sum_{j\in J}(w_j-1). +\] +The right canonical action of a coordinate derivative is minus +differentiation of the coefficient of \(\eta\). A derivative +in a denominator variable increases its order by one with a +nonzero scalar. Because the coefficient in the normal form +is independent of those variables, every term of excess +\(e\) is obtained from a simple term by \(e\) such derivatives. +Conversely, order \(p\) differentiations produce excess at +most \(p\). Thus \(F_p\mathcal K_A\) consists exactly of the +finite sums of excess at most \(p\). This is the good order +filtration generated from \(i_*\omega_H(A)\). + +There is also an increasing filtration \(W\) by the total +number of branch poles. Intrinsically, its step of index +\(-d+k-1\) is the sum of the images of support modules for +subunions of the \(H_i\), localized along sublists of the +\(A_j\), for which the total number of selected branches is +at most \(k\). In the normal form it imposes +\(|I|+|J|\leq k\). The induced double filtration therefore +has +\begin{equation}\label{signed:localized-graded} + \gr^W_{-d+k-1}(\mathcal K_A,F) + \simeq + \bigoplus_{\substack{|I|+|J|=k\\ I\ne\varnothing}} + (i_{I,J})_+(\omega_{H_I\cap A_J},F^{(0)}). +\end{equation} +Here \(F^{(0)}_p\omega_{H_I\cap A_J}=0\) for \(p<0\) and is +\(\omega_{H_I\cap A_J}\) for \(p\geq0\). +Here \(A_\varnothing=\mathscr V\), and empty intersections +contribute zero. To check this formula, retain the polar type +using exactly \(I,J\). Its normal denominators are the +\(c_0-1\) equations \(z_\nu\), the \(|I|\) selected \(x\)'s, +and the \(|J|\) selected \(y\)'s. They are the regular +equations of the smooth stratum \(H_I\cap A_J\), whose +dimension is \(d-k+1\). Right closed transfer has no +codimension shift in \(F\), and the excess counts exactly its +normal derivative orders. Mayer--Vietoris boundary maps for +the \(x\)-subunions and the localization boundary for each +selected \(y\) identify these quotients intrinsically with +the support module of that intersection. For the latter +assertion, localization modulo the nonlocalized module in one +transverse \(y\) direction is its first local cohomology in +that direction. A fixed order of the component labels fixes +the residue signs, so the identifications glue. The smooth- +support graded modules are regular holonomic, and their finite +extension \(\mathcal K_A\) is regular holonomic as well. + +The filtration \(W\) has a real realization. Tensor the +finite-pole de Rham comparisons for the regular support +coordinates with the open localization comparison for the +\(y\) coordinates, whose one-coordinate factor is the +cohomology of a punctured disk. These comparisons are +natural for all sublists. The resulting real support and +localization complexes complexify to the de Rham complexes +just computed. The latter are perverse by regular +holonomicity; exact and faithful complexification gives the +same assertion for the real complexes. Their real perverse +images then give \(W\). +On a graded stratum of dimension \(d-k+1\), the real object +is its dual constant object +\(\R^H[d-k+1](d-k+1)\). Its first right filtration step is +zero and its weight is \(-d+k-1\), in agreement with +Equation~\eqref{signed:localized-graded}. Residue signs and +constant normalizations preserve its real polarization. This +is the real comparison in \cite[Section~11.2]{BL}, with the +punctured-disk factors supplying the additional localization +boundaries. + +The proper-support check also survives localization. If a +holomorphic germ \(f\) vanishes on reduced \(H\), then +\begin{equation}\label{signed:support-condition} + fF_p\mathcal K_A\subset F_{p-1}\mathcal K_A. +\end{equation} +For a generator \(z_\nu\) of \(\mathscr I_H\), multiplication +reduces its pole order or kills the class. Multiplication by +\(x_1\cdots x_t\) also lowers the excess or kills the class, +because the polar type always has \(I\ne\varnothing\). +The \(y\)-poles do not affect this argument. In particular +every graded stratum stays in the proper support \(H\), and +properness is needed only there. + +We have now identified the order filtration, its real pure stratum pieces, +and the proper support of each level. We next pass these data to the base. +Apply Saito's constant-source Kähler direct-image theorem +\cite[Theorem~1 and Remark~1.2]{SaitoKahler} to each smooth +closed stratum in Equation~\eqref{signed:localized-graded}, +using its stipulated relative Kähler form. The resulting +degree-\(j\) direct image of the step indexed by \(\lambda\) +is strict and polarizable real pure of weight \(\lambda+j\). +In the submersion neighborhood, the level-\(F_p\) relative +right de Rham complex has term +\[ + F_{p-e}\mathcal K_A\otimes\bigwedge^e T_{\mathscr V/Y} + \quad\text{in degree }-e. +\] +The simultaneous excess and branch bounds show that passing +to a \(W\)-quotient commutes with taking each such level. + +To prove strictness, use the finite \(W\) spectral sequence +\[ + E_1^{-\lambda,j+\lambda} + =\mathcal H^j q_+\gr^W_\lambda\mathcal K_A + \quad\Longrightarrow\quad \mathcal H^j q_+\mathcal K_A. +\] +Its first-page terms and their level filtrations are strict +pure objects by the preceding application. The differentials +are real and filtration preserving, because the support and +localization comparisons are real and natural. The first +differential preserves weight, hence is a strict morphism of +pure Hodge modules; its kernels and cokernels remain pure. +A higher differential sends \((\lambda,j)\) to +\((\lambda-k,j+1)\) for \(k\geq2\) and strictly lowers +weight. Such a real filtered map is zero. On distinct strict +supports this follows from strict support; on the same dense +smooth support a real map preserves both Hodge filtrations, +and a source vector of type \((p,q)\) would have image in +\(F^p\cap\overline F^q\) of a pure object of smaller weight, +which is zero. Both the unfiltered and each level spectral +sequence therefore degenerate at the second page. Their +comparison is injective there. A least-\(W\)-step argument +on the finite abutment proves injection at every \(F\) level +and strictness of its induced \(W\) filtration. This is the +strict-direct-image argument of \cite[Proposition~11.1]{BL}, +now justified for every graded term with \(A\)-poles. + +For \(p<0\), the relative complex is zero. For \(p=0\), its +only term is \(F_0\mathcal K_A=i_*\omega_H(A)\) in degree +zero. The level injection gives +Equation~\eqref{signed:lowest-step}. + +Suppose finally that \(Y\) is smooth projective. Each strict- +support pure quotient just obtained is of exactly the type in +\cite[Lemma~11.2]{BL}: a polarizable real pure direct-image +piece arising from a dual constant on a smooth Kähler source. +That lemma compares its actual right filtration with the +Sabbah--Schnell pure component extending the same generic +polarized real variation. Its proof uses uniqueness of the +intermediate extension and the reconstruction of \(F\) from +the canonical \(V\)-filtration, so it depends only on this +pure piece, not on the divisor presentation of its source. +It gives the negative-ample vanishing +\[ + \mathbb H^u\bigl(Y,N^{-1}\otimes\gr^F_p\mathrm{DR}_Y Q\bigr)=0 + \quad(u<0) +\] +for each pure quotient \(Q\), by +\cite[Theorem~16.3.10]{SabbahSchnell}. The finite strict +filtration extends this vanishing to \(M^j\). Since its +negative \(F\) levels vanish, its degree-one graded right +de Rham complex is exactly +\[ + \gr^F_1\mathrm{DR}_Y M^j= + \left[F_0M^j\otimes T_Y\xrightarrow{\sigma}\gr^F_1M^j\right] + \quad\text{in degrees }-1,0. +\] +Its negative hypercohomology after tensoring by \(N^{-1}\) +is \(H^0(Y,N^{-1}\otimes\Ker\sigma)\). It vanishes, which +is Equation~\eqref{signed:symbol-vanishing}. This calculation +does not assume that the kernel is locally free. +\end{proof} + +\subsection{Lifting every finite boundary neighborhood} + +Return to the charts and resolution of +Lemma~\ref{signed:root-charts} and +Equation~\eqref{signed:resolved-divisors}. On an empty chart the sheaves +below are zero. On a nonempty chart put \(I=\OO_Z(-S)\), an invertible +ideal. For every integer \(j\) +and positive integer \(k\), put +\[ + \mathcal A_j=I^j/I^{j+1}=\OO_S(-jS),\qquad + \mathcal T_{j,k}=I^j/I^{j+k}. +\] +These are sheaves on the underlying topological space of \(S\). +Pushforward by \(g\) has that meaning for \(\mathcal T_{j,k}\); +it does not require a morphism from a thickening to \(U\). +A local frame of \(R_U^{-1}\) gives a Laurent graded frame +\(u^j\) of \(\mathcal A_j\) for every \(j\). It is a frame +on the associated graded, not an extension of a frame to \(Z\). + +\begin{proposition}[All finite lifting orders]\label{signed:all-orders} +For every root chart, every \(j\in\Z\), and every \(k\geq1\), +the map of sheaves of complex vector spaces on \(U\) +\begin{equation}\label{signed:all-order-surjection} + g_*\mathcal T_{j,k+1}\longrightarrow g_*\mathcal T_{j,k} +\end{equation} +is surjective. The assertion holds on every parameter germ, +including germs supported at special parameters. +\end{proposition} + +\begin{proof} +The assertion is immediate for an empty chart. We induct on \(k\), +simultaneously for all integer \(j\) and all nonempty charts. The layer +algebra and its connecting maps are the +ones in \cite[Lemma~12.1]{BL}; we recall their construction to +specify exactly which obstruction is to be killed. + +Assume all smaller orders. The connecting map for +\[ + 0\longrightarrow\mathcal A_{j+k} + \longrightarrow\mathcal T_{j,k+1} + \longrightarrow\mathcal T_{j,k}\longrightarrow0 +\] +factors uniquely through a map +\begin{equation}\label{signed:obstruction} + \delta_k:g_*\mathcal A_j\longrightarrow R^1g_*\mathcal A_{j+k}. +\end{equation} +Indeed the smaller orders make +\(g_*\mathcal T_{j,k}\to g_*\mathcal A_j\) surjective. For +\(k>1\), the kernel comes from \(g_*\mathcal T_{j+1,k-1}\), +by the leading-layer sequence. The order \(k-1\) in degree +\(j+1\) lifts this kernel into \(g_*\mathcal T_{j,k+1}\), so +the current connecting map kills it. For \(k=1\) the leading +map is the identity. Vanishing of all maps in +Equation~\eqref{signed:obstruction} is equivalent to the +surjectivity at the current order. + +The maps \(\delta_k\) form a degree-\(k\) derivation on the +Laurent graded algebra \(\bigoplus_jg_*\mathcal A_j\), with +values in its first-cohomology module. On a parameter germ, choose a lift +of a leading section \(x\) to \(g_*\mathcal T_{j,k}\), and represent that +lift locally by \(x_\alpha\in I^j\). Then +\(x_\beta-x_\alpha\in I^{j+k}\). +Their differences modulo \(I^{j+k+1}\) represent +\(\delta_k(x)\). For another degree-\(l\) section \(z\), +the product difference is +\[ + x_\beta z_\beta-x_\alpha z_\alpha + =x_\alpha(z_\beta-z_\alpha)+z_\alpha(x_\beta-x_\alpha) + +(x_\beta-x_\alpha)(z_\beta-z_\alpha). +\] +The last term lies in \(I^{j+l+2k}\subset I^{j+l+k+1}\). +This proves the derivation rule. If \(t_1,\ldots,t_b\) are +coordinates on \(U\), choose simultaneous representatives +\(G_{i,\alpha}\) of their lifts to \(g_*\mathcal T_{0,k}\); their +differences lie in \(I^k\). Taylor's +formula modulo their squared differences gives, for any +holomorphic \(F\), +\begin{equation}\label{signed:chain-rule} + \delta_k(F(t))=\sum_i F_{t_i}(t)\delta_k(t_i). +\end{equation} +Both formulas hold on germs, without removing torsion in the +target, and commute with the prescribed germ comparisons. + +Put \(m_k=a+k>0\). An order-\(k\) obstruction raises degree by \(k\), +whereas the residue injection accepts degree \(-a\). We therefore test +leading sections \(x\) of degree \(-m_k=-(a+k)\). Apply the split residue +injection to define +\[ + c(x)=\operatorname{res}_*\delta_k(x),\qquad + e_i(x)=\operatorname{res}_*\bigl(x\delta_k(t_i)\bigr) + \quad\text{in }R^1h_*\omega_H(A). +\] +Both arguments have degree \(-a\), the degree of +\(\OO_S(aS)\) in Equation~\eqref{signed:residue-injection}. +The next construction places these classes in a filtered direct-image +module. For the identity graph, the calculation will give the undivided +relation \(c(x)+\sum_i e_i(x)\partial_{t_i}=0\). Taking its degree-one +symbol gives \(\sigma(\sum_i e_i(x)\otimes\partial_{t_i})=0\). +We will globalize that +symbol on a projective coordinate-power cover, where the boundary root is +an ample line. The adjugate formula expresses the symbol across the +ramification of this cover without dividing by its Jacobian determinant. +The symbol vanishing will then kill the obstructions of the parameter coordinates +on every stalk; the undivided relation will kill \(\delta_k(x)\), and the +derivation rule will recover the other degrees. + +Let \(\psi:U'\to U\) be a holomorphic map between coordinate +opens of dimension \(b\), with coordinates \(w\) on \(U'\), +transverse to every smooth closed stratum \(H_I\cap A_J\). +The identity map is allowed. In \(\widehat Z\times U'\), +the graph support \(H^\flat=H\times_U U'\) is an SNC divisor +in a smooth complete intersection of codimension \(b\). +The residual divisor remains transverse. After shrinking about a +compact fiber, the finite-component Lemma~10.3 of \cite{BL} +gives finite smooth indexing of the graph strata, and their +relative Kähler forms restrict from the product neighborhood. +Complete-intersection adjunction gives the canonical factor +\[ + \omega_{U'}\otimes\psi^*\omega_U^{-1}, +\] +whose coordinate frame we denote by \(dw/dt\). This is a +ratio of canonical frames, with no inverse Jacobian. Pull +the Čech representatives of \(c(x),e_i(x)\) to this graph +and tensor by that frame. Proposition~\ref{signed:localized-hodge} +places the resulting classes \(\widetilde c,\widetilde e_i\) +in \(F_0M^\flat\), where \(M^\flat\) is the degree-one +direct-image module of the localized graph support. + +Write \(J=(\partial\psi_i/\partial w_j)\) and +\(J^\#=\operatorname{adj}(J)\). Because the adjugate is polynomial in +\(J\), the following identity remains meaningful where \(J\) is singular: +\begin{equation}\label{signed:adjugate} + \widetilde c\det J+ + \sum_{i,j}(\widetilde e_i\partial_{w_j})J^\#_{ji}=0 + \quad\text{in }M^\flat. +\end{equation} +We verify the change from \cite[Lemma~12.2]{BL} explicitly. +Choose simultaneous representatives of the lifts to length \(k\) +\[ + x_\alpha\in I^{-a-k},\quad x_\beta-x_\alpha\in I^{-a}, + \qquad G_{i,\beta}-G_{i,\alpha}\in I^k. +\] +After pulling to \(\widehat Z\), put +\(\eta_\alpha=x_\alpha\tau\), using the meromorphic form +defined by Equation~\eqref{signed:adjoint-chart}, and put +\(Q_{i,\alpha}=\psi_i(w)-G_{i,\alpha}\), +\(P_\alpha=\prod_iQ_{i,\alpha}\). On an overlap write +\(\Delta_i=G_{i,\beta}-G_{i,\alpha}\). The insertion and +\(D_S\geq H\) give the precise pole bounds +\begin{align} + \eta_\alpha&\in\omega_{\widehat Z}(H+A+kD_S),\nonumber\\ + \eta_\beta-\eta_\alpha,\quad \eta_\alpha\Delta_i + &\in\omega_{\widehat Z}(H+A),\label{signed:pole-bounds}\\ + \eta_\alpha\Delta_i\Delta_l,\quad + (\eta_\beta-\eta_\alpha)\Delta_i + &\in\omega_{\widehat Z}(H+A-kD_S) + \subset\omega_{\widehat Z}(A).\nonumber +\end{align} +Thus every quadratic or cross difference has no \(H\)-pole, +although it may retain a pole on \(A\). Such a term is zero +in the support module localized along \(A\). + +In that localized graph support module consider the finite +generalized fractions +\[ + S_\alpha=\left[\frac{\eta_\alpha\wedge dw}{P_\alpha}\right]. +\] +The \(H\)-pole is already in the numerator. Each polar class +is killed by a power of a reduced equation of \(H\), and each +\(\Delta_i\) is a multiple of that equation. +Changing \(Q_{i,\alpha}\) to \(Q_{i,\beta}\) therefore has a +finite geometric expansion on each class. The last line of +Equation~\eqref{signed:pole-bounds} kills all terms beyond the +linear terms, and gives the exact equality +\[ + S_\beta-S_\alpha= + \left[\frac{(\eta_\beta-\eta_\alpha)\wedge dw}{P_\alpha}\right] + +\sum_i\left[ + \frac{\eta_\alpha\Delta_i\wedge dw} + {Q_{i,\alpha}P_\alpha}\right]. +\] +Let \(C_\bullet\) be the first cochain. Let \(E_{i,\bullet}\) +be the cochain with the numerator of the \(i\)-th summand and +only denominator \(P_\alpha\), and let \(E_{i,\bullet}^{(l)}\) +have its additional denominator \(Q_{l,\alpha}\). The simple +cochains \(C_\bullet,E_{i,\bullet}\) are cocycles representing +\(\widetilde c,\widetilde e_i\); their possible triple-overlap +errors are precisely the cross terms killed above. The normal +residue frame in this identification is \(dw/dt\). + +The numerator of \(E_{i,\bullet}\), including its \(A\)-poles, +is independent of \(w\). Right differentiation therefore gives +\[ + E_{i,\bullet}\partial_{w_j}=\sum_lJ_{lj}E_{i,\bullet}^{(l)}. +\] +The preceding fraction equality is +\(\check dS_\bullet=C_\bullet+\sum_iE_{i,\bullet}^{(i)}\). +Multiplying by \(\det J\) and using \(JJ^\#=(\det J)\id\) +gives at the cochain level +\[ + \check d(S_\bullet\det J)=C_\bullet\det J+ + \sum_{i,j}(E_{i,\bullet}\partial_{w_j})J^\#_{ji}. +\] +These cochains are in the end term of the relative right de +Rham complex, so a Čech coboundary there is a total +coboundary. Passing to the direct image proves +Equation~\eqref{signed:adjugate}, with \(J^\#_{ji}\) placed +after the right derivative as displayed. Passing to +\(\gr^F_1\) yields +\begin{equation}\label{signed:local-symbol} + \sigma\left(\sum_{j,i}J^\#_{ji}\widetilde e_i + \otimes\partial_{w_j}\right)=0. +\end{equation} +For \(\psi=\id\) the undivided identity is +\begin{equation}\label{signed:undivided} + c(x)+\sum_i e_i(x)\partial_{t_i}=0. +\end{equation} +For \(b=0\) there are no graph equations, and the same +undivided cochain computation gives \(c(x)=0\). + +We now prove that the obstructions of the parameter coordinates vanish on every +stalk, including an obstruction supported only at a special parameter. +Suppose \(b>0\), and fix \(t_*\in P\). We will use one compact graph over +a projective parameter space, obtained by a parameter change unbranched +above \(t_*\). The vanishing of +\(\Hom\) in Equation~\eqref{signed:symbol-vanishing} applies to the entire +coherent symbol kernel, and the local isomorphism +above \(t_*\) will detect vanishing on the full germ there. The +construction of \cite[Proposition~10.5]{BL} gives a coordinate- +power map, here of exponent \(\ell\), +\[ +\begin{gathered} + \psi:P'=\PP^b\longrightarrow P,\qquad + [w_0:\cdots:w_b]\longmapsto[w_0^\ell:\cdots:w_b^\ell],\\ + R=\OO_{P'}(1),\qquad R^\ell\simeq\psi^*\OO_P(1), +\end{gathered} +\] +in suitably chosen coordinates, unbranched over \(t_*\). +We spell out why its geometry also accommodates \(A\). +Choose finitely many relatively compact parameter opens +covering \(\phi(D)\), with the chart resolutions defined on +slightly larger opens. Empty charts contribute no strata. Properness of +the nonempty supports leaves +only finitely many closed smooth strata \(H_I\cap A_J\) meeting +the corresponding compact sets. A general tuple of coordinate +hyperplanes is transverse to the maps of every such stratum, +for every subtuple, and avoids \(t_*\). This follows by Sard's +theorem applied to the incidence with variable hyperplanes; +varying each hyperplane supplies its normal direction. In +these coordinates \(d\psi\) has image the tangent space to +the intersection of its vanishing coordinate hyperplanes. +The transversality condition is consequently +\[ + dh(T_z(H_I\cap A_J))+d\psi(T_wP')=T_{h(z)}P + \quad\text{whenever }h(z)=\psi(w). +\] +It gives the required simultaneous regular graph equations +and SNC transverse residual divisors. + +To obtain a single compact graph support, first form the compact +graph \(\Xi=D\times_P P'\subset X\times P'\). On it the +pullback of \(N_0\) has the specified root \(R\). Lemma~10.1 +of \cite{BL} extends that root to a neighborhood of \(\Xi\). +Perform the full construction of Lemma~\ref{signed:root-charts} +on this neighborhood, including the relative quotient and its ordinary +pullback, and use the canonical sheaf relative to \(P'\) in the periodicity +algebra; the absolute canonical factor is added only when +taking graph residues. Resolve the same ordered total divisor data before imposing +the graph equations. Near a compact graph slice, the root +comparison lemma identifies this construction with the ordinary +product of a local chart with a parameter open, as an ambient +germ preserving the ideal and the adjoint form. Uniqueness +of the comparison as a germ and properness of the graph allow +a base shrink on which the identification holds on the entire +slice. Normalization thus occurs before ramified graph base +change. Smooth-functorial resolution preserves the product +identification. + +The graph cut \(H'\) in this resolved ambient space is compact, +has pure dimension \(n-1\), and is locally an SNC divisor in +a smooth complete intersection of codimension \(b\). Its +ambient projection to \(P'\) is submersive near \(H'\). +Let \(A'\) be the reduced ambient resolved residual divisor, restricted +to a neighborhood of \(H'\). Under each product-germ identification +it is \(A\times U'\); its restriction to the graph complete intersection +is the transverse residual cut. The chosen transversality makes \(A'\) +simultaneously SNC with the graph support equations. +All closed strata \(H'_I\cap A'_J\) are compact and smooth; +splitting their connected components leaves finitely many +indices. The relative metric construction in +Lemma~\ref{signed:root-charts} gives one Kähler form near +\(H'\), and hence the required forms on every stratum. Thus +Proposition~\ref{signed:localized-hodge} applies to this single +global graph. Write \(M'\) for its degree-one module. + +Take \(x=u^{-m_k}\). On a graph chart choose a frame +\(\vartheta\) of \(R\) compatible with the downstairs frame +of \(\OO_P(1)\). The local rule +\begin{equation}\label{signed:global-symbol} + \vartheta^{m_k}\longmapsto + \sum_{j,i}J^\#_{ji}\widetilde e_i(u^{-m_k}) + \otimes\partial_{w_j} +\end{equation} +defines a global morphism +\(R^{m_k}\to F_0M'\otimes T_{P'}\). Here is the transition +check, including the ramification locus. For changes of +coordinates let \(A_t=\partial t'/\partial t\) and +\(B_w=\partial w'/\partial w\), and let the root frame change +by \(\vartheta'=\lambda\vartheta\). Locally \(\lambda\) +descends from a holomorphic unit downstairs: its \(\ell\)-th +power does, and after choosing a local \(\ell\)-th root of +that unit the remaining ratio is a locally constant element +of \(\mu_\ell\). Root germ compatibility and +Equation~\eqref{signed:chain-rule} transform the downstairs +column of \(e_i\)'s by \(\lambda^{m_k}A_t\). No derivative +of \(\lambda\) appears, since \(x\) multiplies outside +\(\delta_k(t_i)\). The absolute canonical factor changes +by \(\det B_w/\det A_t\). Hence +\[ +\begin{aligned} + \widetilde e'&=\frac{\det B_w}{\det A_t}\lambda^{m_k}A_t\widetilde e, + &J'&=A_tJB_w^{-1},\\ + (J')^\#\widetilde e'&=\lambda^{m_k}B_wJ^\#\widetilde e. +\end{aligned} +\] +The adjugate equality is polynomial in \(J\), so holds also +when \(J\) is singular. The last formula, together with the +inverse change of the vector basis, is exactly the transition +of Equation~\eqref{signed:global-symbol}. The graph classes +here are natural Čech pullbacks through ambient germs; no +ramified base-change isomorphism for \(R^1h_*\) is used. + +Equation~\eqref{signed:local-symbol} puts the image of this +morphism in the first-symbol kernel. Since +\(R^{m_k}=\OO_{\PP^b}(a+k)\) is ample, +Equation~\eqref{signed:symbol-vanishing} makes the morphism +zero. At a point above \(t_*\), the map \(\psi\) is locally +biholomorphic and \(J^\#\) is invertible. The ambient germ +comparison identifies the graph classes with the downstairs +classes there. Thus \(e_i(u^{-m_k})=0\) as germs at +\(t_*\), including any class supported there. The point was +arbitrary, so these classes vanish on every chart. + +The split injection in Lemma~\ref{signed:residue} and the +invertible Laurent frame now give \(\delta_k(t_i)=0\). +For every leading section \(x\) of degree \(-m_k\), the +classes \(e_i(x)\) vanish. Equation~\eqref{signed:undivided}, +the injection in Equation~\eqref{signed:lowest-step}, and +then the split residue injection give \(\delta_k(x)=0\). +When \(b=0\), the direct identity \(c(x)=0\) and the same +two injections give this conclusion without a graph test. +In particular, the derivation rule and characteristic zero give +\[ + 0=\delta_k(u^{-m_k})=-m_k u^{-m_k-1}\delta_k(u), + \qquad \delta_k(u)=0. +\] +For any local \(F\in g_*\OO_S\), the degree-\(-m_k\) section +\(Fu^{-m_k}\) then gives +\(0=\delta_k(Fu^{-m_k})=u^{-m_k}\delta_k(F)\). +Every graded section is \(Fu^j\), so \(\delta_k\) vanishes +in every degree. This proves the current order and completes +the induction. +\end{proof} + +\subsection{Compact deformations and the signed theorem} + +We now turn the infinitesimal lifting into actual compact +subspaces. The role of all the integer layers in +Proposition~\ref{signed:all-orders} is that both the equations +of a boundary fiber and a generator of its normal direction +can be lifted compatibly. This is the analytic counterpart of +the compact-family step in \cite[Proposition~5.1]{LI}. + +\begin{lemma}\label{signed:escaping-fibers} +Assume \(00\). +On the selected component the locus where it is singular or +where \(g\) has rank less than \(r\) is proper. Removing its +image, along with the preceding proper bad subsets, under the +finite map leaves a nonempty open \(D_i^\circ\subset D_i\). +Its points have preimages at smooth points of these fiber +components. + +Fix \(x\in D_i^\circ\) and a point above it in the compact +fiber \(F=g^{-1}(t)\), for \(t\in Y\). Work on the single +ordinary chart neighborhood of that fiber from +Lemma~\ref{signed:root-charts}. Take regular coordinates +\(t_1,\ldots,t_r\) on \(Y\) centered at \(t\), extended to +local coordinates of its smooth embedding in \(P\). Their +pullbacks form a regular sequence on \(S\) along \(F\), +by flatness. In addition, \(S\) is Cohen--Macaulay, so +\(F\) is Cohen--Macaulay of pure dimension \(d\). + +Choose once and for all a Hausdorff open neighborhood +\(W\subset Z\) of \(F\), disjoint from \(T\), on which +\(\pi\) is quasi-finite. Set \(I=\OO_Z(-S)\). +Proposition~\ref{signed:all-orders} lets us lift the finitely +many \(t_i\)'s to compatible germs of sections of \(\OO_Z/I^q\), +\(q\geq1\), along \(F\). It also lifts the conormal frame +\(u\in I/I^2\) to compatible germs +\(\widetilde y_q\in I/I^{q+1}\) along \(F\); write \(\widetilde y\) +for this system. At each order choose simultaneous +representatives on a neighborhood of \(F\) contained in +\(W\). These neighborhoods may shrink with the order. +The leading coefficient of \(\widetilde y\) is a unit +frame, so it generates \(I\) on each finite thickening. + +For \(R_q=\C[s]/(s^q)\), cut the lifted \(r\) equations in +the Cartier thickening \(qS\), and map \(s\) to +\(\widetilde y\). Denote the resulting subspace by +\(F_q\). It is flat over the Artin analytic point +\(\Spec R_q\), with closed fiber \(F\). Before cutting, +multiplication by the Cartier generator identifies each +successive \(s\)-layer with \(\OO_S\); this is the +flatness criterion over \(R_q\). Quotienting by lifts of +the closed-fiber regular sequence preserves flatness, by +the local flatness criterion. Compatibility of the lifted +equations gives compatible reductions of all \(F_q\). + +Each of these is an embedded compact deformation in the +same fixed neighborhood \(W\). To see why the shrinking +representative neighborhoods cause no difficulty, the +support of \(F_q\) is the fixed compact set \(F\). Its +ideal on its representative neighborhood glues with the +unit ideal on \(W\setminus F\), since the deformation is +empty on their overlap away from \(F\). This realizes it +as a closed subspace of \(W\times\Spec R_q\). The ideal +is coherent locally on both opens, and the family is proper +over the Artin point because its support is compact. + +The Douady space of compact subspaces of \(W\) represents +proper flat embedded families \cite[Section~9]{Douady1966}. +The compatible \(F_q\)'s thus define a formal arc in its +analytic germ at \([F]\). Embed that germ in a finite +dimensional analytic coordinate space. The arc is a formal +solution of its convergent defining equations. Analytic +Artin approximation \cite[Theorem~1.2]{Artin1968} gives a +convergent arc agreeing modulo \(s^2\). Pull back the +universal family and shrink its disk to obtain +\(\mathscr F\subset W\times\Delta\), proper and flat over +\(\Delta\), with closed fiber \(F\) and the prescribed +first normal displacement. + +The family may be taken to have pure \(d\)-dimensional +fibers throughout the disk. Here is a local justification. +The central fiber is Cohen--Macaulay of dimension \(d\). +Flatness over the regular one-dimensional base makes its +parameter a nonzerodivisor and gives the depth and dimension +of the total local ring as \(d+1\) along that fiber. +Cohen--Macaulayness is open for analytic local rings, and +properness allows a disk shrink for which it holds on the +whole family. Every total component then meets the central +fiber after shrinking, and flatness makes it dominate the +disk; the dimension formula gives it dimension \(d+1\). +Quotient by the nonzerodivisor at each +parameter gives Cohen--Macaulay fibers of dimension \(d\). +The analytic dimension formula and unmixedness of these +local rings give the asserted purity. + +Let \(f\) be a local equation of \(S\). On \(\mathscr F\) +it vanishes on the central fiber, so flatness writes it as +\(f=s h\). Agreement modulo \(s^2\) with the formal +deformation says that \(h|_F\) is a unit: this is precisely +the lifted conormal generator. Finitely many such +neighborhoods cover the compact \(F\). Properness of +\(\mathscr F\to\Delta\) lets us shrink the disk so that +they cover the entire family and all their \(h\)'s are +units; remove the closed image of the complement. Thus a +nonzero fiber misses \(S\). The neighborhood \(W\) already +misses \(T\), and the set equality in Lemma~\ref{signed:root-charts} gives +\(\pi^{-1}(|D|)\cap W=|S|\cap W\). Hence the +nonzero fiber images miss \(D\). + +The induced map \(\Pi:\mathscr F\to X\times\Delta\) is proper: +the source is proper over \(\Delta\) and the target is +Hausdorff over it, so the graph is closed and the projection +is proper. It is quasi-finite by the choice of \(W\), +hence finite. It therefore preserves the dimension of +every fiber component. Every total component through a central +point dominates the disk by flatness, and its punctured part is dense. +Thus every central point, including the point chosen above \(x\), +is a limit of points of nonzero fibers of dimension \(d\). + +Let \(\mathscr Z\subset X\times\Delta\) be the schematic +image of this finite map. Its algebra is the coherent image +of \(\OO_{X\times\Delta}\to\Pi_*\OO_{\mathscr F}\). +Multiplication by a nonzero +germ from the disk is injective on \(\OO_{\mathscr F}\), +by flatness, and remains injective after the exact finite +pushforward. It is therefore injective on the image algebra. +That algebra is torsion-free, hence flat, over each local +discrete valuation ring of the disk. Thus \(\mathscr Z\) +is a proper flat family of subspaces of \(X\). The finite +map to its schematic image is surjective, also on each +fiber as a map of supports. Each \(\mathscr Z_s\) consequently +has pure dimension \(d\). The Douady-to-cycle morphism, in the +form recorded in \cite[Section~3.3]{Fujiki1978}, makes their +fundamental cycles an analytic family in the cycle space +of \(X\). The restriction to a smaller closed disk is +compact. The volumes of these cycles against a Kähler +form are bounded by continuity, proving the last assertion. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{signed:torsion}] +The assertion is immediate in dimension zero, and the case +\(D=0\) follows directly from the signed equivalence. By +Lemma~\ref{signed:boundary-geometry}, the case \(v=0\) +already gives \(L\sim_{\Q}0\), and \(v=n>0\) contradicts +\(a(X)=0\). 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+\node[font=\bfseries] at (3,-3.3) {Prescribed-filtration transport}; +\node[variety] (Z) at (0,-4.4) {$Z^m/\Qbar$\quad(balanced Weil)}; +\node[variety] (B) at (6,-4.4) {$B/K$, $K/K_0$ finite}; +\node[variety] (Z0) at (0,-6.1) {$Z_0^m$}; +\node[variety] (B0) at (6,-6.1) {$B_0$}; +\path (Z) edge node[left,note] {reduction} (Z0); +\path (B) edge node[right,note] {reduction} (B0); +\path (Z0) edge node[above] {$a_{\mathrm{fl}}$} (B0); +\node[note] at (3,-7.05) + {$a_{\mathrm{fl}}^*\Fil^1H^1_{\mathrm{dR}}(B) + =(\mathcal F_Z^1)^{\oplus m}$}; +\end{tikzpicture} diff --git a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/main.tex b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/main.tex index 3b75da36136b6a67ae6353e640a6d85a81634301..e79d6ed2b434f8abf47bee6b9f7efe8123a3aad7 100644 --- a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/main.tex +++ b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/main.tex @@ -1,73 +1,73 @@ -\documentclass[11pt]{article} -\usepackage[a4paper,margin=28mm]{geometry} -\usepackage[T1]{fontenc} -\usepackage{lmodern} -\usepackage{amsmath,amssymb,amsthm,mathtools} -\usepackage{microtype} -\usepackage{enumitem} -\usepackage{booktabs} -\usepackage[dvipsnames]{xcolor} -\usepackage{tikz} -\usetikzlibrary{arrows.meta,positioning} -\usepackage[colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue,urlcolor=MidnightBlue,breaklinks=true]{hyperref} -\hypersetup{pdftitle={Milne's rationality conjecture for abelian varieties}, - pdfsubject={Rational pairings of specialized Hodge classes with Lefschetz classes}, - pdfauthor={OpenAI}} -\setlength{\emergencystretch}{2em} -\setlist[enumerate]{itemsep=3pt,topsep=5pt} -\setlist[itemize]{itemsep=3pt,topsep=5pt} -\newtheorem{theorem}{Theorem}[section] -\newtheorem{proposition}[theorem]{Proposition} -\newtheorem{lemma}[theorem]{Lemma} -\newtheorem{corollary}[theorem]{Corollary} -\theoremstyle{definition} -\newtheorem{definition}[theorem]{Definition} -\theoremstyle{remark} -\newtheorem{remark}[theorem]{Remark} -\numberwithin{equation}{section} -\newcommand{\Q}{\mathbb Q} -\newcommand{\Z}{\mathbb Z} -\newcommand{\R}{\mathbb R} -\newcommand{\C}{\mathbb C} -\newcommand{\F}{\mathbb F} -\newcommand{\Qbar}{\overline{\mathbb Q}} -\newcommand{\Kzero}{K_0} -\newcommand{\Cw}{C_w} -\DeclareMathOperator{\End}{End} -\DeclareMathOperator{\Hom}{Hom} -\DeclareMathOperator{\Tr}{Tr} -\DeclareMathOperator{\Fil}{Fil} -\DeclareMathOperator{\Res}{Res} -\DeclareMathOperator{\GSp}{GSp} -\DeclareMathOperator{\GL}{GL} -\DeclareMathOperator{\SL}{SL} -\DeclareMathOperator{\SU}{SU} -\DeclareMathOperator{\MT}{MT} -\DeclareMathOperator{\Lef}{Lef} -\DeclareMathOperator{\NS}{NS} -\DeclareMathOperator{\Gal}{Gal} -\DeclareMathOperator{\id}{id} -\DeclareMathOperator{\Spec}{Spec} -\title{Milne's rationality conjecture\protect\\ for abelian varieties} -\author{OpenAI} -\date{October 7, 2026} -\begin{document} -\maketitle -\vspace{-1em} -\begin{abstract} -We prove Milne's rationality conjecture for abelian varieties, including -residue characteristic \(2\). After good reduction, the pairing of a -rational Hodge class with any complementary product of divisor classes -on the reduction is the same rational number in every prime-to-\(p\) -realization and in crystalline cohomology. Using the Hodge theorem for CM -abelian varieties, we also show that every such specialized Hodge class -is represented by a single rational algebraic cycle in all these realizations. -\end{abstract} -\input{sections/00_intro} -\input{sections/01_preliminaries} -\input{sections/02_transport} -\input{sections/03_filtration} -\input{sections/04_weil} -\input{sections/05_reduction} -\input{references} -\end{document} +\documentclass[11pt]{article} +\usepackage[a4paper,margin=28mm]{geometry} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{booktabs} +\usepackage[dvipsnames]{xcolor} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage[colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue,urlcolor=MidnightBlue,breaklinks=true]{hyperref} +\hypersetup{pdftitle={Milne's rationality conjecture for abelian varieties}, + pdfsubject={Rational pairings of specialized Hodge classes with Lefschetz classes}, + pdfauthor={OpenAI}} +\setlength{\emergencystretch}{2em} +\setlist[enumerate]{itemsep=3pt,topsep=5pt} +\setlist[itemize]{itemsep=3pt,topsep=5pt} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\newcommand{\Q}{\mathbb Q} +\newcommand{\Z}{\mathbb Z} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\F}{\mathbb F} +\newcommand{\Qbar}{\overline{\mathbb Q}} +\newcommand{\Kzero}{K_0} +\newcommand{\Cw}{C_w} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\Fil}{Fil} +\DeclareMathOperator{\Res}{Res} +\DeclareMathOperator{\GSp}{GSp} +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\SL}{SL} +\DeclareMathOperator{\SU}{SU} +\DeclareMathOperator{\MT}{MT} +\DeclareMathOperator{\Lef}{Lef} +\DeclareMathOperator{\NS}{NS} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\Spec}{Spec} +\title{Milne's rationality conjecture\protect\\ for abelian varieties} +\author{OpenAI} +\date{October 7, 2026} +\begin{document} +\maketitle +\vspace{-1em} +\begin{abstract} +We prove Milne's rationality conjecture for abelian varieties, including +residue characteristic \(2\). After good reduction, the pairing of a +rational Hodge class with any complementary product of divisor classes +on the reduction is the same rational number in every prime-to-\(p\) +realization and in crystalline cohomology. Using the Hodge theorem for CM +abelian varieties, we also show that every such specialized Hodge class +is represented by a single rational algebraic cycle in all these realizations. +\end{abstract} +\input{sections/00_intro} +\input{sections/01_preliminaries} +\input{sections/02_transport} +\input{sections/03_filtration} +\input{sections/04_weil} +\input{sections/05_reduction} +\input{references} +\end{document} diff --git a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/references.tex b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/references.tex index 418e678f30e6d8aa4c0d8c3b513d9ca01a9fce80..aca2d21ca77b497895931a1da82f248f18bc71fb 100644 --- a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/references.tex +++ b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/references.tex @@ -1,274 +1,274 @@ -\begin{thebibliography}{Stacks} - -\bibitem[And92]{Andre1992} -Y.~Andr\'e, -\emph{Une remarque \`a propos des cycles de Hodge de type CM}, -in S.~David (ed.), \emph{S\'eminaire de Th\'eorie des Nombres, -Paris, 1989--90}, Progress in Math. \textbf{102}, -Birkh\"auser Boston, Boston, MA, 1992, 1--7. - -\bibitem[And96]{Andre1996} -Y.~Andr\'e, -\emph{Pour une th\'eorie inconditionnelle des motifs}, -Publ. Math. Inst. Hautes \'{E}tudes Sci. \textbf{83} (1996), 5--49. -\href{https://doi.org/10.1007/BF02698643}{doi:10.1007/BF02698643}. -\href{https://www.numdam.org/item/10.1007/BF02698643.pdf}{Original article}. - -\bibitem[Bla94]{Blasius1994} -D.~Blasius, -\emph{A $p$-adic property of Hodge classes on abelian varieties}, -in \emph{Motives} (Seattle, WA, 1991), Part~2, -U.~Jannsen, S.~Kleiman, and J.-P.~Serre (eds.), -Proc. Sympos. Pure Math. \textbf{55}, Part~2, -American Mathematical Society, Providence, RI, 1994, 293--308. -\href{https://doi.org/10.1090/pspum/055.2/1265557}{doi:10.1090/pspum/055.2/1265557}. -\href{https://archive.mpim-bonn.mpg.de/id/eprint/3859/1/preprint_1992_59.pdf}{Author's preprint}. - -\bibitem[CF00]{ColmezFontaine2000} -P.~Colmez and J.-M.~Fontaine, -\emph{Construction des repr\'esentations $p$-adiques semi-stables}, -Invent. Math. \textbf{140} (2000), no.~1, 1--43. -\href{https://doi.org/10.1007/s002220000042}{doi:10.1007/s002220000042}. -\href{https://www.imo.universite-paris-saclay.fr/~fontaine/CF.pdf}{Author's version}. - -\bibitem[Con03]{Conrad2003} -B.~Conrad, -\emph{Minimal models for elliptic curves}, notes, 20 October 2003. -\href{https://math.stanford.edu/~conrad/vigregroup/vigre03/minimalmodel.pdf}{Author's notes}. - -\bibitem[Del82]{Deligne1982} -P.~Deligne, -\emph{Hodge cycles on abelian varieties} (notes by J.~S.~Milne), -in P.~Deligne, J.~S.~Milne, A.~Ogus, and K.-y.~Shih, -\emph{Hodge Cycles, Motives, and Shimura Varieties}, -Lecture Notes in Math. \textbf{900}, Springer-Verlag, -Berlin--New York, 1982, 9--100. -\href{https://www.jmilne.org/math/Documents/Deligne82.pdf}{Corrected online version}. - -\bibitem[FR05]{FontaineRapoport2005} -J.-M.~Fontaine and M.~Rapoport, -\emph{Existence de filtrations admissibles sur des isocristaux}, -Bull. Soc. Math. France \textbf{133} (2005), no.~1, 73--86. -\href{https://doi.org/10.24033/bsmf.2479}{doi:10.24033/bsmf.2479}. -\href{https://www.numdam.org/article/BSMF_2005__133_1_73_0.pdf}{Published version}. - -\bibitem[GLX26]{GleasonLimXu2026} -I.~Gleason, D.~G.~Lim, and Y.~Xu, -\emph{The connected components of affine Deligne--Lusztig varieties}, -Invent. Math. \textbf{243} (2026), 805--861. -\href{https://doi.org/10.1007/s00222-025-01386-1}{doi:10.1007/s00222-025-01386-1}. -\href{https://arxiv.org/abs/2208.07195v3}{arXiv:2208.07195v3} -(10 November 2025); preprint locators refer to -\href{https://arxiv.org/pdf/2208.07195v3}{this version}. - -\bibitem[Gro71]{Grothendieck1970} -A.~Grothendieck, -\emph{Groupes de Barsotti-Tate et cristaux}, -in \emph{Actes du Congr\`es International des Math\'ematiciens} -(Nice, 1970), Tome~1, Gauthier-Villars, Paris, 1971, 431--436. -\href{https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GroupesB-T.pdf}{Original article}. - -\bibitem[Har08]{Hartl2008} -U.~Hartl, -\emph{On period spaces for $p$-divisible groups}, -C. R. Math. Acad. Sci. Paris \textbf{346} (2008), nos.~21--22, 1123--1128. -\href{https://doi.org/10.1016/j.crma.2008.07.003}{doi:10.1016/j.crma.2008.07.003}. -\href{https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2008.07.003.pdf}{Published version}. - -\bibitem[Ill15]{IllusiePisa} -L.~Illusie, -\emph{Grothendieck at Pisa: crystals and Barsotti--Tate groups}, -in U.~Zannier (ed.), \emph{Colloquium De Giorgi 2013 and 2014}, -Publications of the Scuola Normale Superiore \textbf{5}, -Edizioni della Normale, Pisa, 2015, 79--107. -\href{https://doi.org/10.1007/978-88-7642-515-8_5}{doi:10.1007/978-88-7642-515-8\_5}. -\href{https://www.imo.universite-paris-saclay.fr/~luc.illusie/Illusie-Pisa5.pdf}{Author's version}. - -\bibitem[Kim12]{Kim2012} -W.~Kim, -\emph{The classification of $p$-divisible groups over $2$-adic discrete -valuation rings}, -Math. Res. Lett. \textbf{19} (2012), no.~1, 121--141. -\href{https://doi.org/10.4310/MRL.2012.v19.n1.a10}{doi:10.4310/MRL.2012.v19.n1.a10}. -\href{https://arxiv.org/abs/1007.1904v3}{arXiv:1007.1904v3} -(30 December 2011); preprint locators refer to -\href{https://arxiv.org/pdf/1007.1904v3}{this version}. - -\bibitem[Kis06]{Kisin2006} -M.~Kisin, -\emph{Crystalline representations and $F$-crystals}, -in V.~Ginzburg (ed.), \emph{Algebraic Geometry and Number Theory: -In Honor of Vladimir Drinfeld's 50th Birthday}, -Progress in Math. \textbf{253}, Birkh\"auser Boston, Boston, MA, -2006, 459--496. -\href{https://doi.org/10.1007/978-0-8176-4532-8_7}{doi:10.1007/978-0-8176-4532-8\_7}. - -\bibitem[Kis17]{Kisin2017} -M.~Kisin, -\emph{Mod $p$ points on Shimura varieties of abelian type}, -J. Amer. Math. Soc. \textbf{30} (2017), no.~3, 819--914. -\href{https://people.math.harvard.edu/~kisin/dvifiles/lr.pdf}{Author's version used for all locators}. - -\bibitem[KZ25a]{KisinZhouEarlier} -M.~Kisin and R.~Zhou, -\emph{Independence of $\ell$ for Frobenius conjugacy classes attached to -abelian varieties}, -Ann. of Math. (2) \textbf{202} (2025), no.~3, 1077--1156. -\href{https://doi.org/10.4007/annals.2025.202.3.3}{doi:10.4007/annals.2025.202.3.3}. -\href{https://arxiv.org/abs/2103.09945v2}{arXiv:2103.09945v2} (7 October 2024). -All locators refer to -\href{https://arxiv.org/pdf/2103.09945v2}{arXiv:2103.09945v2}. - -\bibitem[KZ25b]{KisinZhou2025} -M.~Kisin and R.~Zhou, -\emph{Strongly compatible systems associated to semistable abelian varieties}, -preprint, 2025, -\href{https://arxiv.org/abs/2505.02165v1}{arXiv:2505.02165v1} -(deposited 4 May 2025). -\href{https://arxiv.org/pdf/2505.02165v1}{PDF version used for all locators}, -dated 10 June 2025. - -\bibitem[Mar26]{Markman2026} -E.~Markman, -\emph{Secant sheaves and Weil classes on abelian varieties}, -preprint, contribution to the ICM 2026, -\href{https://arxiv.org/abs/2509.23403v2}{arXiv:2509.23403v2} -(11 February 2026). -\href{https://arxiv.org/pdf/2509.23403v2}{Version used for all locators}, -dated 13 February 2026. - -\bibitem[Mil86]{MilneAV} -J.~S.~Milne, -\emph{Abelian varieties}, -in G.~Cornell and J.~H.~Silverman (eds.), \emph{Arithmetic Geometry} -(Storrs, CT, 1984), Springer-Verlag, New York, 1986, 103--150. -\href{https://doi.org/10.1007/978-1-4613-8655-1_5}{doi:10.1007/978-1-4613-8655-1\_5}. -\href{https://www.jmilne.org/math/xnotes/AVs.pdf}{Corrected author's version}, -2 January 2022; original theorem numbering retained. - -\bibitem[Mil99]{Milne1999} -J.~S.~Milne, -\emph{Lefschetz classes on abelian varieties}, -Duke Math. J. \textbf{96} (1999), no.~3, 639--675. -\href{https://www.jmilne.org/math/articles/1999aP.pdf}{Author's version}; -\href{https://www.jmilne.org/math/articles/1999a.html}{author's correction to -the proof of Theorem~5.9}. - -\bibitem[Mil00]{Milne2000} -J.~S.~Milne, -\emph{Towards a proof of the conjecture of Langlands and Rapoport}, -notes for a talk at the Institut Henri Poincar\'e, 28 April 2000, -version dated 30 April 2000. -\href{https://www.jmilne.org/math/articles/2000.pdf}{Author's notes}. - -\bibitem[Mil05]{MilneShimura} -J.~S.~Milne, -\emph{Introduction to Shimura varieties}, -in \emph{Harmonic Analysis, the Trace Formula, and Shimura Varieties}, -Clay Math. Proc. \textbf{4}, American Mathematical Society, -Providence, RI, 2005, 265--378. -\href{https://www.jmilne.org/math/xnotes/svi.pdf}{Author's revision}, -16 September 2017; locators refer to this revision. - -\bibitem[Mil08a]{MilneAIM2007} -J.~S.~Milne, -\emph{The Tate conjecture over finite fields (AIM talk)}, -notes for the American Institute of Mathematics workshop, July 2007, -version~2.2, 7 May 2008. -\href{https://www.jmilne.org/math/articles/2007e.pdf}{Version used for all locators}. - -\bibitem[Mil08b]{MilneAVNotes} -J.~S.~Milne, -\emph{Abelian Varieties}, course notes, version~2.00, 2008, vi+166~pp. -\href{https://www.jmilne.org/math/CourseNotes/AVc.pdf}{Author's course notes}. - -\bibitem[Mil09]{Milne2009} -J.~S.~Milne, -\emph{Rational Tate classes}, -Mosc. Math. J. \textbf{9} (2009), no.~1, 111--141. -\href{https://www.jmilne.org/math/articles/2009b.pdf}{Author's version}; -page numbers refer to this version. - -\bibitem[Mil13]{MilneLEC} -J.~S.~Milne, -\emph{Lectures on \'{E}tale Cohomology}, -course notes, version~2.21, 22 March 2013. -\href{https://www.jmilne.org/math/CourseNotes/LEC.pdf}{Author's notes}. - -\bibitem[MilANT]{MilneANT} -J.~S.~Milne, -\emph{Algebraic Number Theory}, course notes, version~3.08, 19 July 2020. -\href{https://www.jmilne.org/math/CourseNotes/ANTc.pdf}{Author's course notes}. -Locators refer to this version. - -\bibitem[Mil20]{Milne2020} -J.~S.~Milne, -\emph{Hodge classes on abelian varieties}, -preprint, 2020, -\href{https://arxiv.org/abs/2010.08857v2}{arXiv:2010.08857v2} -(11 November 2020). -\href{https://arxiv.org/pdf/2010.08857v2}{Version used for all locators}. - -\bibitem[Mil22]{MilneAG2022} -J.~S.~Milne, -\emph{Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field}, -Cambridge Studies in Advanced Mathematics \textbf{170}, -Cambridge University Press, Cambridge, 2017; corrected reprint, 2022. -\href{https://doi.org/10.1017/9781316711736}{doi:10.1017/9781316711736}. -\href{https://www.jmilne.org/math/Books/iAG2022.pdf}{Corrected 2022 version used for all locators}. - -\bibitem[Mil25]{Milne2025} -J.~S.~Milne, -\emph{Abelian motives and Shimura varieties in nonzero characteristic}, -preprint, version~2.2, 2 November 2025. -\href{https://www.jmilne.org/math/articles/SVp.pdf}{Author's version used for all locators}. - -\bibitem[Moo98]{Moonen1998} -B.~Moonen, -\emph{Models of Shimura varieties in mixed characteristics}, -in A.~J.~Scholl and R.~L.~Taylor (eds.), -\emph{Galois Representations in Arithmetic Algebraic Geometry} -(Durham, 1996), London Math. Soc. Lecture Note Ser. \textbf{254}, -Cambridge University Press, Cambridge, 1998, 267--350. -\href{https://doi.org/10.1017/CBO9780511662010.008}{doi:10.1017/CBO9780511662010.008}. -\href{https://www.math.ru.nl/personal/bmoonen/Papers/SMCfinal.pdf}{Author's version}. - -\bibitem[OAI26]{OpenAICMHodge2026} -OpenAI, -\emph{The rational Hodge conjecture for CM abelian varieties},\newline -OpenAI Math Release preprint -\href{https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf}{OAI:The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026}, -2026. - -\bibitem[PR24]{PappasRapoport2024} -G.~Pappas and M.~Rapoport, -\emph{$p$-adic shtukas and the theory of global and local Shimura varieties}, -Camb. J. Math. \textbf{12} (2024), no.~1, 1--164. -\href{https://doi.org/10.4310/CJM.2024.v12.n1.a1}{doi:10.4310/CJM.2024.v12.n1.a1}. -\href{https://arxiv.org/abs/2106.08270v6}{arXiv:2106.08270v6} -(26 October 2023); preprint page numbers refer to -\href{https://arxiv.org/pdf/2106.08270v6}{this version}. - -\bibitem[Stacks]{StacksProject} -The Stacks Project Authors, -\emph{The Stacks Project}, -\href{https://stacks.math.columbia.edu/tag/0A42}{Lemma~30.28.3, Tag~0A42}, -accessed 23 September 2026. - -\bibitem[Tat67]{Tate1967} -J.~T.~Tate, -\emph{$p$-Divisible Groups}, -in T.~A.~Springer (ed.), \emph{Proceedings of a Conference on Local Fields} -(Driebergen, 1966), Springer-Verlag, Berlin--Heidelberg, 1967, 158--183. -\href{https://doi.org/10.1007/978-3-642-87942-5_12}{doi:10.1007/978-3-642-87942-5\_12}. -\href{https://www.math.purdue.edu/~tongliu/teaching/598/p-divisible.pdf}{Original article}. - -\bibitem[Win97]{Wintenberger1997} -J.-P.~Wintenberger, -\emph{Propri\'et\'es du groupe tannakien des structures de Hodge $p$-adiques -et torseur entre cohomologies cristalline et \'etale}, -Ann. Inst. Fourier (Grenoble) \textbf{47} (1997), no.~5, 1289--1334. -\href{https://doi.org/10.5802/aif.1600}{doi:10.5802/aif.1600}. -\href{https://www.numdam.org/item/10.5802/aif.1600.pdf}{Original article}. - -\end{thebibliography} +\begin{thebibliography}{Stacks} + +\bibitem[And92]{Andre1992} +Y.~Andr\'e, +\emph{Une remarque \`a propos des cycles de Hodge de type CM}, +in S.~David (ed.), \emph{S\'eminaire de Th\'eorie des Nombres, +Paris, 1989--90}, Progress in Math. \textbf{102}, +Birkh\"auser Boston, Boston, MA, 1992, 1--7. + +\bibitem[And96]{Andre1996} +Y.~Andr\'e, +\emph{Pour une th\'eorie inconditionnelle des motifs}, +Publ. Math. Inst. Hautes \'{E}tudes Sci. \textbf{83} (1996), 5--49. +\href{https://doi.org/10.1007/BF02698643}{doi:10.1007/BF02698643}. +\href{https://www.numdam.org/item/10.1007/BF02698643.pdf}{Original article}. + +\bibitem[Bla94]{Blasius1994} +D.~Blasius, +\emph{A $p$-adic property of Hodge classes on abelian varieties}, +in \emph{Motives} (Seattle, WA, 1991), Part~2, +U.~Jannsen, S.~Kleiman, and J.-P.~Serre (eds.), +Proc. Sympos. Pure Math. \textbf{55}, Part~2, +American Mathematical Society, Providence, RI, 1994, 293--308. +\href{https://doi.org/10.1090/pspum/055.2/1265557}{doi:10.1090/pspum/055.2/1265557}. +\href{https://archive.mpim-bonn.mpg.de/id/eprint/3859/1/preprint_1992_59.pdf}{Author's preprint}. + +\bibitem[CF00]{ColmezFontaine2000} +P.~Colmez and J.-M.~Fontaine, +\emph{Construction des repr\'esentations $p$-adiques semi-stables}, +Invent. Math. \textbf{140} (2000), no.~1, 1--43. +\href{https://doi.org/10.1007/s002220000042}{doi:10.1007/s002220000042}. +\href{https://www.imo.universite-paris-saclay.fr/~fontaine/CF.pdf}{Author's version}. + +\bibitem[Con03]{Conrad2003} +B.~Conrad, +\emph{Minimal models for elliptic curves}, notes, 20 October 2003. +\href{https://math.stanford.edu/~conrad/vigregroup/vigre03/minimalmodel.pdf}{Author's notes}. + +\bibitem[Del82]{Deligne1982} +P.~Deligne, +\emph{Hodge cycles on abelian varieties} (notes by J.~S.~Milne), +in P.~Deligne, J.~S.~Milne, A.~Ogus, and K.-y.~Shih, +\emph{Hodge Cycles, Motives, and Shimura Varieties}, +Lecture Notes in Math. \textbf{900}, Springer-Verlag, +Berlin--New York, 1982, 9--100. +\href{https://www.jmilne.org/math/Documents/Deligne82.pdf}{Corrected online version}. + +\bibitem[FR05]{FontaineRapoport2005} +J.-M.~Fontaine and M.~Rapoport, +\emph{Existence de filtrations admissibles sur des isocristaux}, +Bull. Soc. Math. France \textbf{133} (2005), no.~1, 73--86. +\href{https://doi.org/10.24033/bsmf.2479}{doi:10.24033/bsmf.2479}. +\href{https://www.numdam.org/article/BSMF_2005__133_1_73_0.pdf}{Published version}. + +\bibitem[GLX26]{GleasonLimXu2026} +I.~Gleason, D.~G.~Lim, and Y.~Xu, +\emph{The connected components of affine Deligne--Lusztig varieties}, +Invent. Math. \textbf{243} (2026), 805--861. +\href{https://doi.org/10.1007/s00222-025-01386-1}{doi:10.1007/s00222-025-01386-1}. +\href{https://arxiv.org/abs/2208.07195v3}{arXiv:2208.07195v3} +(10 November 2025); preprint locators refer to +\href{https://arxiv.org/pdf/2208.07195v3}{this version}. + +\bibitem[Gro71]{Grothendieck1970} +A.~Grothendieck, +\emph{Groupes de Barsotti-Tate et cristaux}, +in \emph{Actes du Congr\`es International des Math\'ematiciens} +(Nice, 1970), Tome~1, Gauthier-Villars, Paris, 1971, 431--436. +\href{https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GroupesB-T.pdf}{Original article}. + +\bibitem[Har08]{Hartl2008} +U.~Hartl, +\emph{On period spaces for $p$-divisible groups}, +C. R. Math. Acad. Sci. Paris \textbf{346} (2008), nos.~21--22, 1123--1128. +\href{https://doi.org/10.1016/j.crma.2008.07.003}{doi:10.1016/j.crma.2008.07.003}. +\href{https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2008.07.003.pdf}{Published version}. + +\bibitem[Ill15]{IllusiePisa} +L.~Illusie, +\emph{Grothendieck at Pisa: crystals and Barsotti--Tate groups}, +in U.~Zannier (ed.), \emph{Colloquium De Giorgi 2013 and 2014}, +Publications of the Scuola Normale Superiore \textbf{5}, +Edizioni della Normale, Pisa, 2015, 79--107. +\href{https://doi.org/10.1007/978-88-7642-515-8_5}{doi:10.1007/978-88-7642-515-8\_5}. +\href{https://www.imo.universite-paris-saclay.fr/~luc.illusie/Illusie-Pisa5.pdf}{Author's version}. + +\bibitem[Kim12]{Kim2012} +W.~Kim, +\emph{The classification of $p$-divisible groups over $2$-adic discrete +valuation rings}, +Math. Res. Lett. \textbf{19} (2012), no.~1, 121--141. +\href{https://doi.org/10.4310/MRL.2012.v19.n1.a10}{doi:10.4310/MRL.2012.v19.n1.a10}. +\href{https://arxiv.org/abs/1007.1904v3}{arXiv:1007.1904v3} +(30 December 2011); preprint locators refer to +\href{https://arxiv.org/pdf/1007.1904v3}{this version}. + +\bibitem[Kis06]{Kisin2006} +M.~Kisin, +\emph{Crystalline representations and $F$-crystals}, +in V.~Ginzburg (ed.), \emph{Algebraic Geometry and Number Theory: +In Honor of Vladimir Drinfeld's 50th Birthday}, +Progress in Math. \textbf{253}, Birkh\"auser Boston, Boston, MA, +2006, 459--496. +\href{https://doi.org/10.1007/978-0-8176-4532-8_7}{doi:10.1007/978-0-8176-4532-8\_7}. + +\bibitem[Kis17]{Kisin2017} +M.~Kisin, +\emph{Mod $p$ points on Shimura varieties of abelian type}, +J. Amer. Math. Soc. \textbf{30} (2017), no.~3, 819--914. +\href{https://people.math.harvard.edu/~kisin/dvifiles/lr.pdf}{Author's version used for all locators}. + +\bibitem[KZ25a]{KisinZhouEarlier} +M.~Kisin and R.~Zhou, +\emph{Independence of $\ell$ for Frobenius conjugacy classes attached to +abelian varieties}, +Ann. of Math. (2) \textbf{202} (2025), no.~3, 1077--1156. +\href{https://doi.org/10.4007/annals.2025.202.3.3}{doi:10.4007/annals.2025.202.3.3}. +\href{https://arxiv.org/abs/2103.09945v2}{arXiv:2103.09945v2} (7 October 2024). +All locators refer to +\href{https://arxiv.org/pdf/2103.09945v2}{arXiv:2103.09945v2}. + +\bibitem[KZ25b]{KisinZhou2025} +M.~Kisin and R.~Zhou, +\emph{Strongly compatible systems associated to semistable abelian varieties}, +preprint, 2025, +\href{https://arxiv.org/abs/2505.02165v1}{arXiv:2505.02165v1} +(deposited 4 May 2025). +\href{https://arxiv.org/pdf/2505.02165v1}{PDF version used for all locators}, +dated 10 June 2025. + +\bibitem[Mar26]{Markman2026} +E.~Markman, +\emph{Secant sheaves and Weil classes on abelian varieties}, +preprint, contribution to the ICM 2026, +\href{https://arxiv.org/abs/2509.23403v2}{arXiv:2509.23403v2} +(11 February 2026). +\href{https://arxiv.org/pdf/2509.23403v2}{Version used for all locators}, +dated 13 February 2026. + +\bibitem[Mil86]{MilneAV} +J.~S.~Milne, +\emph{Abelian varieties}, +in G.~Cornell and J.~H.~Silverman (eds.), \emph{Arithmetic Geometry} +(Storrs, CT, 1984), Springer-Verlag, New York, 1986, 103--150. +\href{https://doi.org/10.1007/978-1-4613-8655-1_5}{doi:10.1007/978-1-4613-8655-1\_5}. +\href{https://www.jmilne.org/math/xnotes/AVs.pdf}{Corrected author's version}, +2 January 2022; original theorem numbering retained. + +\bibitem[Mil99]{Milne1999} +J.~S.~Milne, +\emph{Lefschetz classes on abelian varieties}, +Duke Math. J. \textbf{96} (1999), no.~3, 639--675. +\href{https://www.jmilne.org/math/articles/1999aP.pdf}{Author's version}; +\href{https://www.jmilne.org/math/articles/1999a.html}{author's correction to +the proof of Theorem~5.9}. + +\bibitem[Mil00]{Milne2000} +J.~S.~Milne, +\emph{Towards a proof of the conjecture of Langlands and Rapoport}, +notes for a talk at the Institut Henri Poincar\'e, 28 April 2000, +version dated 30 April 2000. +\href{https://www.jmilne.org/math/articles/2000.pdf}{Author's notes}. + +\bibitem[Mil05]{MilneShimura} +J.~S.~Milne, +\emph{Introduction to Shimura varieties}, +in \emph{Harmonic Analysis, the Trace Formula, and Shimura Varieties}, +Clay Math. Proc. \textbf{4}, American Mathematical Society, +Providence, RI, 2005, 265--378. +\href{https://www.jmilne.org/math/xnotes/svi.pdf}{Author's revision}, +16 September 2017; locators refer to this revision. + +\bibitem[Mil08a]{MilneAIM2007} +J.~S.~Milne, +\emph{The Tate conjecture over finite fields (AIM talk)}, +notes for the American Institute of Mathematics workshop, July 2007, +version~2.2, 7 May 2008. +\href{https://www.jmilne.org/math/articles/2007e.pdf}{Version used for all locators}. + +\bibitem[Mil08b]{MilneAVNotes} +J.~S.~Milne, +\emph{Abelian Varieties}, course notes, version~2.00, 2008, vi+166~pp. +\href{https://www.jmilne.org/math/CourseNotes/AVc.pdf}{Author's course notes}. + +\bibitem[Mil09]{Milne2009} +J.~S.~Milne, +\emph{Rational Tate classes}, +Mosc. Math. J. \textbf{9} (2009), no.~1, 111--141. +\href{https://www.jmilne.org/math/articles/2009b.pdf}{Author's version}; +page numbers refer to this version. + +\bibitem[Mil13]{MilneLEC} +J.~S.~Milne, +\emph{Lectures on \'{E}tale Cohomology}, +course notes, version~2.21, 22 March 2013. +\href{https://www.jmilne.org/math/CourseNotes/LEC.pdf}{Author's notes}. + +\bibitem[MilANT]{MilneANT} +J.~S.~Milne, +\emph{Algebraic Number Theory}, course notes, version~3.08, 19 July 2020. +\href{https://www.jmilne.org/math/CourseNotes/ANTc.pdf}{Author's course notes}. +Locators refer to this version. + +\bibitem[Mil20]{Milne2020} +J.~S.~Milne, +\emph{Hodge classes on abelian varieties}, +preprint, 2020, +\href{https://arxiv.org/abs/2010.08857v2}{arXiv:2010.08857v2} +(11 November 2020). +\href{https://arxiv.org/pdf/2010.08857v2}{Version used for all locators}. + +\bibitem[Mil22]{MilneAG2022} +J.~S.~Milne, +\emph{Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field}, +Cambridge Studies in Advanced Mathematics \textbf{170}, +Cambridge University Press, Cambridge, 2017; corrected reprint, 2022. +\href{https://doi.org/10.1017/9781316711736}{doi:10.1017/9781316711736}. +\href{https://www.jmilne.org/math/Books/iAG2022.pdf}{Corrected 2022 version used for all locators}. + +\bibitem[Mil25]{Milne2025} +J.~S.~Milne, +\emph{Abelian motives and Shimura varieties in nonzero characteristic}, +preprint, version~2.2, 2 November 2025. +\href{https://www.jmilne.org/math/articles/SVp.pdf}{Author's version used for all locators}. + +\bibitem[Moo98]{Moonen1998} +B.~Moonen, +\emph{Models of Shimura varieties in mixed characteristics}, +in A.~J.~Scholl and R.~L.~Taylor (eds.), +\emph{Galois Representations in Arithmetic Algebraic Geometry} +(Durham, 1996), London Math. Soc. Lecture Note Ser. \textbf{254}, +Cambridge University Press, Cambridge, 1998, 267--350. +\href{https://doi.org/10.1017/CBO9780511662010.008}{doi:10.1017/CBO9780511662010.008}. +\href{https://www.math.ru.nl/personal/bmoonen/Papers/SMCfinal.pdf}{Author's version}. + +\bibitem[OAI26]{OpenAICMHodge2026} +OpenAI, +\emph{The rational Hodge conjecture for CM abelian varieties},\newline +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf}{OAI:The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026}, +2026. + +\bibitem[PR24]{PappasRapoport2024} +G.~Pappas and M.~Rapoport, +\emph{$p$-adic shtukas and the theory of global and local Shimura varieties}, +Camb. J. Math. \textbf{12} (2024), no.~1, 1--164. +\href{https://doi.org/10.4310/CJM.2024.v12.n1.a1}{doi:10.4310/CJM.2024.v12.n1.a1}. +\href{https://arxiv.org/abs/2106.08270v6}{arXiv:2106.08270v6} +(26 October 2023); preprint page numbers refer to +\href{https://arxiv.org/pdf/2106.08270v6}{this version}. + +\bibitem[Stacks]{StacksProject} +The Stacks Project Authors, +\emph{The Stacks Project}, +\href{https://stacks.math.columbia.edu/tag/0A42}{Lemma~30.28.3, Tag~0A42}, +accessed 23 September 2026. + +\bibitem[Tat67]{Tate1967} +J.~T.~Tate, +\emph{$p$-Divisible Groups}, +in T.~A.~Springer (ed.), \emph{Proceedings of a Conference on Local Fields} +(Driebergen, 1966), Springer-Verlag, Berlin--Heidelberg, 1967, 158--183. +\href{https://doi.org/10.1007/978-3-642-87942-5_12}{doi:10.1007/978-3-642-87942-5\_12}. +\href{https://www.math.purdue.edu/~tongliu/teaching/598/p-divisible.pdf}{Original article}. + +\bibitem[Win97]{Wintenberger1997} +J.-P.~Wintenberger, +\emph{Propri\'et\'es du groupe tannakien des structures de Hodge $p$-adiques +et torseur entre cohomologies cristalline et \'etale}, +Ann. Inst. Fourier (Grenoble) \textbf{47} (1997), no.~5, 1289--1334. +\href{https://doi.org/10.5802/aif.1600}{doi:10.5802/aif.1600}. +\href{https://www.numdam.org/item/10.5802/aif.1600.pdf}{Original article}. + +\end{thebibliography} diff --git a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/00_intro.tex b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/00_intro.tex index 4c134f556c9465fd5ba4249eb2a0a44ba93407c0..c926c1c540c138a4b46274f672fc0f4c43547358 100644 --- a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/00_intro.tex +++ b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/00_intro.tex @@ -1,280 +1,280 @@ -\section{Introduction} - -Suppose an abelian variety has good reduction at a prime. A rational -algebraic cycle on the variety specializes to a cycle on the reduction. -Its intersection with any complementary product of divisors has a rational -degree, and every cycle-class realization computes that same number. -Milne's rationality conjecture asks for this conclusion when the original -class is only known to be a rational Hodge class. We prove it for every -residue characteristic. - -The distinction matters because good reduction can acquire divisors that -do not lift to the original variety. Deligne's absolute-Hodge theorem gives -compatible realizations of the Hodge class, but does not by itself identify -its pairing with these new divisors as the degree of a rational cycle. -The conjecture isolates this rationality question from the stronger problem -of representing the Hodge class by an algebraic cycle. - -Fix \(\Qbar\subset\C\), a prime \(p\), and a \(p\)-adic place \(w\) -of \(\Qbar\). Write -\[ - \F=\overline{\mathbb F}_p,\qquad - \Kzero=\operatorname{Frac}W(\F),\qquad - \Cw=\widehat{\Qbar}_w. -\] -We regard \(\Kzero\) and an algebraic closure of it as subfields of -\(\Cw\). Good reduction means good reduction after descent to a number -field and finite extension at the chosen place. - -\begin{theorem}\label{thm:main} -Let \(A/\Qbar\) be an abelian variety of dimension \(d\) with good -reduction \(A_0/\F\) at \(w\). Let \(0\leq r\leq d\), and let -\[ - \gamma\in H^{2r}(A(\C),\Q(r)) -\] -be a rational Hodge class. For every list of Cartier divisors -\(D_1,\ldots,D_{d-r}\) on \(A_0\), there is a number \(q\in\Q\) -such that -\begin{align*} - \Tr_\ell\bigl(\gamma_{0,\ell} - c_{1,\ell}(D_1)\cdots c_{1,\ell}(D_{d-r})\bigr)&=q - &&\text{in }\Q_\ell\quad(\ell\ne p),\\ - \Tr_p\bigl(\gamma_{0,p} - c_{1,\mathrm{cris}}(D_1)\cdots - c_{1,\mathrm{cris}}(D_{d-r})\bigr)&=q - &&\text{in }\Cw. -\end{align*} -Here \(\gamma_{0,\ell}\) is the smooth proper specialization of the -\(\ell\)-adic realization, and \(\gamma_{0,p}\) is the -de Rham--crystalline specialization in -\[ - H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r) - \otimes_{\Kzero}\Cw. -\] -The traces use the top Tate twist and send the class of a zero-cycle to -its degree. If \(r=d\), the divisor product is \(1\). -\end{theorem} - -By linearity the same conclusion holds for every rational Lefschetz class -of complementary codimension, that is, every rational linear combination -of products of divisors. The number \(q\) is common to all realizations -for the fixed data; no comparison between different reduction places is -required. - -\subsection{Origins and consequences} - -Milne introduced a weak-rationality formulation while studying the -reduction of Shimura varieties and the Langlands--Rapoport conjecture -\cite[Conjecture~6.1(A)]{Milne2000}. It asks that a specialized CM -Hodge class whose components lie in the local spans of divisor products -in every realization arise from one rational Lefschetz class. -The pairing formulation appears in his -AIM notes and his study of rational Tate classes -\cite[Section~10]{MilneAIM2007}, \cite[Conjecture~4.1]{Milne2009}. -A previous positive case is a CM abelian variety with simple ordinary -reduction, together with all its powers: in that case every divisor -class on the reduction lifts \cite[Example~4.2]{Milne2009}. - -The broader purpose is to obtain rational cohomological correspondences -for abelian motives in characteristic~\(p\), compatible with reduction -from characteristic zero \cite[Sections~5--6]{Milne2000}; see also -\cite[Introduction and Section~2]{Milne2025}. For \(\ell\ne p\), -Tate classes are twisted \(\ell\)-adic classes fixed by a power of -Frobenius; there is a corresponding \(\Q_p\)-algebra of crystalline -Tate classes. A \emph{good theory of rational Tate classes} chooses -one graded \(\Q\)-algebra on each variety whose scalar extensions -give these Tate algebras. The theory is stable under pullback and -pushforward, contains divisor classes, and contains specializations of -CM Hodge classes \cite[Definitions~2.1 and~3.1]{Milne2009}. -These are cohomological correspondences; their definition does not -assert that they are algebraic. - -Milne proved that the rationality assertion for all CM abelian varieties -is equivalent to the existence of such a good theory on varieties over -\(\F\) whose connected components are products of abelian varieties -and projective spaces \cite[Theorem~4.5]{Milne2009}. -Our proof also expresses each specialized Hodge class as the pullback, -along a rational homomorphism of special fibers, of a specialized Hodge -class from a CM abelian variety. This extends the theory's -specialization property beyond the CM case. -In Corollary~\ref{cor:all-hodge-specialization}, every rational -Hodge class on a good-reduction abelian variety specializes to a unique -class in one fixed good theory. Its crystalline component gives the -stated specialization after extension to \(\Cw\). - -A distinct additional input gives algebraicity on the special fiber. -Combining the same transport with the Hodge theorem for CM abelian -varieties \cite[Theorem~1.1]{OpenAICMHodge2026}, -Corollary~\ref{cor:algebraic-specialization} represents each specialized -Hodge class by one rational algebraic cycle in all the stated -realizations. Its pairing with any complementary algebraic cycle is -therefore a rational intersection number. This consequence uses the CM -Hodge theorem; the proof of Theorem~\ref{thm:main} is independent of it. - -\subsection{The method and its antecedents} - -Three foundational results make this question accessible. Deligne proved -that every Hodge class on an abelian variety is absolute Hodge -\cite[Main Theorem~2.11]{Deligne1982}. Blasius established its compatibility -with \(p\)-adic comparison \cite[Theorems~0.3 and~5.3]{Blasius1994}; -Moonen presents the Blasius--Wintenberger comparison theorem, with -Ogus's simplification, in the perfect-residue-field form needed for -the local auxiliary lifts below \cite[Theorem~5.6.3]{Moonen1998}. Milne's theory of -Lefschetz classes supplies a common rational divisor algebra with perfect -complementary pairing, independent of the Weil cohomology -\cite[Section~5]{Milne1999}. - -The determinant classes used in the proof belong to the theory of Weil -classes developed by Weil, Deligne and Andr\'e; see -\cite[Section~4 and Endnotes~M.10--M.12]{Deligne1982}. -Andr\'e's refinement of the CM reduction expresses Hodge classes by -pullbacks from varieties of split Weil type \cite{Andre1992}; a detailed -account is \cite[Theorem~1]{Milne2020}. We prove directly the balanced -form of that reduction needed here in Lemma~\ref{lem:cm-reduction}. -Balanced signatures assert equal Hodge multiplicities at conjugate -embeddings; split Weil type also requires the associated Hermitian -form to have a half-dimensional isotropic subspace. -Andr\'e's later theory of motivated cycles establishes a broader -unconditional motivic setting for abelian Hodge classes -\cite[Theorem~0.6.2]{Andre1996}, without identifying them with algebraic -cycles. - -For an imaginary quadratic field, Markman has proved algebraicity of -Weil classes on abelian fourfolds of Weil type and on abelian sixfolds -of split Weil type \cite[Theorem~1.2]{Markman2026}. These are -characteristic-zero algebraicity results. The pairing argument here -applies in arbitrary dimension and begins only with the Hodge class -and its good reduction. - -There is also a direct antecedent for the lifting strategy. Milne asked -whether a specialized CM Hodge class and a divisor could be lifted -together, up to isogeny, to a CM variety, and explained the resulting -rationality implication \cite[Question~10.6 and Proposition~10.7]{MilneAIM2007}. -His current account gives a conditional criterion through weak lifting -of Lefschetz classes on CM Weil triples and describes the reduction -from general Hodge classes through CM varieties to Weil classes -\cite[Introduction, Question~2.17, and Theorem~2.18]{Milne2025}. -Our proof follows this general route, using two auxiliary varieties -with different purposes. The -first is CM; the second realizes a prescribed admissible filtration -over a local field and need not be CM. The necessary realization -machinery comes from Grothendieck--Serre--Tate deformation theory, -Tate's extension theorem, and the integral-model and local-Shimura -results of Kisin--Zhou and Pappas--Rapoport -\cite{Grothendieck1970,Tate1967,KisinZhou2025,PappasRapoport2024}. -For the prescribed filtration, we choose a framed local Shimura -point over its admissible period and use integral uniformization; -the required uniformization condition is supplied by -\cite[Corollary~6.3]{GleasonLimXu2026}. We state the precise inputs -where they enter and prove the connecting arguments for the -prescribed flag. - -The contribution along this route is to construct a filtration that -preserves the extra divisor and then obtain a rational divisor identity -on its lift. A separate normalization by rational intersection numbers -makes the tensor transports simultaneous in all realizations. -Kisin--Zhou's compatibility theorem concerns Weil--Deligne -representations across coefficient fields -\cite[Theorem~1.2]{KisinZhou2025}; the additional issue here is the -rational value of a Hodge--divisor intersection. - -The key step is a dichotomy for balanced Weil classes: either all the -complementary Lefschetz pairings vanish, or the specialized class is -itself a rational polynomial in divisors, simultaneously in every -realization. - -\subsection{Proof overview} - -The proof has four stages. The order below describes the deduction; -Sections~\ref{sec:transport}--\ref{sec:weil} establish its reusable -constructions before Section~\ref{sec:reduction} assembles them. - -\paragraph{1. Transport to a CM variety.} -Corollary~\ref{cor:cm-transport} constructs a CM -abelian variety \(C/\Qbar\), a quasi-isogeny -\(a_{\mathrm{CM}}:A_0^n\to C_0\), and a rational Hodge class -\(\kappa\) such that -\[ - \gamma_0=(a_{\mathrm{CM}}j)^*\kappa_0, -\] -where \(j\) includes the first factor. This is one rational algebraic homomorphism on the -special fibers and one equality in every realization. Kisin--Zhou's -special-point theorem supplies the CM lift -\cite[Theorem~2.2.7]{KisinZhou2025}. The local tensors initially determine -similitude scalars; a ratio of rational divisor intersections shows -that all those scalars are the same rational number. This normalization -is what transports the original rational class, rather than a separate -local multiple in each cohomology theory. - -\paragraph{2. Reduce CM classes to balanced Weil classes.} -Lemma~\ref{lem:cm-reduction} constructs rational-idempotent factors of -actual powers of \(C\). Their homology has rank \(2s\) over a CM field -\(E\), with signatures \((s,s)\) at every embedding, and their -polarizations induce conjugation on \(E\). Their Weil classes are the -rational forms whose components, after splitting \(E\), are top -alternating forms on individual \(E\)-eigenspaces. Pullbacks of these -Hodge classes span the given CM Hodge class over \(\Q\). Good reduction -is retained. Having one rational value for each complementary -Lefschetz pairing is preserved by rational pullback -(Lemma~\ref{lem:pairing-functorial}). It therefore remains to prove -that assertion for balanced Weil classes. - -\paragraph{3. Find a divisor and a compatible filtration.} -For a balanced Weil variety \(Z\), let \(\dagger\) denote Rosati on its -reduction and consider -\[ - \mathcal U=\{u\in\End^0(Z_0):u^\dagger=u,\quad - ue=\bar e u\text{ for all }e\in E\}. -\] -Proposition~\ref{prop:weil} proves a dichotomy. If \(\mathcal U\) has no -invertible element, every complementary Lefschetz pairing of a -specialized Weil class is zero. If it has an invertible element, the -specialized Weil class is itself a simultaneous rational Lefschetz -class: one rational polynomial in divisors gives its class in every -realization. For the second branch, such a \(u\) supplies a divisor -\(D^u\) whose alternating form is nondegenerate on each \(E\)-eigenspace. -Theorem~\ref{thm:filtration} constructs a weakly admissible balanced -filtration on the fixed crystalline module that preserves \(E\), \(u\), -and the polarization. Generic transversality has a classical role in -admissible-filtration constructions -\cite[Theorem~1 and Lemmas~1.1--1.2]{FontaineRapoport2005}. -Here bounded-degree avoidance makes all the constrained choices over -one finite extension, and a maximality argument reduces the -weak-admissibility test to subobjects stable under the endomorphisms. - -\paragraph{4. Lift the divisor and extract a rational identity.} -The second case of Proposition~\ref{prop:transport} realizes this flag -on a local auxiliary abelian variety \(B\), with a marked quasi-isogeny -\(a_{\mathrm{fl}}:Z_0^m\to B_0\). The prescribed flag makes the -first-summand projector, its algebraic \(E\)-action, and the transported -divisor filtered. These specified algebraic maps lift to \(B\). -After choosing an abstract complex realization of \(B\), polynomial -coefficient extraction expresses the relevant rational Betti Weil class -as a rational linear combination of divisor products -(Lemma~\ref{lem:weil-divisor-span}). The same rational identity specializes -in every realization. Pullback returns it to \(Z_0\), completing the -case of an invertible element in the dichotomy and, through the first two stages, the -main theorem. - -Figure~\ref{fig:two-transports} separates the two lifts and their -special-fiber markings. Section~\ref{sec:preliminaries} fixes comparisons, -traces, and rational pullback. Section~\ref{sec:transport} proves both -transport alternatives; Section~\ref{sec:filtration} constructs the -prescribed flag; Section~\ref{sec:weil} proves the balanced-Weil dichotomy; -and Section~\ref{sec:reduction} proves the CM reduction and finishes the -argument. - -\begin{figure}[tbp] -\centering -\input{figures/two-transports} -\caption{The two uses of transport. Each lower horizontal arrow is one -actual quasi-isogeny preserving the specified tensors in all -realizations; the vertical arrows denote good reduction. There is no -claimed map between the generic fibers. The upper construction starts -from the original variety. The lower construction is applied separately -to a balanced Weil variety \(Z\) arising in the CM reduction; its lift -\(B\) realizes the chosen filtration \(\mathcal F_Z^{\oplus m}\) under -\(a_{\mathrm{fl}}^*\). The CM and prescribed-filtration auxiliaries need -not coincide.} -\label{fig:two-transports} -\end{figure} +\section{Introduction} + +Suppose an abelian variety has good reduction at a prime. A rational +algebraic cycle on the variety specializes to a cycle on the reduction. +Its intersection with any complementary product of divisors has a rational +degree, and every cycle-class realization computes that same number. +Milne's rationality conjecture asks for this conclusion when the original +class is only known to be a rational Hodge class. We prove it for every +residue characteristic. + +The distinction matters because good reduction can acquire divisors that +do not lift to the original variety. Deligne's absolute-Hodge theorem gives +compatible realizations of the Hodge class, but does not by itself identify +its pairing with these new divisors as the degree of a rational cycle. +The conjecture isolates this rationality question from the stronger problem +of representing the Hodge class by an algebraic cycle. + +Fix \(\Qbar\subset\C\), a prime \(p\), and a \(p\)-adic place \(w\) +of \(\Qbar\). Write +\[ + \F=\overline{\mathbb F}_p,\qquad + \Kzero=\operatorname{Frac}W(\F),\qquad + \Cw=\widehat{\Qbar}_w. +\] +We regard \(\Kzero\) and an algebraic closure of it as subfields of +\(\Cw\). Good reduction means good reduction after descent to a number +field and finite extension at the chosen place. + +\begin{theorem}\label{thm:main} +Let \(A/\Qbar\) be an abelian variety of dimension \(d\) with good +reduction \(A_0/\F\) at \(w\). Let \(0\leq r\leq d\), and let +\[ + \gamma\in H^{2r}(A(\C),\Q(r)) +\] +be a rational Hodge class. For every list of Cartier divisors +\(D_1,\ldots,D_{d-r}\) on \(A_0\), there is a number \(q\in\Q\) +such that +\begin{align*} + \Tr_\ell\bigl(\gamma_{0,\ell} + c_{1,\ell}(D_1)\cdots c_{1,\ell}(D_{d-r})\bigr)&=q + &&\text{in }\Q_\ell\quad(\ell\ne p),\\ + \Tr_p\bigl(\gamma_{0,p} + c_{1,\mathrm{cris}}(D_1)\cdots + c_{1,\mathrm{cris}}(D_{d-r})\bigr)&=q + &&\text{in }\Cw. +\end{align*} +Here \(\gamma_{0,\ell}\) is the smooth proper specialization of the +\(\ell\)-adic realization, and \(\gamma_{0,p}\) is the +de Rham--crystalline specialization in +\[ + H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r) + \otimes_{\Kzero}\Cw. +\] +The traces use the top Tate twist and send the class of a zero-cycle to +its degree. If \(r=d\), the divisor product is \(1\). +\end{theorem} + +By linearity the same conclusion holds for every rational Lefschetz class +of complementary codimension, that is, every rational linear combination +of products of divisors. The number \(q\) is common to all realizations +for the fixed data; no comparison between different reduction places is +required. + +\subsection{Origins and consequences} + +Milne introduced a weak-rationality formulation while studying the +reduction of Shimura varieties and the Langlands--Rapoport conjecture +\cite[Conjecture~6.1(A)]{Milne2000}. It asks that a specialized CM +Hodge class whose components lie in the local spans of divisor products +in every realization arise from one rational Lefschetz class. +The pairing formulation appears in his +AIM notes and his study of rational Tate classes +\cite[Section~10]{MilneAIM2007}, \cite[Conjecture~4.1]{Milne2009}. +A previous positive case is a CM abelian variety with simple ordinary +reduction, together with all its powers: in that case every divisor +class on the reduction lifts \cite[Example~4.2]{Milne2009}. + +The broader purpose is to obtain rational cohomological correspondences +for abelian motives in characteristic~\(p\), compatible with reduction +from characteristic zero \cite[Sections~5--6]{Milne2000}; see also +\cite[Introduction and Section~2]{Milne2025}. For \(\ell\ne p\), +Tate classes are twisted \(\ell\)-adic classes fixed by a power of +Frobenius; there is a corresponding \(\Q_p\)-algebra of crystalline +Tate classes. A \emph{good theory of rational Tate classes} chooses +one graded \(\Q\)-algebra on each variety whose scalar extensions +give these Tate algebras. The theory is stable under pullback and +pushforward, contains divisor classes, and contains specializations of +CM Hodge classes \cite[Definitions~2.1 and~3.1]{Milne2009}. +These are cohomological correspondences; their definition does not +assert that they are algebraic. + +Milne proved that the rationality assertion for all CM abelian varieties +is equivalent to the existence of such a good theory on varieties over +\(\F\) whose connected components are products of abelian varieties +and projective spaces \cite[Theorem~4.5]{Milne2009}. +Our proof also expresses each specialized Hodge class as the pullback, +along a rational homomorphism of special fibers, of a specialized Hodge +class from a CM abelian variety. This extends the theory's +specialization property beyond the CM case. +In Corollary~\ref{cor:all-hodge-specialization}, every rational +Hodge class on a good-reduction abelian variety specializes to a unique +class in one fixed good theory. Its crystalline component gives the +stated specialization after extension to \(\Cw\). + +A distinct additional input gives algebraicity on the special fiber. +Combining the same transport with the Hodge theorem for CM abelian +varieties \cite[Theorem~1.1]{OpenAICMHodge2026}, +Corollary~\ref{cor:algebraic-specialization} represents each specialized +Hodge class by one rational algebraic cycle in all the stated +realizations. Its pairing with any complementary algebraic cycle is +therefore a rational intersection number. This consequence uses the CM +Hodge theorem; the proof of Theorem~\ref{thm:main} is independent of it. + +\subsection{The method and its antecedents} + +Three foundational results make this question accessible. Deligne proved +that every Hodge class on an abelian variety is absolute Hodge +\cite[Main Theorem~2.11]{Deligne1982}. Blasius established its compatibility +with \(p\)-adic comparison \cite[Theorems~0.3 and~5.3]{Blasius1994}; +Moonen presents the Blasius--Wintenberger comparison theorem, with +Ogus's simplification, in the perfect-residue-field form needed for +the local auxiliary lifts below \cite[Theorem~5.6.3]{Moonen1998}. Milne's theory of +Lefschetz classes supplies a common rational divisor algebra with perfect +complementary pairing, independent of the Weil cohomology +\cite[Section~5]{Milne1999}. + +The determinant classes used in the proof belong to the theory of Weil +classes developed by Weil, Deligne and Andr\'e; see +\cite[Section~4 and Endnotes~M.10--M.12]{Deligne1982}. +Andr\'e's refinement of the CM reduction expresses Hodge classes by +pullbacks from varieties of split Weil type \cite{Andre1992}; a detailed +account is \cite[Theorem~1]{Milne2020}. We prove directly the balanced +form of that reduction needed here in Lemma~\ref{lem:cm-reduction}. +Balanced signatures assert equal Hodge multiplicities at conjugate +embeddings; split Weil type also requires the associated Hermitian +form to have a half-dimensional isotropic subspace. +Andr\'e's later theory of motivated cycles establishes a broader +unconditional motivic setting for abelian Hodge classes +\cite[Theorem~0.6.2]{Andre1996}, without identifying them with algebraic +cycles. + +For an imaginary quadratic field, Markman has proved algebraicity of +Weil classes on abelian fourfolds of Weil type and on abelian sixfolds +of split Weil type \cite[Theorem~1.2]{Markman2026}. These are +characteristic-zero algebraicity results. The pairing argument here +applies in arbitrary dimension and begins only with the Hodge class +and its good reduction. + +There is also a direct antecedent for the lifting strategy. Milne asked +whether a specialized CM Hodge class and a divisor could be lifted +together, up to isogeny, to a CM variety, and explained the resulting +rationality implication \cite[Question~10.6 and Proposition~10.7]{MilneAIM2007}. +His current account gives a conditional criterion through weak lifting +of Lefschetz classes on CM Weil triples and describes the reduction +from general Hodge classes through CM varieties to Weil classes +\cite[Introduction, Question~2.17, and Theorem~2.18]{Milne2025}. +Our proof follows this general route, using two auxiliary varieties +with different purposes. The +first is CM; the second realizes a prescribed admissible filtration +over a local field and need not be CM. The necessary realization +machinery comes from Grothendieck--Serre--Tate deformation theory, +Tate's extension theorem, and the integral-model and local-Shimura +results of Kisin--Zhou and Pappas--Rapoport +\cite{Grothendieck1970,Tate1967,KisinZhou2025,PappasRapoport2024}. +For the prescribed filtration, we choose a framed local Shimura +point over its admissible period and use integral uniformization; +the required uniformization condition is supplied by +\cite[Corollary~6.3]{GleasonLimXu2026}. We state the precise inputs +where they enter and prove the connecting arguments for the +prescribed flag. + +The contribution along this route is to construct a filtration that +preserves the extra divisor and then obtain a rational divisor identity +on its lift. A separate normalization by rational intersection numbers +makes the tensor transports simultaneous in all realizations. +Kisin--Zhou's compatibility theorem concerns Weil--Deligne +representations across coefficient fields +\cite[Theorem~1.2]{KisinZhou2025}; the additional issue here is the +rational value of a Hodge--divisor intersection. + +The key step is a dichotomy for balanced Weil classes: either all the +complementary Lefschetz pairings vanish, or the specialized class is +itself a rational polynomial in divisors, simultaneously in every +realization. + +\subsection{Proof overview} + +The proof has four stages. The order below describes the deduction; +Sections~\ref{sec:transport}--\ref{sec:weil} establish its reusable +constructions before Section~\ref{sec:reduction} assembles them. + +\paragraph{1. Transport to a CM variety.} +Corollary~\ref{cor:cm-transport} constructs a CM +abelian variety \(C/\Qbar\), a quasi-isogeny +\(a_{\mathrm{CM}}:A_0^n\to C_0\), and a rational Hodge class +\(\kappa\) such that +\[ + \gamma_0=(a_{\mathrm{CM}}j)^*\kappa_0, +\] +where \(j\) includes the first factor. This is one rational algebraic homomorphism on the +special fibers and one equality in every realization. Kisin--Zhou's +special-point theorem supplies the CM lift +\cite[Theorem~2.2.7]{KisinZhou2025}. The local tensors initially determine +similitude scalars; a ratio of rational divisor intersections shows +that all those scalars are the same rational number. This normalization +is what transports the original rational class, rather than a separate +local multiple in each cohomology theory. + +\paragraph{2. Reduce CM classes to balanced Weil classes.} +Lemma~\ref{lem:cm-reduction} constructs rational-idempotent factors of +actual powers of \(C\). Their homology has rank \(2s\) over a CM field +\(E\), with signatures \((s,s)\) at every embedding, and their +polarizations induce conjugation on \(E\). Their Weil classes are the +rational forms whose components, after splitting \(E\), are top +alternating forms on individual \(E\)-eigenspaces. Pullbacks of these +Hodge classes span the given CM Hodge class over \(\Q\). Good reduction +is retained. Having one rational value for each complementary +Lefschetz pairing is preserved by rational pullback +(Lemma~\ref{lem:pairing-functorial}). It therefore remains to prove +that assertion for balanced Weil classes. + +\paragraph{3. Find a divisor and a compatible filtration.} +For a balanced Weil variety \(Z\), let \(\dagger\) denote Rosati on its +reduction and consider +\[ + \mathcal U=\{u\in\End^0(Z_0):u^\dagger=u,\quad + ue=\bar e u\text{ for all }e\in E\}. +\] +Proposition~\ref{prop:weil} proves a dichotomy. If \(\mathcal U\) has no +invertible element, every complementary Lefschetz pairing of a +specialized Weil class is zero. If it has an invertible element, the +specialized Weil class is itself a simultaneous rational Lefschetz +class: one rational polynomial in divisors gives its class in every +realization. For the second branch, such a \(u\) supplies a divisor +\(D^u\) whose alternating form is nondegenerate on each \(E\)-eigenspace. +Theorem~\ref{thm:filtration} constructs a weakly admissible balanced +filtration on the fixed crystalline module that preserves \(E\), \(u\), +and the polarization. Generic transversality has a classical role in +admissible-filtration constructions +\cite[Theorem~1 and Lemmas~1.1--1.2]{FontaineRapoport2005}. +Here bounded-degree avoidance makes all the constrained choices over +one finite extension, and a maximality argument reduces the +weak-admissibility test to subobjects stable under the endomorphisms. + +\paragraph{4. Lift the divisor and extract a rational identity.} +The second case of Proposition~\ref{prop:transport} realizes this flag +on a local auxiliary abelian variety \(B\), with a marked quasi-isogeny +\(a_{\mathrm{fl}}:Z_0^m\to B_0\). The prescribed flag makes the +first-summand projector, its algebraic \(E\)-action, and the transported +divisor filtered. These specified algebraic maps lift to \(B\). +After choosing an abstract complex realization of \(B\), polynomial +coefficient extraction expresses the relevant rational Betti Weil class +as a rational linear combination of divisor products +(Lemma~\ref{lem:weil-divisor-span}). The same rational identity specializes +in every realization. Pullback returns it to \(Z_0\), completing the +case of an invertible element in the dichotomy and, through the first two stages, the +main theorem. + +Figure~\ref{fig:two-transports} separates the two lifts and their +special-fiber markings. Section~\ref{sec:preliminaries} fixes comparisons, +traces, and rational pullback. Section~\ref{sec:transport} proves both +transport alternatives; Section~\ref{sec:filtration} constructs the +prescribed flag; Section~\ref{sec:weil} proves the balanced-Weil dichotomy; +and Section~\ref{sec:reduction} proves the CM reduction and finishes the +argument. + +\begin{figure}[tbp] +\centering +\input{figures/two-transports} +\caption{The two uses of transport. Each lower horizontal arrow is one +actual quasi-isogeny preserving the specified tensors in all +realizations; the vertical arrows denote good reduction. There is no +claimed map between the generic fibers. The upper construction starts +from the original variety. The lower construction is applied separately +to a balanced Weil variety \(Z\) arising in the CM reduction; its lift +\(B\) realizes the chosen filtration \(\mathcal F_Z^{\oplus m}\) under +\(a_{\mathrm{fl}}^*\). The CM and prescribed-filtration auxiliaries need +not coincide.} +\label{fig:two-transports} +\end{figure} diff --git a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/01_preliminaries.tex b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/01_preliminaries.tex index d3af8d2d840930265d2a1075fbf183832209248b..9d3acf43a4e761d9a1df06f80145976b85d834db 100644 --- a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/01_preliminaries.tex +++ b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/01_preliminaries.tex @@ -1,227 +1,227 @@ -\section{Comparisons and rational Lefschetz classes} -\label{sec:preliminaries} - -We first put the realizations in Theorem~\ref{thm:main} in a common -notation and formulate its conclusion as a property of cohomology classes. -The essential functorial fact is that this property is preserved by rational -pullback. We also record the criterion that will lift an additional divisor -once a suitable Hodge filtration has been realized. - -We work in the category of abelian varieties up to isogeny. Thus -\(\Hom^0(X,Y)=\Hom(X,Y)\otimes_\Z\Q\), and divisors may have rational -coefficients. Pullback by a rational homomorphism is defined on \(H^1\) -and extended to its exterior algebra. If \(Nf\) is an actual homomorphism, -then on \(H^{2r}\) -\begin{equation}\label{eq:rational-pullback} - f^*=N^{-2r}(Nf)^*. -\end{equation} -This convention is the same in all realizations. - -For an abelian variety \(X/\F\), set -\[ - k_\ell=\Q_\ell\quad(\ell\ne p),\qquad k_p=\Cw, -\] -and write \(H_v^{2r}(X)(r)\) for its \(\ell\)-adic cohomology if -\(v=\ell\ne p\), and for crystalline cohomology extended to \(\Cw\) -if \(v=p\). Cup products will usually be written without a cup symbol. -All trace maps and divisor classes have their cycle-class normalizations. -The Betti Tate line \(\Q(r)\) supplies the twists in the other -realizations through comparison. Thus \(\Tr_v\) is the trace on top -cohomology with its top Tate twist and sends the class of a zero-cycle to -its degree. For a good-reduction lift, \(\gamma_{0,\ell}\) denotes -smooth proper specialization of its absolute-Hodge realization, and -\(\gamma_{0,p}\) denotes de Rham--crystalline specialization extended -from \(\Kzero\) to \(\Cw\). We keep first crystalline cohomology -over \(\Kzero\) when constructing filtrations, and specify each -extension of the filtration field separately. - -We use the exterior-algebra description of abelian cohomology and -faithfulness of rational homomorphisms on \(H^1\); see -\cite[Theorems~12.5 and~15.1]{MilneAV} for the prime-to-characteristic -realizations. The uniform Weil-cohomology formalism, including -crystalline cohomology, exterior powers, cycle-normalized traces and -the projection formula, is set out in -\cite[Section~1 and Appendix, Proposition~(A.2)]{Milne1999}. Smooth -proper specialization away from \(p\) is -\cite[Theorem~20.4]{MilneLEC}. - -\subsection{Comparison of Hodge tensors} - -\begin{lemma}\label{lem:comparison} -For an abelian variety with good reduction, the realizations of an -absolute Hodge class are compatible with the de Rham and crystalline -comparison isomorphisms. In particular, the \(p\)-adic crystalline tensor -obtained from its \(p\)-adic \'etale realization agrees, after extension of -scalars, with the de Rham--crystalline realization in -Theorem~\ref{thm:main}. These compatibilities respect tensor operations, -duals, Tate twists, divisor classes, and rational homomorphisms. - -The same assertion applies to auxiliary abelian varieties with good -reduction over a finite extension of \(\Kzero\), using a complex -embedding of an algebraic closure of their field of definition. -\end{lemma} - -\begin{proof} -For abelian varieties over \(\Qbar\), Deligne's absolute-Hodge theorem -\cite[Main Theorem~2.11]{Deligne1982} and Blasius's comparison theorem -\cite[Theorems~0.3 and~5.3]{Blasius1994} give the assertion. The -compatibility of the crystalline comparison with the de Rham comparison -is part of \cite[\S5.1]{Blasius1994}. The formulation for a complete -mixed-characteristic discrete valuation ring with perfect residue field -is given in \cite[Theorem~5.6.3]{Moonen1998}; the paragraph following -that theorem explains removal of the number-field hypothesis. A finite -extension of \(\Kzero\) is within that scope. The stated functorialities -are those of the comparison isomorphisms. -\end{proof} - -An abstract complex embedding used here need not be continuous. To see -that it can be chosen compatibly with \(\Qbar\subset\C\), take a -transcendence basis of \(\overline{\Kzero}/\Qbar\), embed it in a -transcendence basis of \(\C/\Qbar\), and extend algebraically. The -cardinalities permit this construction. No descent of a general -auxiliary lift to \(\Qbar\) is inferred. - -\subsection{The common rational Lefschetz space} - -Let \(\Lef^r(X)\) be the space of rational polynomial expressions in -divisors of codimension \(r\), modulo numerical equivalence. We use the -following established properties: -\begin{enumerate}[label=(\roman*)] - \item its cycle map is injective, and - \(\Lef^r(X)\otimes_\Q k_v\) is the divisor-generated subspace of - \(H_v^{2r}(X)(r)\); - \item intersection and degree give a perfect pairing - \[ - \Lef^r(X)\times\Lef^{\dim X-r}(X)\longrightarrow\Q; - \] - \item pushforward by a homomorphism of abelian varieties preserves the - divisor-generated subspaces in each realization; - \item hard Lefschetz and its inverse restrict to these subspaces. -\end{enumerate} -These are \cite[Proposition~5.2, Corollaries~5.3 and~5.5, -Theorem~5.9]{Milne1999}, with the author's correction to the proof of -Theorem~5.9. Crystalline cohomology is included. Extension of its -coefficient field to \(\Cw\) preserves these statements. We call the -tuple of realizations of a fixed element of \(\Lef^r(X)\) a -\emph{simultaneous rational Lefschetz class}. - -\begin{definition} -A tuple \(\alpha=(\alpha_v)_v\), with -\(\alpha_v\in H_v^{2r}(X)(r)\), has the \emph{Lefschetz pairing -property} if, for every \(\delta\in\Lef^{\dim X-r}(X)\), the numbers -\(\Tr_v(\alpha_v\delta_v)\) are the images of one number in \(\Q\). -\end{definition} - -Simultaneous rational algebraic classes have this property, by the -intersection formula. The property is closed under rational linear -combinations. We shall need the following additional closure property. - -\begin{lemma}\label{lem:pairing-functorial} -If \(\alpha\) has the Lefschetz pairing property on \(Y\) and -\(f\in\Hom^0(X,Y)\), then \(f^*\alpha\) has that property on \(X\). -\end{lemma} - -\begin{proof} -If \(r>\dim Y\), the input class is zero; if \(r>\dim X\), -its pullback is zero. We may therefore suppose that \(f\) is an actual -homomorphism and \(r\leq\min(\dim X,\dim Y)\). -For \(\delta\in\Lef^{\dim X-r}(X)\), the class \(f_*\delta_v\) -lies in the Lefschetz subspace of codimension \(\dim Y-r\) in every -realization. Choose rational bases \(a_i\) of -\(\Lef^{\dim Y-r}(Y)\) and \(b_j\) of \(\Lef^r(Y)\). The matrix -\(M_{ij}=\deg(a_i b_j)\) is an invertible rational matrix. If -\(f_*\delta_v=\sum_i c_{i,v}(a_i)_v\), the projection formula gives -\[ - \sum_i c_{i,v}M_{ij}=\deg(\delta f^*b_j). -\] -The right side is a rational number independent of \(v\). Solving this -one rational system shows that \(c_{i,v}=c_i\in\Q\) for every \(v\). -Thus \(f_*\delta\) is simultaneous rational Lefschetz. Applying the -projection formula once more proves the assertion. For rational \(f\), -use \eqref{eq:rational-pullback}. -\end{proof} - -\subsection{Lifting a prescribed rational homomorphism} - -The second transport construction will produce an abelian lift with a -chosen Hodge filtration. The following criterion then lifts the algebraic -projector, field action, and divisor already specified on its special fiber. -Its input is a rational algebraic homomorphism, rather than an arbitrary -map between cohomology spaces. - -In the next lemma \(R\) is a complete mixed-characteristic discrete -valuation ring finite over \(W(\F)\), with fraction field \(K\). -For an abelian scheme \(\mathcal X/R\), the Hodge filtration is regarded -as a filtration of -\(H^1_{\mathrm{cris}}(X_0/W(\F))[1/p]\otimes_{\Kzero}K\). - -\begin{lemma}\label{lem:lift-hom} -Let \(\mathcal X,\mathcal Y\) be abelian schemes over \(R\), with -special fibers \(X_0,Y_0\), and let -\(f_0\in\Hom^0(X_0,Y_0)\). If its contravariant crystalline map -preserves the Hodge filtrations, then \(f_0\) lifts uniquely to a -rational homomorphism \(\mathcal X\to\mathcal Y\). - -In particular, a symmetric rational homomorphism \(X_0\to X_0^\vee\) -with this property lifts to a rational divisor class on the generic -fiber. Algebraic identities among finitely many such lifted maps remain -valid. -\end{lemma} - -\begin{proof} -The crystalline map already commutes with Frobenius. Full faithfulness -for rational filtered Dieudonn\'e modules gives a rational morphism of -the associated \(p\)-divisible groups; see -\cite{Grothendieck1970} and the perfect-residue-field formulation in -\cite[\S4.2(v)]{IllusiePisa}. Equivalently, crystalline comparison first -gives the rational map on Tate modules. Multiply by an integer \(N\) -that clears both its Tate-lattice denominator and the denominator of -\(f_0\). Tate's extension theorem -\cite[Theorem~4]{Tate1967} extends the resulting integral generic-fiber -map of \(p\)-divisible groups over \(R\). Faithfulness of the -special-fiber Dieudonn\'e functor identifies its reduction with the -map induced by \(Nf_0\). Here equality after inverting \(p\) is -already equality of integral maps, because the integral Dieudonn\'e -modules are torsion free. The integer \(N\) is fixed for the remainder -of the construction. - -The specified-map form of Serre--Tate deformation theory now gives a -lift over each \(R/\mathfrak m_R^n\) -\cite[p.~432]{Grothendieck1970}. One can also deduce it from the -fixed-object formulation \cite[Theorem~2.3.3]{IllusiePisa} as follows. -On \(X_0\times Y_0\) form the shear -\(s_0(x,y)=(x,y+Nf_0(x))\). The lifted \(p\)-divisible-group map gives -the same shear on the product \(p\)-divisible group. View the existing -product deformation with two special-fiber markings differing by -\(s_0\). Full faithfulness lifts the shear to an algebraic isomorphism -between these marked deformations; its second component restricted to -\(X\) lifts \(Nf_0\). Uniqueness makes these maps compatible at every -finite order. - -Formal existence algebraizes the resulting formal map -\cite[Tag~0A42]{StacksProject}: its source is proper over the complete -noetherian ring \(R\), and its target is separated and of finite type. -Dividing by the fixed \(N\) gives the desired rational lift. Rigidity -gives uniqueness and injectivity of specialization on rational -homomorphisms \cite[Proposition~20.1 and Corollary~20.2]{MilneAV}. - -If \(f_0\) is symmetric, injectivity shows that its lift is symmetric. -The usual identification -\[ - \NS(X_K)\otimes\Q - \simeq\Hom(X_K,X_K^\vee)^{\mathrm{sym}}\otimes\Q -\] -gives the rational divisor class, after a finite extension if -necessary \cite[Proposition~17.2]{MilneAV}. Finally an identity between the maps specializes to the -given identity, so injectivity proves it upstairs as well. -\end{proof} - -Only prescribed rational algebraic maps on the special fiber are being -lifted in Lemma~\ref{lem:lift-hom}. An arbitrary \(\Q_p\)-linear map of -\(p\)-divisible groups is not asserted to be a rational algebraic map. -We also use the extension of generic-fiber homomorphisms between abelian -schemes over a discrete valuation ring: their N\'eron mapping property -extends such a homomorphism uniquely over the base -\cite[Definition~5.1 and Example~5.3]{Conrad2003}. Clearing one -rational denominator therefore makes specialization of these maps -well defined. +\section{Comparisons and rational Lefschetz classes} +\label{sec:preliminaries} + +We first put the realizations in Theorem~\ref{thm:main} in a common +notation and formulate its conclusion as a property of cohomology classes. +The essential functorial fact is that this property is preserved by rational +pullback. We also record the criterion that will lift an additional divisor +once a suitable Hodge filtration has been realized. + +We work in the category of abelian varieties up to isogeny. Thus +\(\Hom^0(X,Y)=\Hom(X,Y)\otimes_\Z\Q\), and divisors may have rational +coefficients. Pullback by a rational homomorphism is defined on \(H^1\) +and extended to its exterior algebra. If \(Nf\) is an actual homomorphism, +then on \(H^{2r}\) +\begin{equation}\label{eq:rational-pullback} + f^*=N^{-2r}(Nf)^*. +\end{equation} +This convention is the same in all realizations. + +For an abelian variety \(X/\F\), set +\[ + k_\ell=\Q_\ell\quad(\ell\ne p),\qquad k_p=\Cw, +\] +and write \(H_v^{2r}(X)(r)\) for its \(\ell\)-adic cohomology if +\(v=\ell\ne p\), and for crystalline cohomology extended to \(\Cw\) +if \(v=p\). Cup products will usually be written without a cup symbol. +All trace maps and divisor classes have their cycle-class normalizations. +The Betti Tate line \(\Q(r)\) supplies the twists in the other +realizations through comparison. Thus \(\Tr_v\) is the trace on top +cohomology with its top Tate twist and sends the class of a zero-cycle to +its degree. For a good-reduction lift, \(\gamma_{0,\ell}\) denotes +smooth proper specialization of its absolute-Hodge realization, and +\(\gamma_{0,p}\) denotes de Rham--crystalline specialization extended +from \(\Kzero\) to \(\Cw\). We keep first crystalline cohomology +over \(\Kzero\) when constructing filtrations, and specify each +extension of the filtration field separately. + +We use the exterior-algebra description of abelian cohomology and +faithfulness of rational homomorphisms on \(H^1\); see +\cite[Theorems~12.5 and~15.1]{MilneAV} for the prime-to-characteristic +realizations. The uniform Weil-cohomology formalism, including +crystalline cohomology, exterior powers, cycle-normalized traces and +the projection formula, is set out in +\cite[Section~1 and Appendix, Proposition~(A.2)]{Milne1999}. Smooth +proper specialization away from \(p\) is +\cite[Theorem~20.4]{MilneLEC}. + +\subsection{Comparison of Hodge tensors} + +\begin{lemma}\label{lem:comparison} +For an abelian variety with good reduction, the realizations of an +absolute Hodge class are compatible with the de Rham and crystalline +comparison isomorphisms. In particular, the \(p\)-adic crystalline tensor +obtained from its \(p\)-adic \'etale realization agrees, after extension of +scalars, with the de Rham--crystalline realization in +Theorem~\ref{thm:main}. These compatibilities respect tensor operations, +duals, Tate twists, divisor classes, and rational homomorphisms. + +The same assertion applies to auxiliary abelian varieties with good +reduction over a finite extension of \(\Kzero\), using a complex +embedding of an algebraic closure of their field of definition. +\end{lemma} + +\begin{proof} +For abelian varieties over \(\Qbar\), Deligne's absolute-Hodge theorem +\cite[Main Theorem~2.11]{Deligne1982} and Blasius's comparison theorem +\cite[Theorems~0.3 and~5.3]{Blasius1994} give the assertion. The +compatibility of the crystalline comparison with the de Rham comparison +is part of \cite[\S5.1]{Blasius1994}. The formulation for a complete +mixed-characteristic discrete valuation ring with perfect residue field +is given in \cite[Theorem~5.6.3]{Moonen1998}; the paragraph following +that theorem explains removal of the number-field hypothesis. A finite +extension of \(\Kzero\) is within that scope. The stated functorialities +are those of the comparison isomorphisms. +\end{proof} + +An abstract complex embedding used here need not be continuous. To see +that it can be chosen compatibly with \(\Qbar\subset\C\), take a +transcendence basis of \(\overline{\Kzero}/\Qbar\), embed it in a +transcendence basis of \(\C/\Qbar\), and extend algebraically. The +cardinalities permit this construction. No descent of a general +auxiliary lift to \(\Qbar\) is inferred. + +\subsection{The common rational Lefschetz space} + +Let \(\Lef^r(X)\) be the space of rational polynomial expressions in +divisors of codimension \(r\), modulo numerical equivalence. We use the +following established properties: +\begin{enumerate}[label=(\roman*)] + \item its cycle map is injective, and + \(\Lef^r(X)\otimes_\Q k_v\) is the divisor-generated subspace of + \(H_v^{2r}(X)(r)\); + \item intersection and degree give a perfect pairing + \[ + \Lef^r(X)\times\Lef^{\dim X-r}(X)\longrightarrow\Q; + \] + \item pushforward by a homomorphism of abelian varieties preserves the + divisor-generated subspaces in each realization; + \item hard Lefschetz and its inverse restrict to these subspaces. +\end{enumerate} +These are \cite[Proposition~5.2, Corollaries~5.3 and~5.5, +Theorem~5.9]{Milne1999}, with the author's correction to the proof of +Theorem~5.9. Crystalline cohomology is included. Extension of its +coefficient field to \(\Cw\) preserves these statements. We call the +tuple of realizations of a fixed element of \(\Lef^r(X)\) a +\emph{simultaneous rational Lefschetz class}. + +\begin{definition} +A tuple \(\alpha=(\alpha_v)_v\), with +\(\alpha_v\in H_v^{2r}(X)(r)\), has the \emph{Lefschetz pairing +property} if, for every \(\delta\in\Lef^{\dim X-r}(X)\), the numbers +\(\Tr_v(\alpha_v\delta_v)\) are the images of one number in \(\Q\). +\end{definition} + +Simultaneous rational algebraic classes have this property, by the +intersection formula. The property is closed under rational linear +combinations. We shall need the following additional closure property. + +\begin{lemma}\label{lem:pairing-functorial} +If \(\alpha\) has the Lefschetz pairing property on \(Y\) and +\(f\in\Hom^0(X,Y)\), then \(f^*\alpha\) has that property on \(X\). +\end{lemma} + +\begin{proof} +If \(r>\dim Y\), the input class is zero; if \(r>\dim X\), +its pullback is zero. We may therefore suppose that \(f\) is an actual +homomorphism and \(r\leq\min(\dim X,\dim Y)\). +For \(\delta\in\Lef^{\dim X-r}(X)\), the class \(f_*\delta_v\) +lies in the Lefschetz subspace of codimension \(\dim Y-r\) in every +realization. Choose rational bases \(a_i\) of +\(\Lef^{\dim Y-r}(Y)\) and \(b_j\) of \(\Lef^r(Y)\). The matrix +\(M_{ij}=\deg(a_i b_j)\) is an invertible rational matrix. If +\(f_*\delta_v=\sum_i c_{i,v}(a_i)_v\), the projection formula gives +\[ + \sum_i c_{i,v}M_{ij}=\deg(\delta f^*b_j). +\] +The right side is a rational number independent of \(v\). Solving this +one rational system shows that \(c_{i,v}=c_i\in\Q\) for every \(v\). +Thus \(f_*\delta\) is simultaneous rational Lefschetz. Applying the +projection formula once more proves the assertion. For rational \(f\), +use \eqref{eq:rational-pullback}. +\end{proof} + +\subsection{Lifting a prescribed rational homomorphism} + +The second transport construction will produce an abelian lift with a +chosen Hodge filtration. The following criterion then lifts the algebraic +projector, field action, and divisor already specified on its special fiber. +Its input is a rational algebraic homomorphism, rather than an arbitrary +map between cohomology spaces. + +In the next lemma \(R\) is a complete mixed-characteristic discrete +valuation ring finite over \(W(\F)\), with fraction field \(K\). +For an abelian scheme \(\mathcal X/R\), the Hodge filtration is regarded +as a filtration of +\(H^1_{\mathrm{cris}}(X_0/W(\F))[1/p]\otimes_{\Kzero}K\). + +\begin{lemma}\label{lem:lift-hom} +Let \(\mathcal X,\mathcal Y\) be abelian schemes over \(R\), with +special fibers \(X_0,Y_0\), and let +\(f_0\in\Hom^0(X_0,Y_0)\). If its contravariant crystalline map +preserves the Hodge filtrations, then \(f_0\) lifts uniquely to a +rational homomorphism \(\mathcal X\to\mathcal Y\). + +In particular, a symmetric rational homomorphism \(X_0\to X_0^\vee\) +with this property lifts to a rational divisor class on the generic +fiber. Algebraic identities among finitely many such lifted maps remain +valid. +\end{lemma} + +\begin{proof} +The crystalline map already commutes with Frobenius. Full faithfulness +for rational filtered Dieudonn\'e modules gives a rational morphism of +the associated \(p\)-divisible groups; see +\cite{Grothendieck1970} and the perfect-residue-field formulation in +\cite[\S4.2(v)]{IllusiePisa}. Equivalently, crystalline comparison first +gives the rational map on Tate modules. Multiply by an integer \(N\) +that clears both its Tate-lattice denominator and the denominator of +\(f_0\). Tate's extension theorem +\cite[Theorem~4]{Tate1967} extends the resulting integral generic-fiber +map of \(p\)-divisible groups over \(R\). Faithfulness of the +special-fiber Dieudonn\'e functor identifies its reduction with the +map induced by \(Nf_0\). Here equality after inverting \(p\) is +already equality of integral maps, because the integral Dieudonn\'e +modules are torsion free. The integer \(N\) is fixed for the remainder +of the construction. + +The specified-map form of Serre--Tate deformation theory now gives a +lift over each \(R/\mathfrak m_R^n\) +\cite[p.~432]{Grothendieck1970}. One can also deduce it from the +fixed-object formulation \cite[Theorem~2.3.3]{IllusiePisa} as follows. +On \(X_0\times Y_0\) form the shear +\(s_0(x,y)=(x,y+Nf_0(x))\). The lifted \(p\)-divisible-group map gives +the same shear on the product \(p\)-divisible group. View the existing +product deformation with two special-fiber markings differing by +\(s_0\). Full faithfulness lifts the shear to an algebraic isomorphism +between these marked deformations; its second component restricted to +\(X\) lifts \(Nf_0\). Uniqueness makes these maps compatible at every +finite order. + +Formal existence algebraizes the resulting formal map +\cite[Tag~0A42]{StacksProject}: its source is proper over the complete +noetherian ring \(R\), and its target is separated and of finite type. +Dividing by the fixed \(N\) gives the desired rational lift. Rigidity +gives uniqueness and injectivity of specialization on rational +homomorphisms \cite[Proposition~20.1 and Corollary~20.2]{MilneAV}. + +If \(f_0\) is symmetric, injectivity shows that its lift is symmetric. +The usual identification +\[ + \NS(X_K)\otimes\Q + \simeq\Hom(X_K,X_K^\vee)^{\mathrm{sym}}\otimes\Q +\] +gives the rational divisor class, after a finite extension if +necessary \cite[Proposition~17.2]{MilneAV}. Finally an identity between the maps specializes to the +given identity, so injectivity proves it upstairs as well. +\end{proof} + +Only prescribed rational algebraic maps on the special fiber are being +lifted in Lemma~\ref{lem:lift-hom}. An arbitrary \(\Q_p\)-linear map of +\(p\)-divisible groups is not asserted to be a rational algebraic map. +We also use the extension of generic-fiber homomorphisms between abelian +schemes over a discrete valuation ring: their N\'eron mapping property +extends such a homomorphism uniquely over the base +\cite[Definition~5.1 and Example~5.3]{Conrad2003}. Clearing one +rational denominator therefore makes specialization of these maps +well defined. diff --git a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/02_transport.tex b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/02_transport.tex index 1cb7f952cd111b077becc209aa24b3cd551cca42..5a45d2a38794a314bf4584432704a12e37ffb9c4 100644 --- a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/02_transport.tex +++ b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/02_transport.tex @@ -1,814 +1,814 @@ -\section{Transport through Hodge-type integral models} -\label{sec:transport} - -After taking a power of the original abelian variety, we transport a -rational Hodge class to an auxiliary variety by one quasi-isogeny on -special fibers, with the same equality in every -realization. The auxiliary variety can be chosen either of CM type or -with a prescribed admissible Hodge filtration. These choices serve -different parts of the proof, but share the tensor construction and the -rational normalization in Proposition~\ref{prop:transport} below. - -Throughout this section, a polarization on a rational homology space is -written as a rational alternating form, using the fixed Betti -trivialization of the Tate line. If its similitude character is $c$, -an alternating form $\xi:V^{2r}\to\Q$ of multiplier $c^r$ means that -\begin{equation} - \xi(gv_1,\ldots,gv_{2r})=c(g)^r\xi(v_1,\ldots,v_{2r}). - \label{eq:transport-multiplier} -\end{equation} -Such a form is regarded as a tensor with its codimension-$r$ Tate -twist. All homomorphisms and polarizations may be rational: an -isomorphism in the isogeny category induces the usual isomorphisms on -cohomology, including its tensor constructions. - -We say that filtered isocrystal data have \emph{finite local-field -descent} if they are obtained by scalar extension from an isocrystal -with its tensors over the maximal unramified subfield \(H_0\) of a -finite extension \(H/\Q_p\), together with a filtration over \(H\). -The scalar extension is to \(\Kzero H\). This condition will let us -apply weak admissibility over a finite local field. -For a group in a faithful representation, a tensor-preserving -identification means one respecting a finite family of rational tensors -whose stabilizer is that group, together with their comparison realizations. - -\begin{proposition}[Transport] -\label{prop:transport} -Let $A/\Qbar$ be an abelian variety of positive dimension with good -reduction at $w$, and put $V=H_1(A(\C),\Q)$. Suppose -\[ - G\subset\GSp(V,\psi) -\] -is connected reductive, its Hodge homomorphism $h$ gives the -Hodge structure of $A$, and its conjugacy class $X$ makes this -inclusion a Hodge embedding. Let $\gamma$ be a rational Hodge -class of codimension $r$, represented on $V$ by an alternating -form satisfying \eqref{eq:transport-multiplier}. - -Then there exist a positive integer $n$, an auxiliary -good-reduction abelian variety $B$, a quasi-isogeny -$a:A_0^n\to B_0$, and a rational Hodge class $\kappa$ of -codimension $r$ on $B$ such that, for the first-factor inclusion -$j:A_0\to A_0^n$, -\begin{equation} - (aj)^*\kappa_0=\gamma_0. - \label{eq:transport-class} -\end{equation} -The equality holds simultaneously in every $\ell$-adic -realization with $\ell\ne p$ and in the crystalline realization -after extension to $\Cw$. Either of the following two further -specifications can be imposed. -\begin{enumerate} -\item The variety $B$ is of CM type and is defined over $\Qbar$, -with its indicated good reduction at $w$. -\item Put $D_A=H^1_{\mathrm{cris}}(A_0/W(\F))[1/p]$. Suppose -$\mathcal F_A$ is a filtration in degrees $0,1$ on $D_A$, over a -finite extension of $\Kzero$, which is weakly admissible and has -finite local-field descent together with the specialized tensors -specifying the $G$-structure. Suppose -also that, after a geometric tensor-preserving identification, it -lies in the $G$-orbit of the original Hodge filtration. Then $B$ -can be defined over a finite extension $K/\Kzero$ inside $\Cw$, -and $a$ can be chosen so that -\begin{equation} - a^*\Fil^1H^1_{\mathrm{dR}}(B/K) - =(\mathcal F_A^1)^{\oplus n}. - \label{eq:transport-prescribed-filtration} -\end{equation} -Here the equality is made after a common finite extension and -crystalline--de Rham identification. To discuss Hodge classes on -this $B$, choose a complex embedding of an algebraic closure of -$K$ extending the fixed embedding of $\Qbar$. -\end{enumerate} -In the second case the realizations of $\kappa$ are supplied by -the canonical tensor sections of the Hodge-type model and the -polarization. In the first case these agree with the usual -realizations and specialization of its absolute Hodge class. -\end{proposition} - -We prepare two ingredients for the proof. First, we enlarge the given -Hodge datum so that integral-model theorems apply at every prime while -its rational representation remains a sum of copies of \(V\). -Second, we realize a prescribed admissible filtration by choosing a -framed local Shimura point above its crystalline period and applying -integral uniformization. The resulting abelian lift retains that -point's integral tensors and special-fiber marking. After these -preparations, the proof constructs the transported tensor and normalizes -it using rational divisor intersections. - -\subsection{An auxiliary strongly admissible datum} - -We recall the hypotheses on integral models that we will use. -For a reductive group \(H_p/\Q_p\), a full Bruhat--Tits stabilizer -is the smooth group scheme fixing a point of its extended building. -Its identity component is the associated parahoric group scheme. -Now let \((H,X_H)\) be a Shimura datum over \(\Q\). In the terminology -of \cite[Definition~2.1.6]{KisinZhou2025}, the triple -\((H,X_H,\mathcal H)\) is \emph{strongly admissible} if \((H,X_H)\) -is of Hodge type, \(\mathcal H\) is a full stabilizer of \(H_{\Q_p}\) -with connected special fiber, the centralizer of a maximal -\(\Kzero\)-split torus of \(H_{\Kzero}\) is \(R\)-smooth, and the -relative root system over \(\Kzero\) is reduced when \(p=2\). Here $R$-smoothness is the condition on tori used in -\cite[\S2.4]{KisinZhouEarlier}. We will use it only for products and -extensions of restrictions of scalars of split tori, for which it -holds by \cite[Proposition~2.4.6]{KisinZhouEarlier}. - -\begin{lemma}[Auxiliary Hodge datum] -\label{lem:auxiliary-datum} -Let -\[ - (G,X)\lhook\joinrel\longrightarrow - (\GSp(V,\psi),X_{\mathrm{Siegel}}) -\] -be a Hodge embedding, with $G$ connected and similitude character $c$. -For every prime $p$ there exist a totally real field $P$, a connected -reductive group $J$, and a Hodge embedding -\[ - (J,X_J)\lhook\joinrel\longrightarrow - (\GSp(W,\psi_W),X'_{\mathrm{Siegel}}) -\] -with the following properties. -\begin{enumerate} -\item There are inclusions of Shimura data induced by -\[ - G\longrightarrow J\longrightarrow\Res_{P/\Q}G_P, -\] -where the first map is diagonal and $J$ is the subgroup on which the -similitude multipliers in the restriction of scalars agree. -\item The group $J_{\Q_p}$ is quasi-split and has a full connected -stabilizer $\mathcal J$ making $(J,X_J,\mathcal J)$ strongly admissible. -\item As a rational $J$-representation, -\begin{equation} - W\simeq (V\otimes_{\Q}P)^{\oplus m} - \label{eq:transport-copies} -\end{equation} -for some positive integer $m$. The form $\psi_W$ has multiplier the -common character $c$, and $\mathcal J\to\GL(W_{\Q_p})$ extends to a -closed immersion into $\GL(\Lambda_p)$ for a self-dual $\Z_p$-lattice -$\Lambda_p$. -\end{enumerate} -\end{lemma} - -\begin{proof} -Choose a finite extension of $\Q_p$ splitting $G_{\Q_p}$. Polynomial -approximation, together with Krasner's lemma -\cite[Propositions~7.60--7.61 and Corollary~7.62]{MilneANT}, gives a -totally real number field $P$ whose completions at $p$ split $G_{\Q_p}$. For -example, approximate the minimal polynomial of a primitive element -of a splitting extension at $p$, and a polynomial with distinct real -roots at the real place. The resulting number field can have just -one place above $p$; this additional condition is not needed. - -Define the multiplier fiber product -\begin{equation} - J=(\Res_{P/\Q}G_P) - \mathbin{\times}_{\Res_{P/\Q}\mathbf G_m}\mathbf G_m, - \label{eq:equal-multiplier-group} -\end{equation} -where the last map is diagonal. The Hodge cocharacter, or its -inverse according to convention, pairs to $1$ with $c$. Since $c$ -factors through $G^{\mathrm{ab}}$, its character on this torus is -primitive. Thus $\ker(G^{\mathrm{ab}}\to\mathbf G_m)$ is a torus, -and $\ker(c:G\to\mathbf G_m)$ is connected reductive. The exact -sequence -\[ - 1\longrightarrow\Res_{P/\Q}(\ker c)_P - \longrightarrow J\xrightarrow{c}\mathbf G_m\longrightarrow1 -\] -proves connectedness and reductivity of $J$. Moreover, -\begin{equation} - J^{\mathrm{der}}=\Res_{P/\Q}G_P^{\mathrm{der}}. - \label{eq:auxiliary-derived} -\end{equation} - -The action on $U=V\otimes_\Q P$, with alternating form -$\Tr_{P/\Q}\psi_P$, gives a symplectic embedding of $J$ with -similitude character $c$. The diagonal Hodge homomorphism gives -$X_J$. The adjoint Hodge-type and Cartan-involution axioms follow -componentwise from those for $X$. We also check that extending the -totally real field introduces no rational adjoint factor on which -the Hodge homomorphism is trivial. Write a rational simple adjoint -factor as $\Res_{L/\Q}H$, with $L$ totally real. Every field factor -$L'$ of $P\otimes_\Q L$ is totally real, and every real embedding of -$L$ extends to a real embedding of $L'$. Hence the existence of a -nontrivial Hodge component for $H$ persists in each resulting -rational factor. Finally the diagonal weight is rational and the -trace form is a polarization. This proves that $(J,X_J)$ is a -Shimura datum of Hodge type. - -Here are the local conditions explicitly. Put $P\otimes\Q_p= -\prod_{v\mid p}P_v$ and choose a split maximal torus $T_v$ of -$G_{P_v}$. The character $c$ on $T_v$ has a cocharacter section: -a conjugate of the Hodge cocharacter pairs to $1$ with it, and all -cocharacters of a split torus are defined over $P_v$. The same -argument applies to $G_{P_v}^{\mathrm{ab}}$. Consequently the -maximal torus of $J_{\Q_p}$ obtained from the $T_v$, and the -abelianization of $J_{\Q_p}$, have the forms -\begin{align} - T_J&\simeq\mathbf G_m\times - \prod_{v\mid p}\Res_{P_v/\Q_p}\mathbf G_m^{\operatorname{rk}G-1}, - \label{eq:auxiliary-torus}\\ - J_{\Q_p}^{\mathrm{ab}}&\simeq\mathbf G_m\times - \prod_{v\mid p}\Res_{P_v/\Q_p} - \mathbf G_m^{\dim G^{\mathrm{ab}}-1}. - \label{eq:auxiliary-abelianization} -\end{align} -The split Borels over the $P_v$ give a Borel of $J_{\Q_p}$, so this -group is quasi-split. Its derived group is a product of restrictions -of split groups by \eqref{eq:auxiliary-derived}. The vertices from -the hyperspecial vertices in those split groups give a very special -parahoric of $J_{\Q_p}$. The inertia coinvariants of the cocharacter -module in \eqref{eq:auxiliary-abelianization} are torsion-free, since -that module is a direct sum of permutation modules. The full -stabilizer is therefore connected by -\cite[Lemma~4.2.4]{KisinZhouEarlier}. - -The centralizer of a maximal $\Kzero$-split torus is obtained from -\eqref{eq:auxiliary-torus} by unramified base extension; its torus -factors are still induced split tori and $\mathbf G_m$. -It is $R$-smooth by -\cite[Proposition~2.4.6]{KisinZhouEarlier}. Finally restrictions of -split root systems give reduced relative root systems, also at -$p=2$. These are precisely the strong-admissibility checks in the -construction of \cite[Proposition~5.3.8]{KisinZhou2025}. They use -the stipulated Hodge embedding, not the assertion that $G$ is the -full Mumford--Tate group of a particular point. - -Apply \cite[Proposition~2.1.9]{KisinZhou2025}, starting with the -representation $U$ just constructed. Its proof first tensors $U$ -with an auxiliary totally real $\Q$-vector space, then takes direct -sums, and finally applies the symplectic self-duality construction. -The group acts trivially on that extra vector-space factor. In the -last construction the dual representation twisted by $c$ is -identified with $U$ by its symplectic form. Thus every resulting -rational summand is a copy of $U$, with the same multiplier. This -gives \eqref{eq:transport-copies}, together with $\psi_W$ and -$\Lambda_p$. The assertion of the cited proposition includes -$p=2$; its proof uses the reduced-relative-root condition through -\cite[Lemma~2.1.8]{KisinZhou2025}. -\end{proof} - -Choose a sufficiently small neat prime-to-$p$ level $K^p$ and put -$K_p=\mathcal J(\Z_p)$. The adapted embedding gives the integral -model $\mathcal S_K$ and its universal abelian scheme. A rational -$\Z_{(p)}$-lattice can be chosen with completion $\Lambda_p$. -We use a finite family of tensors defining its stabilizer -$\mathcal J$, as in \cite[\S2.1.4]{KisinZhou2025}. -Any additional rational $J$-invariant tensors may be included in -this family after clearing denominators. Indeed their integral -stabilizer condition already holds on the generic fiber; flatness -of $\mathcal J$ makes it hold on $\mathcal J$ itself. Adding them -therefore changes neither the group scheme nor its tensor -trivializations. - -The added tensors have compatible sections in every realization, as -follows. Use the homological degree-one realizations in this paragraph, -that is, the duals of the corresponding $H^1$, so that their reference -representation is $W$. For such a realization $H_\nu$ over its -coefficient field $k_\nu$, let -\[ - \mathcal P_\nu=\operatorname{Isom}_{\{s_\alpha\}} - (W\otimes k_\nu,H_\nu) -\] -be its torsor of frames preserving the original defining tensors. -For a rational $J$-invariant tensor $u$ and a frame $e$, set -$u_\nu=e(u)$. Two frames differ by an element of $J$, so this -definition is independent of the frame and descends from a -trivialization of the torsor. It also works integrally after clearing -the fixed denominator of $u$. - -The Betti and \'etale frame torsors are supplied by the canonical -tensor sections of \cite[\S2.1.4]{KisinZhou2025}. The construction -of the universal Breuil--Kisin--Fargues module and its structure -tensors gives the crystalline torsor and its compatibility with -those sections -\cite[\S\S2.1.13--2.1.14]{KisinZhou2025}. -Indeed, a comparison map carrying every $s_\alpha$ to its specified -realization carries a frame $e$ to the comparison of $e$; it therefore -carries $e(u)$ to the same tensor evaluated in the new frame. -Thus it preserves $u_\nu$ for every added invariant $u$, without -requiring $u$ to be an expression in the chosen defining tensors. -At a lift over $\Qbar$, these are the realizations of its rational -Hodge tensor, by Lemma~\ref{lem:comparison}. Their crystalline -specializations are independent of the lift because they are -evaluations in the fixed crystalline frame torsor. - -The same observation applies to isogenies. One quasi-isogeny -preserving the defining tensors identifies their frame torsors in -every realization in which those tensors are preserved. It -therefore preserves all the added invariant tensors as well. -This proves the tensor comparison and simultaneous preservation -that will be used below; flatness alone is used only to keep the -integral stabilizer unchanged. - -Specification~\textup{(1)} of Proposition~\ref{prop:transport} -uses the special-point theorem -\cite[Theorem~2.2.7]{KisinZhou2025} directly. The next lemma is -needed only for specification~\textup{(2)}, where the filtration -has been prescribed in advance. - -\subsection{Realizing a prescribed admissible filtration} - -For a point $x\in\mathcal S_K(\F)$ let $C_x$ be the corresponding -abelian variety and put -\[ - \mathbb D_x=H^1_{\mathrm{cris}}(C_x/W(\F)),\qquad - D_x=\mathbb D_x[1/p]. -\] -The crystalline structure tensors are denoted $s_{\alpha,0,x}$; -the tensors away from $p$ are denoted $s_{\alpha,\ell,x}$. -They are the specializations of the canonical tensor sections and -are independent of the chosen lift of $x$. - -The \emph{tensor isogeny class} $\mathcal I_x$ consists of the points -$y\in\mathcal S_K(\F)$ for which a single quasi-isogeny -$C_x\to C_y$ preserves $s_{\alpha,0}$ and every -$s_{\alpha,\ell}$ for $\ell\ne p$. We emphasize that this -definition concerns actual quasi-isogenies of abelian varieties. -It is the class defined in \cite[\S2.2.4]{KisinZhou2025}. - -\begin{lemma}[Realization of an admissible filtration] -\label{lem:admissible-realization} -Let $(J,X_J,\mathcal J)$ be strongly admissible, with $J_{\Q_p}$ -quasi-split, and choose the adapted embedding and neat level above. -Let $x\in\mathcal S_K(\F)$ have a lift to characteristic zero. -Suppose a filtration $\mathcal F$ on $D_x$, after a finite extension -of $\Kzero$, has the following properties: -\begin{enumerate} -\item the resulting filtered isocrystal is admissible; -\item geometrically its filtration is induced by a $J$-valued -cocharacter in the conjugacy class prescribed by $X_J$. -\end{enumerate} -Then, after extending that field finitely, there are $y\in\mathcal I_x$, -a tensor-preserving quasi-isogeny $b:C_x\to C_y$, and a lift -$\widetilde y$ over the extended valuation ring such that -\[ - b^*\Fil^1 H^1_{\mathrm{dR}}(C_{\widetilde y})=\mathcal F^1 -\] -under the crystalline--de Rham identification. - -In particular, condition \textup{(1)} follows if the filtered -isocrystal and its tensor data have finite local-field descent and -$\mathcal F$ is weakly admissible. -\end{lemma} - -\begin{proof} -Write $k=\F$ and let $\breve E$ be the completed local reflex field. -We work over a common finite extension $K$ of $\Kzero$ containing -$\breve E$ and a field of definition of $\mathcal F$. Every further -field extension below is finite. Put $K_p=\mathcal J(\Z_p)$. - -We first fix the relation between the group and cohomological -conventions. Let $\rho:J_{\Q_p}\to\GL(W_{\Q_p})$ be the adapted -homological representation and let $\rho^\vee$ be its dual. Choose -the tensor-preserving identification -$\mathbb D_x\simeq\Lambda_p^\vee\otimes_{\Z_p}W(k)$ of -\cite[\S2.1.14]{KisinZhou2025}. The local shtuka parameter is -$b_x\in J(\Kzero)$, whereas crystalline Frobenius on this -cohomological Dieudonn\'e module is -\[ - F_x=\rho^\vee(\sigma(b_x))\circ\sigma. -\] -The additional $\sigma$ on $b_x$ comes from the comparison with -the vector shtuka: its special-fiber module is -$(\sigma^{-1})^*(\mathbb D_x^\vee)$ -\cite[Example~2.3.4 and Remark~2.3.10]{PappasRapoport2024}. -Write $\{\mu\}$ for the class prescribed by $X_J$; then -$[b_x]\in B(J,\mu^{-1})$. - -For precision, identify the general linear crystalline period -flag with the appropriate filtration variety on $D_x$ by its -marked Rapoport--Zink interpretation. If a deformation -$\mathcal H$ has special-fiber marking -$q:\mathcal H_k\dashrightarrow C_x[p^\infty]$, this identification -sends its period to -$(q^*)^{-1}\Fil^1H^1_{\mathrm{dR}}(\mathcal H)$. -Pushout through the faithful Hodge representation gives the -commutative period square -\[ -\begin{array}{ccc} -M_{\mathcal J,b_x,\mu,K_p}&\longrightarrow&M_{\GL,\rho(b_x),\rho(\mu)}\\ -\pi_{\mathrm{GM}}\downarrow&&\downarrow\pi_{\mathrm{GM}}\\ -\mathcal F_{J,\mu}&\longrightarrow&\operatorname{Flag}(D_x). -\end{array} -\] -The lower map is a closed immersion: the parabolic stabilizing -the faithful filtration intersects $J$ in the parabolic for -$\mu$. Define $f_{\mathcal F}$ to be the unique $J$-flag with -image $\mathcal F$. It exists by condition~\textup{(2)}. The -square uses crystalline periods and the dual Frobenius descent -above, so no extension of $\sigma$ to the ramified filtration -field is chosen. When $q=b^{-1}$ its lower right period is -exactly $b^*\Fil^1$. - -\medskip\noindent -\emph{A framed point over the prescribed period.} -Admissibility of the faithful filtered isocrystal implies that -the image of $f_{\mathcal F}$ in the general linear period flag -variety is in its admissible locus. To see this in the -cohomological coordinates of the square, let $U$ be a crystalline -representation realizing $(D_x,\mathcal F)$. Its crystalline -comparison identifies the associated vector bundle away from -the untilt with $U\otimes\mathcal O$. The filtered de Rham -comparison identifies the modification at the untilt with the -standard lattice $U\otimes B_{\mathrm{dR}}^+$. They glue to the -trivial bundle $U\otimes\mathcal O$ on the Fargues--Fontaine -curve, which is the general linear admissibility condition. -Since $\rho$ is faithful, -$\rho^{-1}(Z_{\GL(W)})\subset Z_J$: an element acting as a scalar -commutes with the faithful image of $J$. The admissible-locus -identity in \cite[Proposition~3.1.1(i), (3.1.9)]{PappasRapoport2024} -therefore puts $f_{\mathcal F}$ in -$\mathcal F_{J,\mu}^{\mathrm{adm}}$. - -This locus is the image of the crystalline period map -\[ - \pi_{\mathrm{GM}}: - M_{\mathcal J,b_x,\mu,K_p} - \longrightarrow\mathcal F_{J,\mu}^{\mathrm{adm}}. -\] -The map is \'{e}tale -\cite[\S3.1.1, (3.1.4)]{PappasRapoport2024}. -Its fiber at the classical point $f_{\mathcal F}$ is nonempty; -an \'{e}tale rigid space over $K$ has a point over a finite -extension of $K$. After making that extension, choose -\begin{equation} - z\in M_{\mathcal J,b_x,\mu,K_p}(K), - \qquad \pi_{\mathrm{GM}}(z)=f_{\mathcal F}. - \label{eq:transport-framed-point} -\end{equation} -In particular $z$ supplies the integral level as well as the -rational isocrystal marking. If $C$ is the completed algebraic -closure of $K$, its moduli description gives an actual -$\mathcal J$-torsor $\mathcal P_z$, a Frobenius isomorphism -bounded by $\mu$, and, for sufficiently large $a$, an isomorphism -\[ - i_a:\mathcal J_{Y_{[a,\infty)}(C^\flat)} - \xrightarrow{\ \sim\ } - \mathcal P_z|_{Y_{[a,\infty)}(C^\flat)}, - \qquad - \varphi_{\mathcal P_z}\circ\varphi^*i_a - =i_a\circ\varphi_{b_x}. -\] -These are precisely the torsor and annular framing in -\cite[\S3.1.1]{PappasRapoport2024}. Its associated -pro-\'{e}tale $\mathcal J(\Z_p)$-torsor gives integral tensor -frames. Thus the integral structure and Frobenius-compatible -framing are part of the chosen point, with all their tensor -compatibilities. - -\medskip\noindent -\emph{Integral realization and its special fiber.} -Our adapted datum is of global Hodge type: the group scheme is -a full connected Bruhat--Tits stabilizer, and the chosen Hodge -embedding is integral on $\Lambda_p$. -The center also satisfies the split-rank hypothesis of -\cite[\S4.1]{PappasRapoport2024}. Indeed its real subgroup of -multiplier one preserves the positive definite form -$\psi_W(v,h(i)w)$ and is compact. Its real split rank is -therefore at most one, while the rational weight torus supplies -one split central direction over $\Q$; both split ranks are one. -The model here is the one in -\cite[\S2.1.11 and Theorem~2.1.12]{KisinZhou2025}, obtained -from this adapted Hodge embedding and satisfying the -Pappas--Rapoport characterization. Condition $(U_x)$ holds -by \cite[Corollary~6.3]{GleasonLimXu2026}; see also -\cite[Remark~4.10.4]{PappasRapoport2024}. Consequently -\cite[Theorem~4.10.6]{PappasRapoport2024} represents the integral -local Shimura variety by a normal formal scheme -$\mathfrak M_{\mathcal J,b_x,\mu}$ and gives the morphism -\[ - \Theta_x:\mathfrak M_{\mathcal J,b_x,\mu} - \longrightarrow\widehat{\mathcal S_K}. -\] -More precisely, \cite[\S4.10.2, (4.10.2)--(4.10.6)]{PappasRapoport2024} -identifies this formal scheme with the tensor-compatible open and -closed part of the pullback of the Siegel Rapoport--Zink space. -This construction identifies the pullback of the universal -$\mathcal J$-shtuka with its tautological framed shtuka and the -faithful vector realization with the universal $p$-divisible group. - -Its rigid generic fiber is the local Shimura variety -\cite[Proposition~3.5.1(i)]{PappasRapoport2024}. -The classical point $z$ of this rigid generic fiber gives -$\operatorname{Spf}\mathcal O_K\to\mathfrak M_{\mathcal J,b_x,\mu}$. -Indeed it lies in the generic fiber of a formal affine chart; -for a special formal chart one may take an affinoid member of -its standard generic-fiber covering containing $z$. -Evaluation of the formal chart's ring is integral, and its -ideal of definition maps to topologically nilpotent elements of -$\mathcal O_K$. Let $z_0$ be the special point and put -$y=\Theta_x(z_0)$ and $\widetilde y=\Theta_x(z)$. -The formal map $\widetilde y$ gives a point of the integral model -over $\mathcal O_K$ by restricting to an affine neighborhood of -$y$. Pullback of the universal abelian scheme gives -$C_{\widetilde y}$ and its $p$-divisible group $\mathcal H_z$. - -We record explicitly the integral comparison at this point. -For a uniformizer $\pi$ of $K$, put -$\mathfrak S=W(k)[[u]]$, with $u\mapsto\pi$, and let -$E_\pi(u)$ generate its kernel. The crystalline representation -and integral level of $z$ give the $\mathcal J$-Breuil--Kisin -torsor of \cite[\S3.5]{PappasRapoport2024}, with -\[ - \Phi:\varphi^*\mathcal P_{\mathrm{BK}}[1/E_\pi] - \xrightarrow{\ \sim\ } - \mathcal P_{\mathrm{BK}}[1/E_\pi]. -\] -Its pushout to $\GL(\Lambda_p)$ is the Breuil--Kisin module of -$\mathcal H_z$. Reduction at $u=0$, followed by the dual -Frobenius descent fixed above, therefore gives the actual -Dieudonn\'e lattice of $\mathcal H_{z,k}$. -Proposition~3.5.1 of \cite{PappasRapoport2024} compares the two -specializations of this already constructed framed point. In -the coordinates of \cite[Theorem~2.2.5]{KisinZhou2025} it gives -\[ - g\in X_{\mathcal J}(\mu^{-1},b_x),\qquad - \mathbb D(\mathcal H_{z,k}) - =\rho^\vee(\sigma(g))\mathbb D_x\subset D_x. -\] -Indeed the Frobenius matrix on this lattice, written in its -induced basis, is -\[ - \rho^\vee\!\left( - \sigma(g^{-1}b_x\sigma(g))\right)\circ\sigma, -\] -which explains the same Frobenius descent in the -Deligne--Lusztig condition and in the lattice formula. - -\medskip\noindent -\emph{The abelian marking and the filtration.} -Keep the identity prime-to-$p$ coordinate in the uniformization. -The description of its special fiber in -\cite[\S4.10.3]{PappasRapoport2024}, or equivalently the -isogeny-class map of \cite[Theorem~2.2.5]{KisinZhou2025}, gives -$y=i_x(g)$ and a single abelian quasi-isogeny -$b:C_x\to C_y$ preserving the crystalline tensors and every -prime-to-$p$ tensor. Its crystalline marking is the one just -used for $\mathcal H_z$, because the uniformization is induced -by the marked Siegel Rapoport--Zink space. - -The crystalline period map of that space is the Hodge filtration -of its universal deformation pulled back by its marking. -Compatibility of $\Theta_x$ with the tautological shtuka and -\eqref{eq:transport-framed-point} consequently give -\[ - b^*\Fil^1H^1_{\mathrm{dR}}(C_{\widetilde y}/K) - =\mathcal F^1. -\] -The deformation criterion -\cite[Proposition~2.2.3]{KisinZhou2025} is satisfied concretely: -the filtration is of type $\mu$, and the crystalline structure -tensors compare with the integral \'{e}tale tensors in the -$\mathcal J(\Z_p)$-frames of $z$. The bounded integral shtuka -and its annular framing were obtained before using this -criterion or comparing specializations. All cited global -Hodge-type and strongly admissible results apply at $p=2$ as -well. - -Finally, under finite local-field descent, the weak-admissibility -inequalities restrict to the descended Frobenius subobjects. -Colmez--Fontaine's theorem -\cite[Theorem~A]{ColmezFontaine2000} makes those descended data -admissible, and scalar extension supplies condition~\textup{(1)}. -\end{proof} - -\subsection{Proof of the transport proposition} - -\begin{proof}[Proof of Proposition~\ref{prop:transport}] -\medskip\noindent -\emph{The original point and its reduction.} -Apply Lemma~\ref{lem:auxiliary-datum}. Choose a rational -$J$-equivariant identification -$W=(V\otimes_\Q P)^{\oplus m}$ and set $n=m[P:\Q]$. -Choose a basis of $P$ whose first vector is $1$. The resulting -identification of the Hodge structure at the diagonal point $h$ -with $V^{\oplus n}$ makes the first-factor map equal to -\begin{equation} - v\longmapsto(v\otimes1,0,\ldots,0). - \label{eq:transport-first-factor} -\end{equation} -The rational homology/isogeny correspondence -\cite[Corollary~6.9]{MilneShimura} therefore makes the abelian variety -\(C\) at that point isogenous to \(A^n\). Fix a rational isogeny -\(f:A^n\to C\) realizing this identification. - -We explain the field and reduction assertions for this starting -point. The Hodge embedding maps it to the Siegel point of the -polarized variety $C$ with finite level. The abelian variety $C$ -is defined over $\Qbar$: clearing a denominator in $f$ realizes -it as a quotient of $A^n$ by a finite subgroup of its torsion, and -every such subgroup is defined over $\Qbar$. Its rational -polarization and its finite level also descend to $\Qbar$, since -homomorphisms between abelian varieties over $\Qbar$ do not -enlarge after extending algebraically closed fields -\cite[Corollary~20.4]{MilneAV}. The Hodge -embedding induces a finite morphism onto its -image at sufficiently small finite level -\cite[Theorem~5.16]{MilneShimura}. Its fiber over a -$\Qbar$-point therefore consists of $\Qbar$-points. This places -the diagonal point on the canonical model over $\Qbar$. - -After a finite number-field extension the variety $C$ has good -reduction, since it is isogenous to $A^n$ -\cite[Chapter~IV, Theorem~3.5]{MilneAVNotes}. Its polarization and -prime-to-$p$ level extend after a further finite extension. -The integral Hodge-type model is the normalization of the closure -of the generic fiber in the adapted Siegel model. The valuative -criterion for this finite normalization therefore extends our -point to its integral model. Denote the reduction by $x$. -The rational isogeny $f$ specializes to a quasi-isogeny -$f_0:A_0^n\to C_x$. - -\medskip\noindent -\emph{The trace tensor and its descent.} -Next extend the alternating form defining $\gamma$ -$P$-multilinearly and take its trace. Composing with projection -onto the first copy in \eqref{eq:transport-copies} gives -\begin{equation} - \eta=\Tr_{P/\Q}(\gamma_P)\circ - (\operatorname{pr}_1)^{\otimes 2r} - \in\bigwedge^{2r}W^*. - \label{eq:transport-trace-tensor} -\end{equation} -For $r=0$ this is interpreted as the scalar trace. -The equal-multiplier condition implies that $\eta$ has multiplier -$c^r$ under $J$. On \eqref{eq:transport-first-factor}, its -restriction is -\begin{equation} - j^*f^*\eta_C=[P:\Q]\gamma. - \label{eq:transport-trace-restriction} -\end{equation} -The class $\eta_C$ is understood with its Tate twist. - -We turn this semi-invariant into the untwisted tensors to which -the integral-model theorems directly apply. Write -$\psi_W^{-1}\in\bigwedge^2W$ for the inverse symplectic -bivector, and include -\begin{equation} - u_\psi=\psi_W\otimes\psi_W^{-1},\qquad - u_\eta=\eta\otimes(\psi_W^{-1})^{\otimes r} - \label{eq:transport-untwisted-tensors} -\end{equation} -among the structure tensors. Both are $J$-invariant tensors in -ordinary tensor constructions of $W$ and $W^*$; no separately -trivialized Tate object is needed. They can be made integral and -added to the defining family as explained above. Their -crystalline and prime-to-$p$ realizations at $x$ are the -specializations from the original point, with the comparison -conventions fixed above. - -\phantomsection -\label{par:transport-tensor-descent} -The entire finite family of structure tensors, including -$u_\psi$ and $u_\eta$, has finite local-field descent. To see -this, choose a finite extension $L/\Q_p$ over whose valuation -ring the original number-field point is defined, and let $L_0$ -be its maximal unramified subfield. The canonical \'etale -tensor sections of \cite[\S2.1.4]{KisinZhou2025}, pulled back -to that point, are $\Gal(\overline L/L)$-invariant. Applying -the tensor functor $D_{\mathrm{cris},L}$ to these morphisms -from the unit object places them in the crystalline tensor -spaces over $L_0$. Their extension to $\Kzero$ is exactly -the family $s_{\alpha,0,x}$, by good-reduction comparison. -The map $f_0$ also descends after a finite residue-field -extension, so this descent is retained under its marking. -Taking a compositum with the finite field of definition of a -prescribed filtration therefore gives one finite local field -for all the data. No tensor-preserving trivialization over -$L_0$ is being asserted or needed. - -\medskip\noindent -\emph{The two choices of lift.} -For specification~(1), apply -\cite[Theorem~2.2.7]{KisinZhou2025} to $x$. It gives a point -$y\in\mathcal I_x$ with a special-point lift and a -tensor-preserving quasi-isogeny $b:C_x\to C_y$. -The special-point lift supplies a CM abelian variety $B$. Special -points on the canonical Shimura variety are algebraic over its -reflex field by the canonical-model theory -\cite[Definition~12.8 and Proposition~14.14]{MilneShimura}, so $B$ -and its level data can be taken over $\Qbar$. The given local lift specifies their embedding at $w$ -and their good reduction $B_0=C_y$. - -For specification~(2), identify $D_x$ with $D_A^{\oplus n}$ using -$f_0^*$ and prescribe the direct-sum filtration -$\mathcal F_A^{\oplus n}$. It is weakly admissible, with the -required finite local-field descent. It is also in the required -$J$-cocharacter orbit. Indeed a geometric element of the -transported $G$ taking the original filtration on $D_A$ to -$\mathcal F_A$ acts diagonally on all the copies of $V$ in $W$; -its image in $J$ takes the original filtration on $D_x$ to the -prescribed one. The tensor-preserving identifications used here -come from the initial point and comparison, so they include the -$J$-structure tensors as well. Geometric conjugacy can be -realized over a finite extension, since the corresponding -isomorphism scheme is of finite type. -Lemma~\ref{lem:admissible-realization} now gives $y$, its lift -$B$, and a tensor-preserving quasi-isogeny $b:C_x\to B_0$ with -the prescribed marked filtration. This proves -\eqref{eq:transport-prescribed-filtration} for -$a=bf_0$. - -\medskip\noindent -\emph{A common rational normalization.} -It remains, in both specifications, to recover the twisted class -and normalize it by a single rational scalar. Choose a complex -description of the lifted Shimura point. Its rational marking by -$W$ supplies a rational Hodge class $\eta_B$ corresponding to -$\eta$, and a rational divisor class $\theta_B$ -corresponding to $\psi_W$. Write $\theta_C$ for the analogous -class on $C$ and put -\[ - \theta_*=f_0^*\theta_{C,0}\in\NS(A_0^n)\otimes_\Z\Q. -\] -The class $\theta_*$ is rational ample. Tensor preservation of -$u_\psi$ says that, in each realization, -\begin{equation} - a^*\theta_{B,0}=\lambda_v\theta_*, - \label{eq:transport-local-similitudes} -\end{equation} -with a nonzero local scalar $\lambda_v$. Let $D=n\dim A$. -Pairing with $\theta_*^{D-1}$ gives -\begin{equation} - \lambda_v= - \frac{\deg\bigl(a^*\theta_{B,0}\cdot\theta_*^{D-1}\bigr)} - {\deg(\theta_*^D)}=:\lambda\in\Q^\times. - \label{eq:transport-rational-similitude} -\end{equation} -Both intersection numbers refer to rational divisor classes on -the same abelian variety; their cycle-class traces are identical -in every realization. The denominator is positive because -$\theta_*$ is ample, and the numerator is nonzero by -\eqref{eq:transport-local-similitudes}. Thus all the scalars in -\eqref{eq:transport-local-similitudes}, including the crystalline -one, are the single rational number $\lambda$. - -Preservation of $u_\eta$, together with -\eqref{eq:transport-rational-similitude}, then yields -\begin{equation} - a^*\eta_{B,0}=\lambda^r f_0^*\eta_{C,0} - \label{eq:transport-eta-scaling} -\end{equation} -simultaneously. In fact the inverse bivector scales by -$\lambda^{-1}$, so that the equality for the untwisted tensor -in \eqref{eq:transport-untwisted-tensors} gives exactly -\eqref{eq:transport-eta-scaling}. Set -\[ - \kappa=\frac{\lambda^{-r}}{[P:\Q]}\eta_B. -\] -Specializing \eqref{eq:transport-trace-restriction} and using -\eqref{eq:transport-eta-scaling} proves -\eqref{eq:transport-class}. - -For the arbitrary local lift, the crystalline realization just -used is unambiguous: the canonical model supplies the untwisted -tensor $u_\eta$, and the algebraic polarization supplies its Tate -line and inverse bivector. The comparison isomorphism and these -operations define $\eta_B$ in the crystalline realization and -agree with its canonical \'{e}tale tensors. Consequently any -identity expressing this class by algebraic divisor operations -specializes with the same rational coefficients in all -realizations. For the $\Qbar$-valued CM lift, these are also the -usual absolute-Hodge and de Rham realizations. This completes -the two constructions. -\end{proof} - -\begin{corollary}[CM transport of a Hodge class] -\label{cor:cm-transport} -Let \(A/\Qbar\) be an abelian variety of positive dimension with -good reduction at \(w\), and let -\(\gamma\in H^{2r}(A(\C),\Q(r))\) be a rational Hodge class, -where \(0\leq r\leq\dim A\). There are a positive integer \(n\), -a CM abelian variety \(B/\Qbar\) with good reduction at \(w\), -a rational Hodge class \(\kappa\) of codimension \(r\) on \(B\), -and a quasi-isogeny \(a:A_0^n\to B_0\) such that -\[ - \gamma_0=(aj)^*\kappa_0, -\] -where \(j:A_0\to A_0^n\) is the first-factor inclusion. This is an -equality of the usual absolute-Hodge specializations in every -prime-to-\(p\) realization and in crystalline cohomology extended -to \(\Cw\). -\end{corollary} - -\begin{proof} -Choose a polarization \(\psi\) on \(V=H_1(A(\C),\Q)\), and let -\(G=\MT(V)\). The Mumford--Tate group is connected reductive and -is characterized by its Hodge tensors -\cite[Section~3, Propositions~3.4 and~3.6]{Deligne1982}. -The Hodge homomorphism and its \(G(\R)\)-conjugacy class define -the Mumford--Tate Shimura datum of Hodge type, with faithful -symplectic representation -\[ - G\hookrightarrow\GSp(V,\psi). -\] -The polarization supplies the rational weight and the Cartan -involution condition. The definition of the Mumford--Tate group -excludes an adjoint factor on which the Hodge homomorphism is -trivial. The class \(\gamma\), with its Tate twist, is fixed by -\(G\). Equivalently, its alternating form on \(V\) has multiplier -\(c^r\), where \(c\) is the similitude character. Thus the first -alternative of Proposition~\ref{prop:transport} applies and gives -the asserted data and simultaneous equality. -\end{proof} +\section{Transport through Hodge-type integral models} +\label{sec:transport} + +After taking a power of the original abelian variety, we transport a +rational Hodge class to an auxiliary variety by one quasi-isogeny on +special fibers, with the same equality in every +realization. The auxiliary variety can be chosen either of CM type or +with a prescribed admissible Hodge filtration. These choices serve +different parts of the proof, but share the tensor construction and the +rational normalization in Proposition~\ref{prop:transport} below. + +Throughout this section, a polarization on a rational homology space is +written as a rational alternating form, using the fixed Betti +trivialization of the Tate line. If its similitude character is $c$, +an alternating form $\xi:V^{2r}\to\Q$ of multiplier $c^r$ means that +\begin{equation} + \xi(gv_1,\ldots,gv_{2r})=c(g)^r\xi(v_1,\ldots,v_{2r}). + \label{eq:transport-multiplier} +\end{equation} +Such a form is regarded as a tensor with its codimension-$r$ Tate +twist. All homomorphisms and polarizations may be rational: an +isomorphism in the isogeny category induces the usual isomorphisms on +cohomology, including its tensor constructions. + +We say that filtered isocrystal data have \emph{finite local-field +descent} if they are obtained by scalar extension from an isocrystal +with its tensors over the maximal unramified subfield \(H_0\) of a +finite extension \(H/\Q_p\), together with a filtration over \(H\). +The scalar extension is to \(\Kzero H\). This condition will let us +apply weak admissibility over a finite local field. +For a group in a faithful representation, a tensor-preserving +identification means one respecting a finite family of rational tensors +whose stabilizer is that group, together with their comparison realizations. + +\begin{proposition}[Transport] +\label{prop:transport} +Let $A/\Qbar$ be an abelian variety of positive dimension with good +reduction at $w$, and put $V=H_1(A(\C),\Q)$. Suppose +\[ + G\subset\GSp(V,\psi) +\] +is connected reductive, its Hodge homomorphism $h$ gives the +Hodge structure of $A$, and its conjugacy class $X$ makes this +inclusion a Hodge embedding. Let $\gamma$ be a rational Hodge +class of codimension $r$, represented on $V$ by an alternating +form satisfying \eqref{eq:transport-multiplier}. + +Then there exist a positive integer $n$, an auxiliary +good-reduction abelian variety $B$, a quasi-isogeny +$a:A_0^n\to B_0$, and a rational Hodge class $\kappa$ of +codimension $r$ on $B$ such that, for the first-factor inclusion +$j:A_0\to A_0^n$, +\begin{equation} + (aj)^*\kappa_0=\gamma_0. + \label{eq:transport-class} +\end{equation} +The equality holds simultaneously in every $\ell$-adic +realization with $\ell\ne p$ and in the crystalline realization +after extension to $\Cw$. Either of the following two further +specifications can be imposed. +\begin{enumerate} +\item The variety $B$ is of CM type and is defined over $\Qbar$, +with its indicated good reduction at $w$. +\item Put $D_A=H^1_{\mathrm{cris}}(A_0/W(\F))[1/p]$. Suppose +$\mathcal F_A$ is a filtration in degrees $0,1$ on $D_A$, over a +finite extension of $\Kzero$, which is weakly admissible and has +finite local-field descent together with the specialized tensors +specifying the $G$-structure. Suppose +also that, after a geometric tensor-preserving identification, it +lies in the $G$-orbit of the original Hodge filtration. Then $B$ +can be defined over a finite extension $K/\Kzero$ inside $\Cw$, +and $a$ can be chosen so that +\begin{equation} + a^*\Fil^1H^1_{\mathrm{dR}}(B/K) + =(\mathcal F_A^1)^{\oplus n}. + \label{eq:transport-prescribed-filtration} +\end{equation} +Here the equality is made after a common finite extension and +crystalline--de Rham identification. To discuss Hodge classes on +this $B$, choose a complex embedding of an algebraic closure of +$K$ extending the fixed embedding of $\Qbar$. +\end{enumerate} +In the second case the realizations of $\kappa$ are supplied by +the canonical tensor sections of the Hodge-type model and the +polarization. In the first case these agree with the usual +realizations and specialization of its absolute Hodge class. +\end{proposition} + +We prepare two ingredients for the proof. First, we enlarge the given +Hodge datum so that integral-model theorems apply at every prime while +its rational representation remains a sum of copies of \(V\). +Second, we realize a prescribed admissible filtration by choosing a +framed local Shimura point above its crystalline period and applying +integral uniformization. The resulting abelian lift retains that +point's integral tensors and special-fiber marking. After these +preparations, the proof constructs the transported tensor and normalizes +it using rational divisor intersections. + +\subsection{An auxiliary strongly admissible datum} + +We recall the hypotheses on integral models that we will use. +For a reductive group \(H_p/\Q_p\), a full Bruhat--Tits stabilizer +is the smooth group scheme fixing a point of its extended building. +Its identity component is the associated parahoric group scheme. +Now let \((H,X_H)\) be a Shimura datum over \(\Q\). In the terminology +of \cite[Definition~2.1.6]{KisinZhou2025}, the triple +\((H,X_H,\mathcal H)\) is \emph{strongly admissible} if \((H,X_H)\) +is of Hodge type, \(\mathcal H\) is a full stabilizer of \(H_{\Q_p}\) +with connected special fiber, the centralizer of a maximal +\(\Kzero\)-split torus of \(H_{\Kzero}\) is \(R\)-smooth, and the +relative root system over \(\Kzero\) is reduced when \(p=2\). Here $R$-smoothness is the condition on tori used in +\cite[\S2.4]{KisinZhouEarlier}. We will use it only for products and +extensions of restrictions of scalars of split tori, for which it +holds by \cite[Proposition~2.4.6]{KisinZhouEarlier}. + +\begin{lemma}[Auxiliary Hodge datum] +\label{lem:auxiliary-datum} +Let +\[ + (G,X)\lhook\joinrel\longrightarrow + (\GSp(V,\psi),X_{\mathrm{Siegel}}) +\] +be a Hodge embedding, with $G$ connected and similitude character $c$. +For every prime $p$ there exist a totally real field $P$, a connected +reductive group $J$, and a Hodge embedding +\[ + (J,X_J)\lhook\joinrel\longrightarrow + (\GSp(W,\psi_W),X'_{\mathrm{Siegel}}) +\] +with the following properties. +\begin{enumerate} +\item There are inclusions of Shimura data induced by +\[ + G\longrightarrow J\longrightarrow\Res_{P/\Q}G_P, +\] +where the first map is diagonal and $J$ is the subgroup on which the +similitude multipliers in the restriction of scalars agree. +\item The group $J_{\Q_p}$ is quasi-split and has a full connected +stabilizer $\mathcal J$ making $(J,X_J,\mathcal J)$ strongly admissible. +\item As a rational $J$-representation, +\begin{equation} + W\simeq (V\otimes_{\Q}P)^{\oplus m} + \label{eq:transport-copies} +\end{equation} +for some positive integer $m$. The form $\psi_W$ has multiplier the +common character $c$, and $\mathcal J\to\GL(W_{\Q_p})$ extends to a +closed immersion into $\GL(\Lambda_p)$ for a self-dual $\Z_p$-lattice +$\Lambda_p$. +\end{enumerate} +\end{lemma} + +\begin{proof} +Choose a finite extension of $\Q_p$ splitting $G_{\Q_p}$. Polynomial +approximation, together with Krasner's lemma +\cite[Propositions~7.60--7.61 and Corollary~7.62]{MilneANT}, gives a +totally real number field $P$ whose completions at $p$ split $G_{\Q_p}$. For +example, approximate the minimal polynomial of a primitive element +of a splitting extension at $p$, and a polynomial with distinct real +roots at the real place. The resulting number field can have just +one place above $p$; this additional condition is not needed. + +Define the multiplier fiber product +\begin{equation} + J=(\Res_{P/\Q}G_P) + \mathbin{\times}_{\Res_{P/\Q}\mathbf G_m}\mathbf G_m, + \label{eq:equal-multiplier-group} +\end{equation} +where the last map is diagonal. The Hodge cocharacter, or its +inverse according to convention, pairs to $1$ with $c$. Since $c$ +factors through $G^{\mathrm{ab}}$, its character on this torus is +primitive. Thus $\ker(G^{\mathrm{ab}}\to\mathbf G_m)$ is a torus, +and $\ker(c:G\to\mathbf G_m)$ is connected reductive. The exact +sequence +\[ + 1\longrightarrow\Res_{P/\Q}(\ker c)_P + \longrightarrow J\xrightarrow{c}\mathbf G_m\longrightarrow1 +\] +proves connectedness and reductivity of $J$. Moreover, +\begin{equation} + J^{\mathrm{der}}=\Res_{P/\Q}G_P^{\mathrm{der}}. + \label{eq:auxiliary-derived} +\end{equation} + +The action on $U=V\otimes_\Q P$, with alternating form +$\Tr_{P/\Q}\psi_P$, gives a symplectic embedding of $J$ with +similitude character $c$. The diagonal Hodge homomorphism gives +$X_J$. The adjoint Hodge-type and Cartan-involution axioms follow +componentwise from those for $X$. We also check that extending the +totally real field introduces no rational adjoint factor on which +the Hodge homomorphism is trivial. Write a rational simple adjoint +factor as $\Res_{L/\Q}H$, with $L$ totally real. Every field factor +$L'$ of $P\otimes_\Q L$ is totally real, and every real embedding of +$L$ extends to a real embedding of $L'$. Hence the existence of a +nontrivial Hodge component for $H$ persists in each resulting +rational factor. Finally the diagonal weight is rational and the +trace form is a polarization. This proves that $(J,X_J)$ is a +Shimura datum of Hodge type. + +Here are the local conditions explicitly. Put $P\otimes\Q_p= +\prod_{v\mid p}P_v$ and choose a split maximal torus $T_v$ of +$G_{P_v}$. The character $c$ on $T_v$ has a cocharacter section: +a conjugate of the Hodge cocharacter pairs to $1$ with it, and all +cocharacters of a split torus are defined over $P_v$. The same +argument applies to $G_{P_v}^{\mathrm{ab}}$. Consequently the +maximal torus of $J_{\Q_p}$ obtained from the $T_v$, and the +abelianization of $J_{\Q_p}$, have the forms +\begin{align} + T_J&\simeq\mathbf G_m\times + \prod_{v\mid p}\Res_{P_v/\Q_p}\mathbf G_m^{\operatorname{rk}G-1}, + \label{eq:auxiliary-torus}\\ + J_{\Q_p}^{\mathrm{ab}}&\simeq\mathbf G_m\times + \prod_{v\mid p}\Res_{P_v/\Q_p} + \mathbf G_m^{\dim G^{\mathrm{ab}}-1}. + \label{eq:auxiliary-abelianization} +\end{align} +The split Borels over the $P_v$ give a Borel of $J_{\Q_p}$, so this +group is quasi-split. Its derived group is a product of restrictions +of split groups by \eqref{eq:auxiliary-derived}. The vertices from +the hyperspecial vertices in those split groups give a very special +parahoric of $J_{\Q_p}$. The inertia coinvariants of the cocharacter +module in \eqref{eq:auxiliary-abelianization} are torsion-free, since +that module is a direct sum of permutation modules. The full +stabilizer is therefore connected by +\cite[Lemma~4.2.4]{KisinZhouEarlier}. + +The centralizer of a maximal $\Kzero$-split torus is obtained from +\eqref{eq:auxiliary-torus} by unramified base extension; its torus +factors are still induced split tori and $\mathbf G_m$. +It is $R$-smooth by +\cite[Proposition~2.4.6]{KisinZhouEarlier}. Finally restrictions of +split root systems give reduced relative root systems, also at +$p=2$. These are precisely the strong-admissibility checks in the +construction of \cite[Proposition~5.3.8]{KisinZhou2025}. They use +the stipulated Hodge embedding, not the assertion that $G$ is the +full Mumford--Tate group of a particular point. + +Apply \cite[Proposition~2.1.9]{KisinZhou2025}, starting with the +representation $U$ just constructed. Its proof first tensors $U$ +with an auxiliary totally real $\Q$-vector space, then takes direct +sums, and finally applies the symplectic self-duality construction. +The group acts trivially on that extra vector-space factor. In the +last construction the dual representation twisted by $c$ is +identified with $U$ by its symplectic form. Thus every resulting +rational summand is a copy of $U$, with the same multiplier. This +gives \eqref{eq:transport-copies}, together with $\psi_W$ and +$\Lambda_p$. The assertion of the cited proposition includes +$p=2$; its proof uses the reduced-relative-root condition through +\cite[Lemma~2.1.8]{KisinZhou2025}. +\end{proof} + +Choose a sufficiently small neat prime-to-$p$ level $K^p$ and put +$K_p=\mathcal J(\Z_p)$. The adapted embedding gives the integral +model $\mathcal S_K$ and its universal abelian scheme. A rational +$\Z_{(p)}$-lattice can be chosen with completion $\Lambda_p$. +We use a finite family of tensors defining its stabilizer +$\mathcal J$, as in \cite[\S2.1.4]{KisinZhou2025}. +Any additional rational $J$-invariant tensors may be included in +this family after clearing denominators. Indeed their integral +stabilizer condition already holds on the generic fiber; flatness +of $\mathcal J$ makes it hold on $\mathcal J$ itself. Adding them +therefore changes neither the group scheme nor its tensor +trivializations. + +The added tensors have compatible sections in every realization, as +follows. Use the homological degree-one realizations in this paragraph, +that is, the duals of the corresponding $H^1$, so that their reference +representation is $W$. For such a realization $H_\nu$ over its +coefficient field $k_\nu$, let +\[ + \mathcal P_\nu=\operatorname{Isom}_{\{s_\alpha\}} + (W\otimes k_\nu,H_\nu) +\] +be its torsor of frames preserving the original defining tensors. +For a rational $J$-invariant tensor $u$ and a frame $e$, set +$u_\nu=e(u)$. Two frames differ by an element of $J$, so this +definition is independent of the frame and descends from a +trivialization of the torsor. It also works integrally after clearing +the fixed denominator of $u$. + +The Betti and \'etale frame torsors are supplied by the canonical +tensor sections of \cite[\S2.1.4]{KisinZhou2025}. The construction +of the universal Breuil--Kisin--Fargues module and its structure +tensors gives the crystalline torsor and its compatibility with +those sections +\cite[\S\S2.1.13--2.1.14]{KisinZhou2025}. +Indeed, a comparison map carrying every $s_\alpha$ to its specified +realization carries a frame $e$ to the comparison of $e$; it therefore +carries $e(u)$ to the same tensor evaluated in the new frame. +Thus it preserves $u_\nu$ for every added invariant $u$, without +requiring $u$ to be an expression in the chosen defining tensors. +At a lift over $\Qbar$, these are the realizations of its rational +Hodge tensor, by Lemma~\ref{lem:comparison}. Their crystalline +specializations are independent of the lift because they are +evaluations in the fixed crystalline frame torsor. + +The same observation applies to isogenies. One quasi-isogeny +preserving the defining tensors identifies their frame torsors in +every realization in which those tensors are preserved. It +therefore preserves all the added invariant tensors as well. +This proves the tensor comparison and simultaneous preservation +that will be used below; flatness alone is used only to keep the +integral stabilizer unchanged. + +Specification~\textup{(1)} of Proposition~\ref{prop:transport} +uses the special-point theorem +\cite[Theorem~2.2.7]{KisinZhou2025} directly. The next lemma is +needed only for specification~\textup{(2)}, where the filtration +has been prescribed in advance. + +\subsection{Realizing a prescribed admissible filtration} + +For a point $x\in\mathcal S_K(\F)$ let $C_x$ be the corresponding +abelian variety and put +\[ + \mathbb D_x=H^1_{\mathrm{cris}}(C_x/W(\F)),\qquad + D_x=\mathbb D_x[1/p]. +\] +The crystalline structure tensors are denoted $s_{\alpha,0,x}$; +the tensors away from $p$ are denoted $s_{\alpha,\ell,x}$. +They are the specializations of the canonical tensor sections and +are independent of the chosen lift of $x$. + +The \emph{tensor isogeny class} $\mathcal I_x$ consists of the points +$y\in\mathcal S_K(\F)$ for which a single quasi-isogeny +$C_x\to C_y$ preserves $s_{\alpha,0}$ and every +$s_{\alpha,\ell}$ for $\ell\ne p$. We emphasize that this +definition concerns actual quasi-isogenies of abelian varieties. +It is the class defined in \cite[\S2.2.4]{KisinZhou2025}. + +\begin{lemma}[Realization of an admissible filtration] +\label{lem:admissible-realization} +Let $(J,X_J,\mathcal J)$ be strongly admissible, with $J_{\Q_p}$ +quasi-split, and choose the adapted embedding and neat level above. +Let $x\in\mathcal S_K(\F)$ have a lift to characteristic zero. +Suppose a filtration $\mathcal F$ on $D_x$, after a finite extension +of $\Kzero$, has the following properties: +\begin{enumerate} +\item the resulting filtered isocrystal is admissible; +\item geometrically its filtration is induced by a $J$-valued +cocharacter in the conjugacy class prescribed by $X_J$. +\end{enumerate} +Then, after extending that field finitely, there are $y\in\mathcal I_x$, +a tensor-preserving quasi-isogeny $b:C_x\to C_y$, and a lift +$\widetilde y$ over the extended valuation ring such that +\[ + b^*\Fil^1 H^1_{\mathrm{dR}}(C_{\widetilde y})=\mathcal F^1 +\] +under the crystalline--de Rham identification. + +In particular, condition \textup{(1)} follows if the filtered +isocrystal and its tensor data have finite local-field descent and +$\mathcal F$ is weakly admissible. +\end{lemma} + +\begin{proof} +Write $k=\F$ and let $\breve E$ be the completed local reflex field. +We work over a common finite extension $K$ of $\Kzero$ containing +$\breve E$ and a field of definition of $\mathcal F$. Every further +field extension below is finite. Put $K_p=\mathcal J(\Z_p)$. + +We first fix the relation between the group and cohomological +conventions. Let $\rho:J_{\Q_p}\to\GL(W_{\Q_p})$ be the adapted +homological representation and let $\rho^\vee$ be its dual. Choose +the tensor-preserving identification +$\mathbb D_x\simeq\Lambda_p^\vee\otimes_{\Z_p}W(k)$ of +\cite[\S2.1.14]{KisinZhou2025}. The local shtuka parameter is +$b_x\in J(\Kzero)$, whereas crystalline Frobenius on this +cohomological Dieudonn\'e module is +\[ + F_x=\rho^\vee(\sigma(b_x))\circ\sigma. +\] +The additional $\sigma$ on $b_x$ comes from the comparison with +the vector shtuka: its special-fiber module is +$(\sigma^{-1})^*(\mathbb D_x^\vee)$ +\cite[Example~2.3.4 and Remark~2.3.10]{PappasRapoport2024}. +Write $\{\mu\}$ for the class prescribed by $X_J$; then +$[b_x]\in B(J,\mu^{-1})$. + +For precision, identify the general linear crystalline period +flag with the appropriate filtration variety on $D_x$ by its +marked Rapoport--Zink interpretation. If a deformation +$\mathcal H$ has special-fiber marking +$q:\mathcal H_k\dashrightarrow C_x[p^\infty]$, this identification +sends its period to +$(q^*)^{-1}\Fil^1H^1_{\mathrm{dR}}(\mathcal H)$. +Pushout through the faithful Hodge representation gives the +commutative period square +\[ +\begin{array}{ccc} +M_{\mathcal J,b_x,\mu,K_p}&\longrightarrow&M_{\GL,\rho(b_x),\rho(\mu)}\\ +\pi_{\mathrm{GM}}\downarrow&&\downarrow\pi_{\mathrm{GM}}\\ +\mathcal F_{J,\mu}&\longrightarrow&\operatorname{Flag}(D_x). +\end{array} +\] +The lower map is a closed immersion: the parabolic stabilizing +the faithful filtration intersects $J$ in the parabolic for +$\mu$. Define $f_{\mathcal F}$ to be the unique $J$-flag with +image $\mathcal F$. It exists by condition~\textup{(2)}. The +square uses crystalline periods and the dual Frobenius descent +above, so no extension of $\sigma$ to the ramified filtration +field is chosen. When $q=b^{-1}$ its lower right period is +exactly $b^*\Fil^1$. + +\medskip\noindent +\emph{A framed point over the prescribed period.} +Admissibility of the faithful filtered isocrystal implies that +the image of $f_{\mathcal F}$ in the general linear period flag +variety is in its admissible locus. To see this in the +cohomological coordinates of the square, let $U$ be a crystalline +representation realizing $(D_x,\mathcal F)$. Its crystalline +comparison identifies the associated vector bundle away from +the untilt with $U\otimes\mathcal O$. The filtered de Rham +comparison identifies the modification at the untilt with the +standard lattice $U\otimes B_{\mathrm{dR}}^+$. They glue to the +trivial bundle $U\otimes\mathcal O$ on the Fargues--Fontaine +curve, which is the general linear admissibility condition. +Since $\rho$ is faithful, +$\rho^{-1}(Z_{\GL(W)})\subset Z_J$: an element acting as a scalar +commutes with the faithful image of $J$. The admissible-locus +identity in \cite[Proposition~3.1.1(i), (3.1.9)]{PappasRapoport2024} +therefore puts $f_{\mathcal F}$ in +$\mathcal F_{J,\mu}^{\mathrm{adm}}$. + +This locus is the image of the crystalline period map +\[ + \pi_{\mathrm{GM}}: + M_{\mathcal J,b_x,\mu,K_p} + \longrightarrow\mathcal F_{J,\mu}^{\mathrm{adm}}. +\] +The map is \'{e}tale +\cite[\S3.1.1, (3.1.4)]{PappasRapoport2024}. +Its fiber at the classical point $f_{\mathcal F}$ is nonempty; +an \'{e}tale rigid space over $K$ has a point over a finite +extension of $K$. After making that extension, choose +\begin{equation} + z\in M_{\mathcal J,b_x,\mu,K_p}(K), + \qquad \pi_{\mathrm{GM}}(z)=f_{\mathcal F}. + \label{eq:transport-framed-point} +\end{equation} +In particular $z$ supplies the integral level as well as the +rational isocrystal marking. If $C$ is the completed algebraic +closure of $K$, its moduli description gives an actual +$\mathcal J$-torsor $\mathcal P_z$, a Frobenius isomorphism +bounded by $\mu$, and, for sufficiently large $a$, an isomorphism +\[ + i_a:\mathcal J_{Y_{[a,\infty)}(C^\flat)} + \xrightarrow{\ \sim\ } + \mathcal P_z|_{Y_{[a,\infty)}(C^\flat)}, + \qquad + \varphi_{\mathcal P_z}\circ\varphi^*i_a + =i_a\circ\varphi_{b_x}. +\] +These are precisely the torsor and annular framing in +\cite[\S3.1.1]{PappasRapoport2024}. Its associated +pro-\'{e}tale $\mathcal J(\Z_p)$-torsor gives integral tensor +frames. Thus the integral structure and Frobenius-compatible +framing are part of the chosen point, with all their tensor +compatibilities. + +\medskip\noindent +\emph{Integral realization and its special fiber.} +Our adapted datum is of global Hodge type: the group scheme is +a full connected Bruhat--Tits stabilizer, and the chosen Hodge +embedding is integral on $\Lambda_p$. +The center also satisfies the split-rank hypothesis of +\cite[\S4.1]{PappasRapoport2024}. Indeed its real subgroup of +multiplier one preserves the positive definite form +$\psi_W(v,h(i)w)$ and is compact. Its real split rank is +therefore at most one, while the rational weight torus supplies +one split central direction over $\Q$; both split ranks are one. +The model here is the one in +\cite[\S2.1.11 and Theorem~2.1.12]{KisinZhou2025}, obtained +from this adapted Hodge embedding and satisfying the +Pappas--Rapoport characterization. Condition $(U_x)$ holds +by \cite[Corollary~6.3]{GleasonLimXu2026}; see also +\cite[Remark~4.10.4]{PappasRapoport2024}. Consequently +\cite[Theorem~4.10.6]{PappasRapoport2024} represents the integral +local Shimura variety by a normal formal scheme +$\mathfrak M_{\mathcal J,b_x,\mu}$ and gives the morphism +\[ + \Theta_x:\mathfrak M_{\mathcal J,b_x,\mu} + \longrightarrow\widehat{\mathcal S_K}. +\] +More precisely, \cite[\S4.10.2, (4.10.2)--(4.10.6)]{PappasRapoport2024} +identifies this formal scheme with the tensor-compatible open and +closed part of the pullback of the Siegel Rapoport--Zink space. +This construction identifies the pullback of the universal +$\mathcal J$-shtuka with its tautological framed shtuka and the +faithful vector realization with the universal $p$-divisible group. + +Its rigid generic fiber is the local Shimura variety +\cite[Proposition~3.5.1(i)]{PappasRapoport2024}. +The classical point $z$ of this rigid generic fiber gives +$\operatorname{Spf}\mathcal O_K\to\mathfrak M_{\mathcal J,b_x,\mu}$. +Indeed it lies in the generic fiber of a formal affine chart; +for a special formal chart one may take an affinoid member of +its standard generic-fiber covering containing $z$. +Evaluation of the formal chart's ring is integral, and its +ideal of definition maps to topologically nilpotent elements of +$\mathcal O_K$. Let $z_0$ be the special point and put +$y=\Theta_x(z_0)$ and $\widetilde y=\Theta_x(z)$. +The formal map $\widetilde y$ gives a point of the integral model +over $\mathcal O_K$ by restricting to an affine neighborhood of +$y$. Pullback of the universal abelian scheme gives +$C_{\widetilde y}$ and its $p$-divisible group $\mathcal H_z$. + +We record explicitly the integral comparison at this point. +For a uniformizer $\pi$ of $K$, put +$\mathfrak S=W(k)[[u]]$, with $u\mapsto\pi$, and let +$E_\pi(u)$ generate its kernel. The crystalline representation +and integral level of $z$ give the $\mathcal J$-Breuil--Kisin +torsor of \cite[\S3.5]{PappasRapoport2024}, with +\[ + \Phi:\varphi^*\mathcal P_{\mathrm{BK}}[1/E_\pi] + \xrightarrow{\ \sim\ } + \mathcal P_{\mathrm{BK}}[1/E_\pi]. +\] +Its pushout to $\GL(\Lambda_p)$ is the Breuil--Kisin module of +$\mathcal H_z$. Reduction at $u=0$, followed by the dual +Frobenius descent fixed above, therefore gives the actual +Dieudonn\'e lattice of $\mathcal H_{z,k}$. +Proposition~3.5.1 of \cite{PappasRapoport2024} compares the two +specializations of this already constructed framed point. In +the coordinates of \cite[Theorem~2.2.5]{KisinZhou2025} it gives +\[ + g\in X_{\mathcal J}(\mu^{-1},b_x),\qquad + \mathbb D(\mathcal H_{z,k}) + =\rho^\vee(\sigma(g))\mathbb D_x\subset D_x. +\] +Indeed the Frobenius matrix on this lattice, written in its +induced basis, is +\[ + \rho^\vee\!\left( + \sigma(g^{-1}b_x\sigma(g))\right)\circ\sigma, +\] +which explains the same Frobenius descent in the +Deligne--Lusztig condition and in the lattice formula. + +\medskip\noindent +\emph{The abelian marking and the filtration.} +Keep the identity prime-to-$p$ coordinate in the uniformization. +The description of its special fiber in +\cite[\S4.10.3]{PappasRapoport2024}, or equivalently the +isogeny-class map of \cite[Theorem~2.2.5]{KisinZhou2025}, gives +$y=i_x(g)$ and a single abelian quasi-isogeny +$b:C_x\to C_y$ preserving the crystalline tensors and every +prime-to-$p$ tensor. Its crystalline marking is the one just +used for $\mathcal H_z$, because the uniformization is induced +by the marked Siegel Rapoport--Zink space. + +The crystalline period map of that space is the Hodge filtration +of its universal deformation pulled back by its marking. +Compatibility of $\Theta_x$ with the tautological shtuka and +\eqref{eq:transport-framed-point} consequently give +\[ + b^*\Fil^1H^1_{\mathrm{dR}}(C_{\widetilde y}/K) + =\mathcal F^1. +\] +The deformation criterion +\cite[Proposition~2.2.3]{KisinZhou2025} is satisfied concretely: +the filtration is of type $\mu$, and the crystalline structure +tensors compare with the integral \'{e}tale tensors in the +$\mathcal J(\Z_p)$-frames of $z$. The bounded integral shtuka +and its annular framing were obtained before using this +criterion or comparing specializations. All cited global +Hodge-type and strongly admissible results apply at $p=2$ as +well. + +Finally, under finite local-field descent, the weak-admissibility +inequalities restrict to the descended Frobenius subobjects. +Colmez--Fontaine's theorem +\cite[Theorem~A]{ColmezFontaine2000} makes those descended data +admissible, and scalar extension supplies condition~\textup{(1)}. +\end{proof} + +\subsection{Proof of the transport proposition} + +\begin{proof}[Proof of Proposition~\ref{prop:transport}] +\medskip\noindent +\emph{The original point and its reduction.} +Apply Lemma~\ref{lem:auxiliary-datum}. Choose a rational +$J$-equivariant identification +$W=(V\otimes_\Q P)^{\oplus m}$ and set $n=m[P:\Q]$. +Choose a basis of $P$ whose first vector is $1$. The resulting +identification of the Hodge structure at the diagonal point $h$ +with $V^{\oplus n}$ makes the first-factor map equal to +\begin{equation} + v\longmapsto(v\otimes1,0,\ldots,0). + \label{eq:transport-first-factor} +\end{equation} +The rational homology/isogeny correspondence +\cite[Corollary~6.9]{MilneShimura} therefore makes the abelian variety +\(C\) at that point isogenous to \(A^n\). Fix a rational isogeny +\(f:A^n\to C\) realizing this identification. + +We explain the field and reduction assertions for this starting +point. The Hodge embedding maps it to the Siegel point of the +polarized variety $C$ with finite level. The abelian variety $C$ +is defined over $\Qbar$: clearing a denominator in $f$ realizes +it as a quotient of $A^n$ by a finite subgroup of its torsion, and +every such subgroup is defined over $\Qbar$. Its rational +polarization and its finite level also descend to $\Qbar$, since +homomorphisms between abelian varieties over $\Qbar$ do not +enlarge after extending algebraically closed fields +\cite[Corollary~20.4]{MilneAV}. The Hodge +embedding induces a finite morphism onto its +image at sufficiently small finite level +\cite[Theorem~5.16]{MilneShimura}. Its fiber over a +$\Qbar$-point therefore consists of $\Qbar$-points. This places +the diagonal point on the canonical model over $\Qbar$. + +After a finite number-field extension the variety $C$ has good +reduction, since it is isogenous to $A^n$ +\cite[Chapter~IV, Theorem~3.5]{MilneAVNotes}. Its polarization and +prime-to-$p$ level extend after a further finite extension. +The integral Hodge-type model is the normalization of the closure +of the generic fiber in the adapted Siegel model. The valuative +criterion for this finite normalization therefore extends our +point to its integral model. Denote the reduction by $x$. +The rational isogeny $f$ specializes to a quasi-isogeny +$f_0:A_0^n\to C_x$. + +\medskip\noindent +\emph{The trace tensor and its descent.} +Next extend the alternating form defining $\gamma$ +$P$-multilinearly and take its trace. Composing with projection +onto the first copy in \eqref{eq:transport-copies} gives +\begin{equation} + \eta=\Tr_{P/\Q}(\gamma_P)\circ + (\operatorname{pr}_1)^{\otimes 2r} + \in\bigwedge^{2r}W^*. + \label{eq:transport-trace-tensor} +\end{equation} +For $r=0$ this is interpreted as the scalar trace. +The equal-multiplier condition implies that $\eta$ has multiplier +$c^r$ under $J$. On \eqref{eq:transport-first-factor}, its +restriction is +\begin{equation} + j^*f^*\eta_C=[P:\Q]\gamma. + \label{eq:transport-trace-restriction} +\end{equation} +The class $\eta_C$ is understood with its Tate twist. + +We turn this semi-invariant into the untwisted tensors to which +the integral-model theorems directly apply. Write +$\psi_W^{-1}\in\bigwedge^2W$ for the inverse symplectic +bivector, and include +\begin{equation} + u_\psi=\psi_W\otimes\psi_W^{-1},\qquad + u_\eta=\eta\otimes(\psi_W^{-1})^{\otimes r} + \label{eq:transport-untwisted-tensors} +\end{equation} +among the structure tensors. Both are $J$-invariant tensors in +ordinary tensor constructions of $W$ and $W^*$; no separately +trivialized Tate object is needed. They can be made integral and +added to the defining family as explained above. Their +crystalline and prime-to-$p$ realizations at $x$ are the +specializations from the original point, with the comparison +conventions fixed above. + +\phantomsection +\label{par:transport-tensor-descent} +The entire finite family of structure tensors, including +$u_\psi$ and $u_\eta$, has finite local-field descent. To see +this, choose a finite extension $L/\Q_p$ over whose valuation +ring the original number-field point is defined, and let $L_0$ +be its maximal unramified subfield. The canonical \'etale +tensor sections of \cite[\S2.1.4]{KisinZhou2025}, pulled back +to that point, are $\Gal(\overline L/L)$-invariant. Applying +the tensor functor $D_{\mathrm{cris},L}$ to these morphisms +from the unit object places them in the crystalline tensor +spaces over $L_0$. Their extension to $\Kzero$ is exactly +the family $s_{\alpha,0,x}$, by good-reduction comparison. +The map $f_0$ also descends after a finite residue-field +extension, so this descent is retained under its marking. +Taking a compositum with the finite field of definition of a +prescribed filtration therefore gives one finite local field +for all the data. No tensor-preserving trivialization over +$L_0$ is being asserted or needed. + +\medskip\noindent +\emph{The two choices of lift.} +For specification~(1), apply +\cite[Theorem~2.2.7]{KisinZhou2025} to $x$. It gives a point +$y\in\mathcal I_x$ with a special-point lift and a +tensor-preserving quasi-isogeny $b:C_x\to C_y$. +The special-point lift supplies a CM abelian variety $B$. Special +points on the canonical Shimura variety are algebraic over its +reflex field by the canonical-model theory +\cite[Definition~12.8 and Proposition~14.14]{MilneShimura}, so $B$ +and its level data can be taken over $\Qbar$. The given local lift specifies their embedding at $w$ +and their good reduction $B_0=C_y$. + +For specification~(2), identify $D_x$ with $D_A^{\oplus n}$ using +$f_0^*$ and prescribe the direct-sum filtration +$\mathcal F_A^{\oplus n}$. It is weakly admissible, with the +required finite local-field descent. It is also in the required +$J$-cocharacter orbit. Indeed a geometric element of the +transported $G$ taking the original filtration on $D_A$ to +$\mathcal F_A$ acts diagonally on all the copies of $V$ in $W$; +its image in $J$ takes the original filtration on $D_x$ to the +prescribed one. The tensor-preserving identifications used here +come from the initial point and comparison, so they include the +$J$-structure tensors as well. Geometric conjugacy can be +realized over a finite extension, since the corresponding +isomorphism scheme is of finite type. +Lemma~\ref{lem:admissible-realization} now gives $y$, its lift +$B$, and a tensor-preserving quasi-isogeny $b:C_x\to B_0$ with +the prescribed marked filtration. This proves +\eqref{eq:transport-prescribed-filtration} for +$a=bf_0$. + +\medskip\noindent +\emph{A common rational normalization.} +It remains, in both specifications, to recover the twisted class +and normalize it by a single rational scalar. Choose a complex +description of the lifted Shimura point. Its rational marking by +$W$ supplies a rational Hodge class $\eta_B$ corresponding to +$\eta$, and a rational divisor class $\theta_B$ +corresponding to $\psi_W$. Write $\theta_C$ for the analogous +class on $C$ and put +\[ + \theta_*=f_0^*\theta_{C,0}\in\NS(A_0^n)\otimes_\Z\Q. +\] +The class $\theta_*$ is rational ample. Tensor preservation of +$u_\psi$ says that, in each realization, +\begin{equation} + a^*\theta_{B,0}=\lambda_v\theta_*, + \label{eq:transport-local-similitudes} +\end{equation} +with a nonzero local scalar $\lambda_v$. Let $D=n\dim A$. +Pairing with $\theta_*^{D-1}$ gives +\begin{equation} + \lambda_v= + \frac{\deg\bigl(a^*\theta_{B,0}\cdot\theta_*^{D-1}\bigr)} + {\deg(\theta_*^D)}=:\lambda\in\Q^\times. + \label{eq:transport-rational-similitude} +\end{equation} +Both intersection numbers refer to rational divisor classes on +the same abelian variety; their cycle-class traces are identical +in every realization. The denominator is positive because +$\theta_*$ is ample, and the numerator is nonzero by +\eqref{eq:transport-local-similitudes}. Thus all the scalars in +\eqref{eq:transport-local-similitudes}, including the crystalline +one, are the single rational number $\lambda$. + +Preservation of $u_\eta$, together with +\eqref{eq:transport-rational-similitude}, then yields +\begin{equation} + a^*\eta_{B,0}=\lambda^r f_0^*\eta_{C,0} + \label{eq:transport-eta-scaling} +\end{equation} +simultaneously. In fact the inverse bivector scales by +$\lambda^{-1}$, so that the equality for the untwisted tensor +in \eqref{eq:transport-untwisted-tensors} gives exactly +\eqref{eq:transport-eta-scaling}. Set +\[ + \kappa=\frac{\lambda^{-r}}{[P:\Q]}\eta_B. +\] +Specializing \eqref{eq:transport-trace-restriction} and using +\eqref{eq:transport-eta-scaling} proves +\eqref{eq:transport-class}. + +For the arbitrary local lift, the crystalline realization just +used is unambiguous: the canonical model supplies the untwisted +tensor $u_\eta$, and the algebraic polarization supplies its Tate +line and inverse bivector. The comparison isomorphism and these +operations define $\eta_B$ in the crystalline realization and +agree with its canonical \'{e}tale tensors. Consequently any +identity expressing this class by algebraic divisor operations +specializes with the same rational coefficients in all +realizations. For the $\Qbar$-valued CM lift, these are also the +usual absolute-Hodge and de Rham realizations. This completes +the two constructions. +\end{proof} + +\begin{corollary}[CM transport of a Hodge class] +\label{cor:cm-transport} +Let \(A/\Qbar\) be an abelian variety of positive dimension with +good reduction at \(w\), and let +\(\gamma\in H^{2r}(A(\C),\Q(r))\) be a rational Hodge class, +where \(0\leq r\leq\dim A\). There are a positive integer \(n\), +a CM abelian variety \(B/\Qbar\) with good reduction at \(w\), +a rational Hodge class \(\kappa\) of codimension \(r\) on \(B\), +and a quasi-isogeny \(a:A_0^n\to B_0\) such that +\[ + \gamma_0=(aj)^*\kappa_0, +\] +where \(j:A_0\to A_0^n\) is the first-factor inclusion. This is an +equality of the usual absolute-Hodge specializations in every +prime-to-\(p\) realization and in crystalline cohomology extended +to \(\Cw\). +\end{corollary} + +\begin{proof} +Choose a polarization \(\psi\) on \(V=H_1(A(\C),\Q)\), and let +\(G=\MT(V)\). The Mumford--Tate group is connected reductive and +is characterized by its Hodge tensors +\cite[Section~3, Propositions~3.4 and~3.6]{Deligne1982}. +The Hodge homomorphism and its \(G(\R)\)-conjugacy class define +the Mumford--Tate Shimura datum of Hodge type, with faithful +symplectic representation +\[ + G\hookrightarrow\GSp(V,\psi). +\] +The polarization supplies the rational weight and the Cartan +involution condition. The definition of the Mumford--Tate group +excludes an adjoint factor on which the Hodge homomorphism is +trivial. The class \(\gamma\), with its Tate twist, is fixed by +\(G\). Equivalently, its alternating form on \(V\) has multiplier +\(c^r\), where \(c\) is the similitude character. Thus the first +alternative of Proposition~\ref{prop:transport} applies and gives +the asserted data and simultaneous equality. +\end{proof} diff --git a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/03_filtration.tex b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/03_filtration.tex index c14d2e86b1d777b7944ce9e765adaa60aa3b7cdf..85d69e5ad42e88cef57434146f71ae747dd44e31 100644 --- a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/03_filtration.tex +++ b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/03_filtration.tex @@ -1,482 +1,482 @@ -\section{An endomorphism-stable weakly admissible filtration} -\label{sec:filtration} - -We construct a filtration on first crystalline cohomology that preserves an -additional divisor. The construction takes place entirely on the given -isocrystal. Its realization by an abelian lift will be a separate step. -Generic transversality is a basic ingredient of the admissible-filtration -existence theorems of Fontaine--Rapoport -\cite[Theorem~1 and Lemmas~1.1--1.2]{FontaineRapoport2005}. -We impose the additional endomorphism and polarization constraints -explicitly. The bounded-degree argument below chooses the required -coordinates over a finite algebraic extension, while testing every -subisocrystal over the fixed coefficient field. - -Throughout this section, $\Kzero=\operatorname{Frac}W(\overline{\F}_p)$, -and valuations are normalized by $v(p)=1$. If $D$ is an isocrystal over -$\Kzero$ and $K/\Kzero$ is finite, a subspace $L\subset D_K$ defines the -one-step filtration -\[ - \Fil^iD_K= - \begin{cases} - D_K,&i\leq 0,\\ - L,&i=1,\\ - 0,&i\geq 2. - \end{cases} -\] -For an isocrystal subobject $P\subset D$, its Hodge number for this -filtration is $t_H(P)=\dim_K(P_K\cap L)$. Its Newton number $t_N(P)$ is -the sum of its slopes with multiplicities. Weak admissibility means that -$t_H(D)=t_N(D)$ and $t_H(P)\leq t_N(P)$ for every isocrystal subobject -$P$ over $\Kzero$. - -Finite local-field descent is understood as in Section~\ref{sec:transport}: -the isocrystal and its indicated tensors descend to the maximal unramified -subfield of a finite extension of \(\Q_p\), and the filtration descends -to that finite extension. - -\begin{theorem}[The filtration construction] -\label{thm:filtration} -Let $A_0/\overline{\F}_p$ be an abelian variety equipped with a rational -action of a CM field $E$ and a polarization whose Rosati involution -restricts to conjugation on $E$. Suppose that -\[ - D=H^1_{\mathrm{cris}}(A_0/W(\overline{\F}_p))[1/p] -\] -has rank $2s$ over $E\otimes_\Q\Kzero$, where $s\geq 1$. Suppose also -that there is an invertible $u\in\End^0(A_0)$ such that -\[ - u^\dagger=u,\qquad ue=\bar e u\quad(e\in E). -\] -Let $\beta$ be the dual polarization pairing on $D$, with values in -the isocrystal line of slope $1$. - -Then there are a finite extension $K/\Kzero$ and a subspace $L\subset -D_K$ with the following properties: -\begin{enumerate} -\item The one-step filtration with $\Fil^1D_K=L$ is weakly admissible. -\item $L$ is stable under the cohomological actions of $E$ and $u$, -and is Lagrangian for $\beta$. -\item After extending to an algebraically closed field $\Omega/K$, -one has $\dim_\Omega L_\tau=s$ for every embedding -$\tau:E\hookrightarrow\Omega$. -\item The isocrystal, its indicated tensors, and this filtration have -finite local field descent. -\end{enumerate} -Moreover, geometrically $L$ belongs to the balanced polarized -cocharacter orbit. More explicitly, define -\[ - G_D=\left\{g\in\GSp(D,\beta)\cap - \GL_{E\otimes_\Q\Kzero}(D): - \det_{E\otimes_\Q\Kzero}(g)=c(g)^s\right\}, -\] -where the scalar on the right is viewed in -$(E\otimes_\Q\Kzero)^\times$. The balanced $E$-stable -$\beta$-Lagrangians form a single $G_D(\Omega)$-orbit. Each is the -weight-$1$ space of a cocharacter of $G_{D,\Omega}$ with weights $0,1$. -In particular $L$ belongs to the orbit of any original filtration of -this type under a tensor-preserving comparison. -\end{theorem} - -The proof will construct \(L\) from Lagrangians in finitely many -symplectic spaces. The two lemmas below supply the required simultaneous -choice: minimum intersection with a subspace is ensured by a nonzero -minor of bounded degree, and all such minors over a fixed coefficient -field can be made nonzero over one finite extension. - -\begin{lemma}[Bounded-degree avoidance over a finite extension] -\label{lem:avoidance} -Let $K_1/\Kzero$ be a finite extension, and let $r,b\geq 1$ be integers. -There exist an element $\theta$ algebraic over $\Q_p$ and a point -\[ - (x_0,\ldots,x_{r-1})\in K_1(\theta)^r -\] -at which every nonzero polynomial in $K_1[X_0,\ldots,X_{r-1}]$ of total -degree at most $b$ is nonzero. The coordinates $x_i$ can all be chosen -in $\Q_p(\theta)$. -\end{lemma} - -\begin{proof} -Choose an integer $N>b$ and put -\[ - B=b\sum_{i=0}^{r-1}N^i. -\] -Let $e$ be the ramification index of $K_1/\Kzero$, so that -$v(K_1^\times)=\frac1e\Z$. Choose a prime $q>\max(B,e)$, and choose -$\theta$ with $\theta^q=p$ in an algebraic closure of $K_1$. The value -$v(\theta)=1/q$ has order $q$ modulo $\frac1e\Z$. Consequently the -ramification index, and hence the degree, of $K_1(\theta)/K_1$ is at -least $q$. The polynomial $X^q-p$ gives the reverse inequality, so -\[ - [K_1(\theta):K_1]=q>B. -\] - -Set $x_i=\theta^{N^i}$. Under the substitution $X_i=T^{N^i}$, -distinct monomials of total degree at most $b$ have distinct exponents: -each exponent of an individual variable is smaller than $N$, and -uniqueness follows from base-$N$ expansion. Thus a nonzero polynomial -of total degree at most $b$ becomes a nonzero polynomial in $T$ of -degree at most $B$. It cannot vanish at $\theta$, whose degree over -$K_1$ exceeds $B$. -\end{proof} - -\begin{remark} -The coefficient field in Lemma~\ref{lem:avoidance} is the fixed field -$K_1$. The conclusion concerns all bounded-degree polynomials over that -field, with no restriction on their cardinality. It would be false with -$K_1(\theta)$ in place of $K_1$ as the coefficient field. -\end{remark} - -\begin{lemma}[Intersections with Lagrangians] -\label{lem:lagrangian-intersection} -Let $M$ be a symplectic space of dimension $2m$ over a field $k$ of -characteristic zero, and let $W\subset M$ have dimension $a$. A nonempty -open subset of the Lagrangian Grassmannian consists of subspaces $L$ -such that -\[ - \dim(L\cap W)=\max(0,a-m). -\] -On any fixed symplectic affine big cell, at least one nonzero polynomial -of degree at most $m$ has nonvanishing locus contained in this open -subset. This degree bound is independent of $W$. -\end{lemma} - -\begin{proof} -We first work over an algebraic closure. The Lagrangian Grassmannian -is geometrically integral of dimension $m(m+1)/2$; its affine big cells -are parametrized by symmetric $m\times m$ matrices. If $1\leq a\leq m$, -consider the projective incidence variety of pairs $(\ell,L)$ with -$\ell\subset W$ a line and $\ell\subset L$ a Lagrangian. The fiber over -$\ell$ is the Lagrangian Grassmannian of the symplectic space -$\ell^\perp/\ell$, and therefore the incidence variety has dimension -\[ - a-1+\frac{m(m-1)}2<\frac{m(m+1)}2. -\] -Its image is a proper closed subset. Its complement consists precisely -of the Lagrangians missing $W$. The case $a=0$ is immediate. For -$a>m$, use -\[ - \dim(L\cap W)=a-m+\dim(L\cap W^\perp) -\] -and apply the preceding argument to $W^\perp$. - -For the assertion on equations, choose coordinates on the fixed big -cell in which $L$ is the column space of -\[ - \begin{pmatrix}I_m\\ T\end{pmatrix},\qquad T=T^{\mathsf t}. -\] -The minimum-intersection condition says that the composite of this -matrix with the quotient map $M\longrightarrow M/W$ has rank -$\min(m,2m-a)$. Its entries are affine linear polynomials in the -independent entries of $T$. The nonempty open subset already proved -meets this big cell, so some minor of the required size is a nonzero -polynomial over $k$. Its degree is at most $m$, and its nonvanishing -implies the required rank. If the required rank is zero, use the -constant polynomial $1$. -\end{proof} - - -The two lemmas reduce simultaneous transversality to a uniform bound on -the degrees of the relevant minors. We now identify the symplectic -spaces on which to apply them and the inequalities they must enforce. - -\begin{proof}[Proof of Theorem~\ref{thm:filtration}] -We write $e$ and $u$ also for their cohomological actions on $D$. -These actions commute with Frobenius and satisfy -\begin{equation} -\label{eq:filtration-adjoints} - \beta(ux,y)=\beta(x,uy),\qquad - \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u. -\end{equation} -There is no change to the last relation from contravariance. Indeed, -pulling back $ue=\bar e u$ gives $e^*u^*=u^*\bar e^*$; replacing $e$ -by $\bar e$ gives $u^*e^*=\bar e^*u^*$. Likewise, if $J$ is the -homological polarization matrix, $U^{\mathsf t}J=JU$ implies -$UJ^{-1}=J^{-1}U^{\mathsf t}$, which is the self-adjointness of the -cohomological action $U^{\mathsf t}$ for the dual pairing. The -Rosati-conjugate adjoint identity for $E$ follows in the same way. - -The pullback $U^{\mathsf t}$ of an individual endomorphism is -distinct from the contragredient group action $g\mapsto g^{-\mathsf t}$. -If a homological similitude $g$ has multiplier $c_V$ and satisfies -$\det_E(g)=c_V^s$, its dual action has multiplier $c_D=c_V^{-1}$ -and satisfies $\det_E(g^{-\mathsf t})=c_D^s$. -Thus $G_D$ is the dual form of the homological group defined by this -determinant condition, with its cohomological multiplier. - -\medskip\noindent -\emph{Reduction to the stable subobjects.} -First consider any $E,u$-stable candidate $L\subset D_K$, where -$K/\Kzero$ is finite. We show that the weak-admissibility inequalities -need only be tested on $E,u$-stable subobjects. This resembles the use of canonical -destabilizing objects in \cite[Section~2]{FontaineRapoport2005}, -but we use total defect, not defect divided by dimension, and give -the required elementary argument. -For any isocrystal subobject $P\subset D$ over $\Kzero$, set -\[ - f(P)=t_H(P)-t_N(P). -\] -These values form a finite set: the Hodge number is a bounded integer, -and the Newton number is determined by bounded integer multiplicities -among the finitely many slopes of $D$. Let $a$ be their maximum. -Newton numbers are additive in short exact sequences, while -\[ - (P_K\cap L)+(Q_K\cap L)\subset (P+Q)_K\cap L. -\] -Consequently -\begin{equation} -\label{eq:filtration-supermodularity} - f(P)+f(Q)\leq f(P\cap Q)+f(P+Q). -\end{equation} -If $P$ and $Q$ both attain $a$, the right-hand terms are each at -most $a$, so both attain $a$. Thus maximizers are closed under -intersection and sum. Choose a maximizer $P_{\max}$ of largest -dimension. Adding any other maximizer to it shows that the other -one is contained in it. Hence $P_{\max}$ is the unique largest -maximizer. - -The operators $u$ and $e\in E^\times$ are filtration-preserving -isocrystal automorphisms. They preserve $f$ and hence fix -$P_{\max}$, by its uniqueness. Thus $P_{\max}$ is $E,u$-stable. -Consequently, if $t_H(D')\leq t_N(D')$ for every $E,u$-stable -subobject $D'$, then $a=f(P_{\max})\leq0$. All the -weak-admissibility inequalities follow. We must therefore construct -an $E,u$-stable Lagrangian with these restricted inequalities and -$t_H(D)=t_N(D)$. - -\medskip\noindent -\emph{Semisimplicity and symplectic blocks.} -The Rosati trace form -\[ - \langle a,b\rangle=\Tr(ab^\dagger) - \quad\text{on }\End^0(A_0)\otimes_\Q\R -\] -is positive definite \cite[Theorem~17.3]{MilneAV}. Left multiplication -by $u$ is self-adjoint for this form, since trace cyclicity gives -\[ - \langle ua,b\rangle=\Tr(uab^\dagger) - =\Tr(ab^\dagger u)=\langle a,ub\rangle. -\] -It is therefore diagonalizable. Its minimal polynomial equals the -minimal polynomial of $u$ in the endomorphism algebra, as is seen by -applying a polynomial in the multiplication operator to the identity. -Thus $u$, and hence $u^2$, acts semisimply on $D$. The relation in -\eqref{eq:filtration-adjoints} also shows that $u^2$ commutes with $E$. - -Put -\[ - B(x,y)=\beta(x,uy). -\] -This is a nondegenerate alternating pairing. Indeed, invertibility is -immediate, and -$B(y,x)=-\beta(ux,y)=-B(x,y)$. In addition, -\[ - B(ex,y)=\beta(x,\bar euy)=\beta(x,uey)=B(x,ey), - \qquad B(u^2x,y)=B(x,u^2y). -\] -After a finite splitting extension $K_1/\Kzero$, write -\[ - D_{K_1}=\bigoplus_{\tau,z}M_{\tau,z}, - \qquad - M_{\tau,z}=\{x:ex=\tau(e)x\ (e\in E),\ u^2x=zx\}. -\] -We omit zero summands. The self-adjoint identities for $B$ show that -distinct summands are orthogonal. Its restriction to each summand is -therefore nondegenerate. Write $\dim M_{\tau,z}=2m_{\tau,z}$. - -For $\bar\tau=\tau\circ\overline{(\cdot)}$, the operator $u$ maps -$M_{\tau,z}$ isomorphically to $M_{\bar\tau,z}$. On these spaces, -\begin{equation} -\label{eq:filtration-u-similitude} - B(ux,uy)=zB(x,y). -\end{equation} -Here $z\neq 0$, and $\tau\neq\bar\tau$. Choose a $B$-Lagrangian -$L_{\tau,z}$ in one member of each opposite pair, and prescribe -\begin{equation} -\label{eq:filtration-opposite-choice} - L_{\bar\tau,z}=uL_{\tau,z}. -\end{equation} -Equation~\eqref{eq:filtration-u-similitude} shows that the latter is -Lagrangian as well. Their direct sum $L$ is $E$-stable and $u$-stable; -on applying $u$ twice, one obtains $zL_{\tau,z}=L_{\tau,z}$. Since -\[ - \beta(x,y)=B(x,u^{-1}y), -\] -$L$ is also $\beta$-Lagrangian. Its dimension at each $E$-embedding -is $s$. It remains to make the independent choices appropriately. - -\medskip\noindent -\emph{The inherited Newton grading.} -Use the canonical slope decomposition -\[ - D=\bigoplus_\lambda D_\lambda. -\] -The isocrystal category over $\overline{\F}_p$ is semisimple, and -this decomposition is functorial -\cite[\S3.2, Proposition~3.3]{ColmezFontaine2000}. The slopes of $D$ -belong to $[0,1]$: its integral Dieudonn\'e operators \(F,V\) -satisfy \(FV=p\), so both the slopes of \(F\) and those of \(V\) -are nonnegative. All the endomorphisms above preserve every -$D_\lambda$, and the projectors onto the simultaneous eigenspaces -preserve its scalar extension. Thus each block has a grading -\[ - M_{\tau,z}=\bigoplus_\lambda M_{\tau,z,\lambda},\qquad - M_{\tau,z,\lambda}=M_{\tau,z}\cap(D_\lambda)_{K_1}. -\] -We use this grading even if Frobenius permutes the labels $(\tau,z)$. -No extension of Frobenius to $K_1$, and no isocrystal structure on an -individual block, is required. - -Before scalar extension, $B$ is a pairing of isocrystals into the -slope-$1$ line. Hence $B(D_\lambda,D_\mu)=0$ unless -$\lambda+\mu=1$. Nondegeneracy on an individual block $M$ then gives -perfect pairings between $M_\lambda$ and $M_{1-\lambda}$. In -particular, if $\dim M=2m$, -\begin{equation} -\label{eq:filtration-block-newton} - \sum_\lambda\lambda\dim M_\lambda=m. -\end{equation} - -Let $D'\subset D$ be an isocrystal subobject over $\Kzero$ stable -under $E$ and $u$. Its intersections $M'=D'_{K_1}\cap M$ decompose -$D'_{K_1}$ and inherit the same slope grading. Put $a=\dim M'$. -The slope sum of the graded quotient $M/M'$ is at most $2m-a$, -because all its slopes belong to $[0,1]$. By -\eqref{eq:filtration-block-newton}, -\begin{equation} -\label{eq:filtration-block-bound} - \sum_\lambda\lambda\dim M'_\lambda - \geq \max(0,a-m). -\end{equation} -The additional bound by zero follows from the nonnegative slopes. -The right side is also the smallest possible dimension of the -intersection of a Lagrangian in $M$ with $M'$. -Thus attaining this minimum in every block will give the -weak-admissibility inequalities for all the stable subobjects. - -\medskip\noindent -\emph{Simultaneous choices and finite local field descent.} -We may arrange the splitting field and coordinates used above as -follows. Choose a finite field $k_0\subset\overline{\F}_p$ over -which $A_0$, its polarization, $u$, and a rational basis of $E$ are -defined. The underlying isocrystal and the indicated tensors then -descend to $\operatorname{Frac}W(k_0)$. Choose a finite local field -$H/\Q_p$ containing this field and splitting the actions of $E$ and -$u^2$. The eigenspaces and their symplectic bases can be constructed -over $H$ by linear algebra. We take $K_1=\Kzero H$. In particular, -the independent Lagrangian Grassmannians, fixed symplectic big cells -in them, and the maps induced by $u$ all have coordinates over $H$. - -Take the product of these big cells, one for each independent choice -in \eqref{eq:filtration-opposite-choice}, and let $r$ be its number -of affine coordinates. For every $E,u$-stable isocrystal subobject -$D'$ over $\Kzero$ and each independent block $M$, the space -$M'=D'_{K_1}\cap M$ is defined over the \emph{fixed} field $K_1$. -Lemma~\ref{lem:lagrangian-intersection} supplies a nonzero polynomial -over $K_1$, of degree at most $m$, whose nonvanishing ensures -\begin{equation} -\label{eq:filtration-minimal-intersection} - \dim(L_{\tau,z}\cap M')=\max(0,\dim M'-m). -\end{equation} -We regard this as a polynomial on the whole product by ignoring the -other coordinates. A single bound $b=\max m_{\tau,z}$ works for all -these polynomials, independently of $D'$. On the opposite block, -the same intersection equality follows from $u$-stability of $D'$ -and \eqref{eq:filtration-opposite-choice}. - -Apply Lemma~\ref{lem:avoidance} to $K_1,r,b$. Its one point satisfies -\eqref{eq:filtration-minimal-intersection} for every one of the -subobjects just specified, simultaneously. This use of the lemma -does not assume that the collection of subobjects is countable or -that its members descend to $H$. Their definition over $K_1$ and -the uniform polynomial degree bound are the only requirements. -There are nonzero blocks with $m\geq1$, so $r,b\geq1$ as required. - -Put $K=K_1(\theta)=\Kzero H(\theta)$. The resulting Lagrangian is -defined over $H(\theta)$ in the chosen coordinates, and $H(\theta)$ -is finite over $\Q_p$. Extending the original isocrystal to the -maximal unramified subfield of $H(\theta)$ therefore gives finite -local field descent of the complete filtered data. In particular, -the construction has not obtained its generic point by introducing -transcendental parameters. - -Summing \eqref{eq:filtration-minimal-intersection} and -\eqref{eq:filtration-block-bound} over all blocks gives -\begin{equation} -\label{eq:filtration-stable-test} - t_H(D')\leq t_N(D') - \quad\text{for every $E,u$-stable isocrystal subobject $D'\subset D$.} -\end{equation} -For $D$ itself, both numbers equal $\frac12\dim D$ by -\eqref{eq:filtration-block-newton}. The reduction to stable subobjects -proved at the start now gives all the weak-admissibility inequalities -for subobjects of $D$ over $\Kzero$. Each subobject over the smaller unramified -field of descent extends to one of these subobjects. Hence the -descended filtered isocrystal is weakly admissible as well. - -\medskip\noindent -\emph{The geometric cocharacter orbit.} -Over $\Omega$, the polarization pairs $D_\tau$ only with -$D_{\bar\tau}$. A balanced $E$-stable Lagrangian is determined by an -$s$-dimensional subspace of $D_\tau$ for one member of each pair; -the subspace in the opposite member is its annihilator. The -subgroup of $G_D$ with multiplier $1$ contains the product of -$\SL(D_\tau)$ acting on opposite spaces by the inverse dual -representations. These special linear groups are transitive on -the $s$-dimensional subspaces. Thus all the indicated Lagrangians -form one $G_D(\Omega)$-orbit. - -Finally choose an $E$-stable $\beta$-Lagrangian complement to one -such $L$. Such a complement is obtained separately on each pair -of opposite embedding spaces. Acting by $t$ on $L$ and by $1$ -on the complement gives a cocharacter with multiplier $t$ and -$E$-determinant $t^s$ at each embedding. It lies in $G_D$ and has -the stated filtration. This proves the final assertions. -\end{proof} - -\begin{corollary} -\label{cor:filtration-admissible} -The filtration of Theorem~\ref{thm:filtration}, with its finite local -field descent, is admissible: the descended filtered isocrystal is -the filtered crystalline module of a crystalline representation. -\end{corollary} - -\begin{proof} -Apply the weakly-admissible-implies-admissible theorem of -Colmez--Fontaine with monodromy operator zero -\cite[Theorem~A]{ColmezFontaine2000}. -\end{proof} - -\begin{corollary}[The divisor filtration condition] -\label{cor:filtration-divisor} -Let \(D,L\) be as in Theorem~\ref{thm:filtration}, and let \(\psi\) -and \(U\) denote the homological polarization and the homological -action of \(u\) on \(D_K^\vee\). The untwisted divisor form -\(\psi(\cdot,U\cdot)\) belongs to -\[ - \Fil^1\bigl(\bigwedge^2D_K\bigr)=L\wedge D_K, -\] -and hence to filtration degree zero after its codimension-one Tate -twist. If a marked power is realized by an abelian lift, the -corresponding pulled-back divisor satisfies the filtration condition -for its symmetric homomorphism in Lemma~\ref{lem:lift-hom}. -\end{corollary} - -\begin{proof} -Let \(H\subset D_K^\vee\) be the annihilator of \(L\). It is -Lagrangian for \(\psi\) and stable under \(U\): for \(a\in L\) -and \(h\in H\), -\[ - a(Uh)=(u^*a)(h)=0. -\] -Therefore \(\psi(h,Uh')=0\) for \(h,h'\in H\). The divisor form -thus lies in -\[ - \ker\bigl(\bigwedge^2D_K\longrightarrow\bigwedge^2H^\vee\bigr) - =L\wedge D_K. -\] -This proves the assertion on the fixed filtered vector space. After -realization of a marked power, pullback respects this filtration -condition, which is precisely the condition on the associated -symmetric homomorphism in Lemma~\ref{lem:lift-hom}. -\end{proof} +\section{An endomorphism-stable weakly admissible filtration} +\label{sec:filtration} + +We construct a filtration on first crystalline cohomology that preserves an +additional divisor. The construction takes place entirely on the given +isocrystal. Its realization by an abelian lift will be a separate step. +Generic transversality is a basic ingredient of the admissible-filtration +existence theorems of Fontaine--Rapoport +\cite[Theorem~1 and Lemmas~1.1--1.2]{FontaineRapoport2005}. +We impose the additional endomorphism and polarization constraints +explicitly. The bounded-degree argument below chooses the required +coordinates over a finite algebraic extension, while testing every +subisocrystal over the fixed coefficient field. + +Throughout this section, $\Kzero=\operatorname{Frac}W(\overline{\F}_p)$, +and valuations are normalized by $v(p)=1$. If $D$ is an isocrystal over +$\Kzero$ and $K/\Kzero$ is finite, a subspace $L\subset D_K$ defines the +one-step filtration +\[ + \Fil^iD_K= + \begin{cases} + D_K,&i\leq 0,\\ + L,&i=1,\\ + 0,&i\geq 2. + \end{cases} +\] +For an isocrystal subobject $P\subset D$, its Hodge number for this +filtration is $t_H(P)=\dim_K(P_K\cap L)$. Its Newton number $t_N(P)$ is +the sum of its slopes with multiplicities. Weak admissibility means that +$t_H(D)=t_N(D)$ and $t_H(P)\leq t_N(P)$ for every isocrystal subobject +$P$ over $\Kzero$. + +Finite local-field descent is understood as in Section~\ref{sec:transport}: +the isocrystal and its indicated tensors descend to the maximal unramified +subfield of a finite extension of \(\Q_p\), and the filtration descends +to that finite extension. + +\begin{theorem}[The filtration construction] +\label{thm:filtration} +Let $A_0/\overline{\F}_p$ be an abelian variety equipped with a rational +action of a CM field $E$ and a polarization whose Rosati involution +restricts to conjugation on $E$. Suppose that +\[ + D=H^1_{\mathrm{cris}}(A_0/W(\overline{\F}_p))[1/p] +\] +has rank $2s$ over $E\otimes_\Q\Kzero$, where $s\geq 1$. Suppose also +that there is an invertible $u\in\End^0(A_0)$ such that +\[ + u^\dagger=u,\qquad ue=\bar e u\quad(e\in E). +\] +Let $\beta$ be the dual polarization pairing on $D$, with values in +the isocrystal line of slope $1$. + +Then there are a finite extension $K/\Kzero$ and a subspace $L\subset +D_K$ with the following properties: +\begin{enumerate} +\item The one-step filtration with $\Fil^1D_K=L$ is weakly admissible. +\item $L$ is stable under the cohomological actions of $E$ and $u$, +and is Lagrangian for $\beta$. +\item After extending to an algebraically closed field $\Omega/K$, +one has $\dim_\Omega L_\tau=s$ for every embedding +$\tau:E\hookrightarrow\Omega$. +\item The isocrystal, its indicated tensors, and this filtration have +finite local field descent. +\end{enumerate} +Moreover, geometrically $L$ belongs to the balanced polarized +cocharacter orbit. More explicitly, define +\[ + G_D=\left\{g\in\GSp(D,\beta)\cap + \GL_{E\otimes_\Q\Kzero}(D): + \det_{E\otimes_\Q\Kzero}(g)=c(g)^s\right\}, +\] +where the scalar on the right is viewed in +$(E\otimes_\Q\Kzero)^\times$. The balanced $E$-stable +$\beta$-Lagrangians form a single $G_D(\Omega)$-orbit. Each is the +weight-$1$ space of a cocharacter of $G_{D,\Omega}$ with weights $0,1$. +In particular $L$ belongs to the orbit of any original filtration of +this type under a tensor-preserving comparison. +\end{theorem} + +The proof will construct \(L\) from Lagrangians in finitely many +symplectic spaces. The two lemmas below supply the required simultaneous +choice: minimum intersection with a subspace is ensured by a nonzero +minor of bounded degree, and all such minors over a fixed coefficient +field can be made nonzero over one finite extension. + +\begin{lemma}[Bounded-degree avoidance over a finite extension] +\label{lem:avoidance} +Let $K_1/\Kzero$ be a finite extension, and let $r,b\geq 1$ be integers. +There exist an element $\theta$ algebraic over $\Q_p$ and a point +\[ + (x_0,\ldots,x_{r-1})\in K_1(\theta)^r +\] +at which every nonzero polynomial in $K_1[X_0,\ldots,X_{r-1}]$ of total +degree at most $b$ is nonzero. The coordinates $x_i$ can all be chosen +in $\Q_p(\theta)$. +\end{lemma} + +\begin{proof} +Choose an integer $N>b$ and put +\[ + B=b\sum_{i=0}^{r-1}N^i. +\] +Let $e$ be the ramification index of $K_1/\Kzero$, so that +$v(K_1^\times)=\frac1e\Z$. Choose a prime $q>\max(B,e)$, and choose +$\theta$ with $\theta^q=p$ in an algebraic closure of $K_1$. The value +$v(\theta)=1/q$ has order $q$ modulo $\frac1e\Z$. Consequently the +ramification index, and hence the degree, of $K_1(\theta)/K_1$ is at +least $q$. The polynomial $X^q-p$ gives the reverse inequality, so +\[ + [K_1(\theta):K_1]=q>B. +\] + +Set $x_i=\theta^{N^i}$. Under the substitution $X_i=T^{N^i}$, +distinct monomials of total degree at most $b$ have distinct exponents: +each exponent of an individual variable is smaller than $N$, and +uniqueness follows from base-$N$ expansion. Thus a nonzero polynomial +of total degree at most $b$ becomes a nonzero polynomial in $T$ of +degree at most $B$. It cannot vanish at $\theta$, whose degree over +$K_1$ exceeds $B$. +\end{proof} + +\begin{remark} +The coefficient field in Lemma~\ref{lem:avoidance} is the fixed field +$K_1$. The conclusion concerns all bounded-degree polynomials over that +field, with no restriction on their cardinality. It would be false with +$K_1(\theta)$ in place of $K_1$ as the coefficient field. +\end{remark} + +\begin{lemma}[Intersections with Lagrangians] +\label{lem:lagrangian-intersection} +Let $M$ be a symplectic space of dimension $2m$ over a field $k$ of +characteristic zero, and let $W\subset M$ have dimension $a$. A nonempty +open subset of the Lagrangian Grassmannian consists of subspaces $L$ +such that +\[ + \dim(L\cap W)=\max(0,a-m). +\] +On any fixed symplectic affine big cell, at least one nonzero polynomial +of degree at most $m$ has nonvanishing locus contained in this open +subset. This degree bound is independent of $W$. +\end{lemma} + +\begin{proof} +We first work over an algebraic closure. The Lagrangian Grassmannian +is geometrically integral of dimension $m(m+1)/2$; its affine big cells +are parametrized by symmetric $m\times m$ matrices. If $1\leq a\leq m$, +consider the projective incidence variety of pairs $(\ell,L)$ with +$\ell\subset W$ a line and $\ell\subset L$ a Lagrangian. The fiber over +$\ell$ is the Lagrangian Grassmannian of the symplectic space +$\ell^\perp/\ell$, and therefore the incidence variety has dimension +\[ + a-1+\frac{m(m-1)}2<\frac{m(m+1)}2. +\] +Its image is a proper closed subset. Its complement consists precisely +of the Lagrangians missing $W$. The case $a=0$ is immediate. For +$a>m$, use +\[ + \dim(L\cap W)=a-m+\dim(L\cap W^\perp) +\] +and apply the preceding argument to $W^\perp$. + +For the assertion on equations, choose coordinates on the fixed big +cell in which $L$ is the column space of +\[ + \begin{pmatrix}I_m\\ T\end{pmatrix},\qquad T=T^{\mathsf t}. +\] +The minimum-intersection condition says that the composite of this +matrix with the quotient map $M\longrightarrow M/W$ has rank +$\min(m,2m-a)$. Its entries are affine linear polynomials in the +independent entries of $T$. The nonempty open subset already proved +meets this big cell, so some minor of the required size is a nonzero +polynomial over $k$. Its degree is at most $m$, and its nonvanishing +implies the required rank. If the required rank is zero, use the +constant polynomial $1$. +\end{proof} + + +The two lemmas reduce simultaneous transversality to a uniform bound on +the degrees of the relevant minors. We now identify the symplectic +spaces on which to apply them and the inequalities they must enforce. + +\begin{proof}[Proof of Theorem~\ref{thm:filtration}] +We write $e$ and $u$ also for their cohomological actions on $D$. +These actions commute with Frobenius and satisfy +\begin{equation} +\label{eq:filtration-adjoints} + \beta(ux,y)=\beta(x,uy),\qquad + \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u. +\end{equation} +There is no change to the last relation from contravariance. Indeed, +pulling back $ue=\bar e u$ gives $e^*u^*=u^*\bar e^*$; replacing $e$ +by $\bar e$ gives $u^*e^*=\bar e^*u^*$. Likewise, if $J$ is the +homological polarization matrix, $U^{\mathsf t}J=JU$ implies +$UJ^{-1}=J^{-1}U^{\mathsf t}$, which is the self-adjointness of the +cohomological action $U^{\mathsf t}$ for the dual pairing. The +Rosati-conjugate adjoint identity for $E$ follows in the same way. + +The pullback $U^{\mathsf t}$ of an individual endomorphism is +distinct from the contragredient group action $g\mapsto g^{-\mathsf t}$. +If a homological similitude $g$ has multiplier $c_V$ and satisfies +$\det_E(g)=c_V^s$, its dual action has multiplier $c_D=c_V^{-1}$ +and satisfies $\det_E(g^{-\mathsf t})=c_D^s$. +Thus $G_D$ is the dual form of the homological group defined by this +determinant condition, with its cohomological multiplier. + +\medskip\noindent +\emph{Reduction to the stable subobjects.} +First consider any $E,u$-stable candidate $L\subset D_K$, where +$K/\Kzero$ is finite. We show that the weak-admissibility inequalities +need only be tested on $E,u$-stable subobjects. This resembles the use of canonical +destabilizing objects in \cite[Section~2]{FontaineRapoport2005}, +but we use total defect, not defect divided by dimension, and give +the required elementary argument. +For any isocrystal subobject $P\subset D$ over $\Kzero$, set +\[ + f(P)=t_H(P)-t_N(P). +\] +These values form a finite set: the Hodge number is a bounded integer, +and the Newton number is determined by bounded integer multiplicities +among the finitely many slopes of $D$. Let $a$ be their maximum. +Newton numbers are additive in short exact sequences, while +\[ + (P_K\cap L)+(Q_K\cap L)\subset (P+Q)_K\cap L. +\] +Consequently +\begin{equation} +\label{eq:filtration-supermodularity} + f(P)+f(Q)\leq f(P\cap Q)+f(P+Q). +\end{equation} +If $P$ and $Q$ both attain $a$, the right-hand terms are each at +most $a$, so both attain $a$. Thus maximizers are closed under +intersection and sum. Choose a maximizer $P_{\max}$ of largest +dimension. Adding any other maximizer to it shows that the other +one is contained in it. Hence $P_{\max}$ is the unique largest +maximizer. + +The operators $u$ and $e\in E^\times$ are filtration-preserving +isocrystal automorphisms. They preserve $f$ and hence fix +$P_{\max}$, by its uniqueness. Thus $P_{\max}$ is $E,u$-stable. +Consequently, if $t_H(D')\leq t_N(D')$ for every $E,u$-stable +subobject $D'$, then $a=f(P_{\max})\leq0$. All the +weak-admissibility inequalities follow. We must therefore construct +an $E,u$-stable Lagrangian with these restricted inequalities and +$t_H(D)=t_N(D)$. + +\medskip\noindent +\emph{Semisimplicity and symplectic blocks.} +The Rosati trace form +\[ + \langle a,b\rangle=\Tr(ab^\dagger) + \quad\text{on }\End^0(A_0)\otimes_\Q\R +\] +is positive definite \cite[Theorem~17.3]{MilneAV}. Left multiplication +by $u$ is self-adjoint for this form, since trace cyclicity gives +\[ + \langle ua,b\rangle=\Tr(uab^\dagger) + =\Tr(ab^\dagger u)=\langle a,ub\rangle. +\] +It is therefore diagonalizable. Its minimal polynomial equals the +minimal polynomial of $u$ in the endomorphism algebra, as is seen by +applying a polynomial in the multiplication operator to the identity. +Thus $u$, and hence $u^2$, acts semisimply on $D$. The relation in +\eqref{eq:filtration-adjoints} also shows that $u^2$ commutes with $E$. + +Put +\[ + B(x,y)=\beta(x,uy). +\] +This is a nondegenerate alternating pairing. Indeed, invertibility is +immediate, and +$B(y,x)=-\beta(ux,y)=-B(x,y)$. In addition, +\[ + B(ex,y)=\beta(x,\bar euy)=\beta(x,uey)=B(x,ey), + \qquad B(u^2x,y)=B(x,u^2y). +\] +After a finite splitting extension $K_1/\Kzero$, write +\[ + D_{K_1}=\bigoplus_{\tau,z}M_{\tau,z}, + \qquad + M_{\tau,z}=\{x:ex=\tau(e)x\ (e\in E),\ u^2x=zx\}. +\] +We omit zero summands. The self-adjoint identities for $B$ show that +distinct summands are orthogonal. Its restriction to each summand is +therefore nondegenerate. Write $\dim M_{\tau,z}=2m_{\tau,z}$. + +For $\bar\tau=\tau\circ\overline{(\cdot)}$, the operator $u$ maps +$M_{\tau,z}$ isomorphically to $M_{\bar\tau,z}$. On these spaces, +\begin{equation} +\label{eq:filtration-u-similitude} + B(ux,uy)=zB(x,y). +\end{equation} +Here $z\neq 0$, and $\tau\neq\bar\tau$. Choose a $B$-Lagrangian +$L_{\tau,z}$ in one member of each opposite pair, and prescribe +\begin{equation} +\label{eq:filtration-opposite-choice} + L_{\bar\tau,z}=uL_{\tau,z}. +\end{equation} +Equation~\eqref{eq:filtration-u-similitude} shows that the latter is +Lagrangian as well. Their direct sum $L$ is $E$-stable and $u$-stable; +on applying $u$ twice, one obtains $zL_{\tau,z}=L_{\tau,z}$. Since +\[ + \beta(x,y)=B(x,u^{-1}y), +\] +$L$ is also $\beta$-Lagrangian. Its dimension at each $E$-embedding +is $s$. It remains to make the independent choices appropriately. + +\medskip\noindent +\emph{The inherited Newton grading.} +Use the canonical slope decomposition +\[ + D=\bigoplus_\lambda D_\lambda. +\] +The isocrystal category over $\overline{\F}_p$ is semisimple, and +this decomposition is functorial +\cite[\S3.2, Proposition~3.3]{ColmezFontaine2000}. The slopes of $D$ +belong to $[0,1]$: its integral Dieudonn\'e operators \(F,V\) +satisfy \(FV=p\), so both the slopes of \(F\) and those of \(V\) +are nonnegative. All the endomorphisms above preserve every +$D_\lambda$, and the projectors onto the simultaneous eigenspaces +preserve its scalar extension. Thus each block has a grading +\[ + M_{\tau,z}=\bigoplus_\lambda M_{\tau,z,\lambda},\qquad + M_{\tau,z,\lambda}=M_{\tau,z}\cap(D_\lambda)_{K_1}. +\] +We use this grading even if Frobenius permutes the labels $(\tau,z)$. +No extension of Frobenius to $K_1$, and no isocrystal structure on an +individual block, is required. + +Before scalar extension, $B$ is a pairing of isocrystals into the +slope-$1$ line. Hence $B(D_\lambda,D_\mu)=0$ unless +$\lambda+\mu=1$. Nondegeneracy on an individual block $M$ then gives +perfect pairings between $M_\lambda$ and $M_{1-\lambda}$. In +particular, if $\dim M=2m$, +\begin{equation} +\label{eq:filtration-block-newton} + \sum_\lambda\lambda\dim M_\lambda=m. +\end{equation} + +Let $D'\subset D$ be an isocrystal subobject over $\Kzero$ stable +under $E$ and $u$. Its intersections $M'=D'_{K_1}\cap M$ decompose +$D'_{K_1}$ and inherit the same slope grading. Put $a=\dim M'$. +The slope sum of the graded quotient $M/M'$ is at most $2m-a$, +because all its slopes belong to $[0,1]$. By +\eqref{eq:filtration-block-newton}, +\begin{equation} +\label{eq:filtration-block-bound} + \sum_\lambda\lambda\dim M'_\lambda + \geq \max(0,a-m). +\end{equation} +The additional bound by zero follows from the nonnegative slopes. +The right side is also the smallest possible dimension of the +intersection of a Lagrangian in $M$ with $M'$. +Thus attaining this minimum in every block will give the +weak-admissibility inequalities for all the stable subobjects. + +\medskip\noindent +\emph{Simultaneous choices and finite local field descent.} +We may arrange the splitting field and coordinates used above as +follows. Choose a finite field $k_0\subset\overline{\F}_p$ over +which $A_0$, its polarization, $u$, and a rational basis of $E$ are +defined. The underlying isocrystal and the indicated tensors then +descend to $\operatorname{Frac}W(k_0)$. Choose a finite local field +$H/\Q_p$ containing this field and splitting the actions of $E$ and +$u^2$. The eigenspaces and their symplectic bases can be constructed +over $H$ by linear algebra. We take $K_1=\Kzero H$. In particular, +the independent Lagrangian Grassmannians, fixed symplectic big cells +in them, and the maps induced by $u$ all have coordinates over $H$. + +Take the product of these big cells, one for each independent choice +in \eqref{eq:filtration-opposite-choice}, and let $r$ be its number +of affine coordinates. For every $E,u$-stable isocrystal subobject +$D'$ over $\Kzero$ and each independent block $M$, the space +$M'=D'_{K_1}\cap M$ is defined over the \emph{fixed} field $K_1$. +Lemma~\ref{lem:lagrangian-intersection} supplies a nonzero polynomial +over $K_1$, of degree at most $m$, whose nonvanishing ensures +\begin{equation} +\label{eq:filtration-minimal-intersection} + \dim(L_{\tau,z}\cap M')=\max(0,\dim M'-m). +\end{equation} +We regard this as a polynomial on the whole product by ignoring the +other coordinates. A single bound $b=\max m_{\tau,z}$ works for all +these polynomials, independently of $D'$. On the opposite block, +the same intersection equality follows from $u$-stability of $D'$ +and \eqref{eq:filtration-opposite-choice}. + +Apply Lemma~\ref{lem:avoidance} to $K_1,r,b$. Its one point satisfies +\eqref{eq:filtration-minimal-intersection} for every one of the +subobjects just specified, simultaneously. This use of the lemma +does not assume that the collection of subobjects is countable or +that its members descend to $H$. Their definition over $K_1$ and +the uniform polynomial degree bound are the only requirements. +There are nonzero blocks with $m\geq1$, so $r,b\geq1$ as required. + +Put $K=K_1(\theta)=\Kzero H(\theta)$. The resulting Lagrangian is +defined over $H(\theta)$ in the chosen coordinates, and $H(\theta)$ +is finite over $\Q_p$. Extending the original isocrystal to the +maximal unramified subfield of $H(\theta)$ therefore gives finite +local field descent of the complete filtered data. In particular, +the construction has not obtained its generic point by introducing +transcendental parameters. + +Summing \eqref{eq:filtration-minimal-intersection} and +\eqref{eq:filtration-block-bound} over all blocks gives +\begin{equation} +\label{eq:filtration-stable-test} + t_H(D')\leq t_N(D') + \quad\text{for every $E,u$-stable isocrystal subobject $D'\subset D$.} +\end{equation} +For $D$ itself, both numbers equal $\frac12\dim D$ by +\eqref{eq:filtration-block-newton}. The reduction to stable subobjects +proved at the start now gives all the weak-admissibility inequalities +for subobjects of $D$ over $\Kzero$. Each subobject over the smaller unramified +field of descent extends to one of these subobjects. Hence the +descended filtered isocrystal is weakly admissible as well. + +\medskip\noindent +\emph{The geometric cocharacter orbit.} +Over $\Omega$, the polarization pairs $D_\tau$ only with +$D_{\bar\tau}$. A balanced $E$-stable Lagrangian is determined by an +$s$-dimensional subspace of $D_\tau$ for one member of each pair; +the subspace in the opposite member is its annihilator. The +subgroup of $G_D$ with multiplier $1$ contains the product of +$\SL(D_\tau)$ acting on opposite spaces by the inverse dual +representations. These special linear groups are transitive on +the $s$-dimensional subspaces. Thus all the indicated Lagrangians +form one $G_D(\Omega)$-orbit. + +Finally choose an $E$-stable $\beta$-Lagrangian complement to one +such $L$. Such a complement is obtained separately on each pair +of opposite embedding spaces. Acting by $t$ on $L$ and by $1$ +on the complement gives a cocharacter with multiplier $t$ and +$E$-determinant $t^s$ at each embedding. It lies in $G_D$ and has +the stated filtration. This proves the final assertions. +\end{proof} + +\begin{corollary} +\label{cor:filtration-admissible} +The filtration of Theorem~\ref{thm:filtration}, with its finite local +field descent, is admissible: the descended filtered isocrystal is +the filtered crystalline module of a crystalline representation. +\end{corollary} + +\begin{proof} +Apply the weakly-admissible-implies-admissible theorem of +Colmez--Fontaine with monodromy operator zero +\cite[Theorem~A]{ColmezFontaine2000}. +\end{proof} + +\begin{corollary}[The divisor filtration condition] +\label{cor:filtration-divisor} +Let \(D,L\) be as in Theorem~\ref{thm:filtration}, and let \(\psi\) +and \(U\) denote the homological polarization and the homological +action of \(u\) on \(D_K^\vee\). The untwisted divisor form +\(\psi(\cdot,U\cdot)\) belongs to +\[ + \Fil^1\bigl(\bigwedge^2D_K\bigr)=L\wedge D_K, +\] +and hence to filtration degree zero after its codimension-one Tate +twist. If a marked power is realized by an abelian lift, the +corresponding pulled-back divisor satisfies the filtration condition +for its symmetric homomorphism in Lemma~\ref{lem:lift-hom}. +\end{corollary} + +\begin{proof} +Let \(H\subset D_K^\vee\) be the annihilator of \(L\). It is +Lagrangian for \(\psi\) and stable under \(U\): for \(a\in L\) +and \(h\in H\), +\[ + a(Uh)=(u^*a)(h)=0. +\] +Therefore \(\psi(h,Uh')=0\) for \(h,h'\in H\). The divisor form +thus lies in +\[ + \ker\bigl(\bigwedge^2D_K\longrightarrow\bigwedge^2H^\vee\bigr) + =L\wedge D_K. +\] +This proves the assertion on the fixed filtered vector space. After +realization of a marked power, pullback respects this filtration +condition, which is precisely the condition on the associated +symmetric homomorphism in Lemma~\ref{lem:lift-hom}. +\end{proof} diff --git a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/04_weil.tex b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/04_weil.tex index e3fd56b907ba992b5537800caf08805364902de0..99f61f696a7e9ac935e0915cea3309d500c3c780 100644 --- a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/04_weil.tex +++ b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/04_weil.tex @@ -1,439 +1,439 @@ -\section{Balanced Weil classes}\label{sec:weil} - -Let $A/\Qbar$ have good reduction at the fixed place $w$, and let a CM -field $E$ act on $A$ up to isogeny. We write $e\mapsto\bar e$ for its -complex conjugation. Suppose that a polarization $\lambda$ induces this -involution on $E$. Put -\[ - V=H_1(A(\C),\Q),\qquad \dim_E V=2s,\qquad - g=\dim A=s[E:\Q], -\] -where $s\geq 1$. We assume that the Hodge multiplicities at every -embedding of $E$ are $(s,s)$. The polarization is an alternating form -$\psi:V\times V\longrightarrow\Q(1)$ satisfying -\[ - \psi(ex,y)=\psi(x,\bar e y). -\] - -\begin{definition}\label{def:weil-space} -The rational space of \emph{Weil classes} associated with $E$ is the -subspace -\[ - W_E^s(A)\subset - \bigwedge\nolimits_{\Q}^{2s} V^\vee\otimes\Q(s) - =H^{2s}(A(\C),\Q(s)) -\] -whose scalar extension to a splitting field $\Omega$ of $E$ is -\begin{equation}\label{eq:weil-pure-space} - W_E^s(A)\otimes_\Q\Omega - =\bigoplus_{\tau:E\hookrightarrow\Omega} - \bigwedge\nolimits_\Omega^{2s}V_\tau^\vee - \otimes\Q(s)_\Omega. -\end{equation} -Here each summand is extended by zero on the other eigenspaces. The -right side is Galois stable, so it defines the indicated rational -subspace. -\end{definition} - -The balanced multiplicities imply that every element of $W_E^s(A)$ is -a Hodge class: each determinant in \eqref{eq:weil-pure-space} has -untwisted type $(s,s)$. We use its compatible realizations as an -absolute Hodge class; see \cite{Deligne1982}. The same eigenspace -description of its support holds after specialization, since the -$E$-action specializes compatibly. - -Our goal is the Lefschetz pairing property for these specialized -classes. In fact, the reduction satisfies a dichotomy determined by -its rational endomorphisms. Let $\dagger$ be Rosati for the reduced -polarization and put -\begin{equation}\label{eq:weil-U} - \mathcal U= - \{u\in\End^0(A_0):u^\dagger=u,\quad - ue=\bar e u\ \text{for every }e\in E\}. -\end{equation} - -\begin{proposition}\label{prop:weil} -Every $\gamma\in W_E^s(A)$ satisfies the Lefschetz pairing property -at $w$. More precisely, if $\mathcal U$ in \eqref{eq:weil-U} has no -invertible element, all complementary pairings are zero. If it has -an invertible element, $\gamma_0$ is a simultaneous rational -Lefschetz class. -\end{proposition} - -We first prove the vanishing alternative. For the other alternative, -an invertible $u$ supplies a divisor whose form is nondegenerate on -each $E$-eigenspace separately. We will lift this divisor and use it -to obtain a rational identity for the determinant classes. - -\subsection{The orthogonality alternative} - -\begin{lemma}\label{lem:weil-orthogonality} -If $\mathcal U$ contains no invertible element, every specialization of -a class in $W_E^s(A)$ is orthogonal to all complementary Lefschetz -classes, in each cohomological realization. -\end{lemma} - -\begin{proof} -We prove the contrapositive. A nonzero pairing will produce a divisor -form nondegenerate on one $E$-eigenspace. We will then show that a -rational endomorphism in $\mathcal U$ with this property is invertible. - -Fix one realization and extend its coefficient field to an -algebraically closed field $\Omega$. At the crystalline component we -also include the coefficient extension used for the specialized -class. In this proof, $V_\tau$ also denotes the corresponding -eigenspace in the dual of first cohomology of $A_0$, identified by -specialization and comparison. Write $L^j_\Omega(A_0)$ for the $\Omega$-span of products of -$j$ divisor classes. Since $[E:\Q]\geq2$, we have $g-2s\geq0$. -Hard Lefschetz on the Lefschetz algebra gives -\begin{equation}\label{eq:weil-hard-lefschetz} - L^{g-s}_\Omega(A_0) - =\psi_0^{g-2s}L^s_\Omega(A_0). -\end{equation} -The restriction of hard Lefschetz to this algebra, including for -crystalline cohomology, follows from the Lefschetz correspondences -of \cite[Theorem~5.9]{Milne1999}. - -Suppose a Weil class pairs nontrivially with -$D_1\cdots D_s\psi_0^{g-2s}$, and choose a pure term, supported on -$V_\tau$, whose contribution is nonzero. This term already has -degree $2s$ on $V_\tau$, so no further factor in a nonzero top-degree -product can use that space. A factor from $\psi_0$ which uses -$V_{\bar\tau}$ also uses $V_\tau$; hence the polarization factors -contribute to neither space in this pair. All $2s$ degrees on -$V_{\bar\tau}$ must therefore come from the $s$ divisor factors. -Each has degree two, so each must contribute its restriction to -$V_{\bar\tau}$, and those restrictions have nonzero product. -The coefficient of $t_1\cdots t_s$ in -\[ - (t_1D_1+\cdots+t_sD_s)^s\big|_{V_{\bar\tau}} -\] -is $s!$ times that product. Some $\Omega$-linear combination of the -divisors therefore restricts to a nondegenerate alternating form on -$V_{\bar\tau}$. - -We identify which endomorphisms can give these restrictions. Let -$p_\xi\in E\otimes_\Q\Omega$ be the eigenspace projectors. Rosati -satisfies $p_\xi^\dagger=p_{\bar\xi}$. A divisor corresponds under -$\psi_0$ to a Rosati-symmetric endomorphism $t$. Define -\begin{equation}\label{eq:weil-block-projection} - \Pi(t)=\sum_\xi p_{\bar\xi}t p_\xi. -\end{equation} -This operation takes place inside -$\End^0(A_0)\otimes_\Q\Omega$. Moreover, -\[ - (p_{\bar\xi}t p_\xi)^\dagger - =p_{\bar\xi}t^\dagger p_\xi, - \qquad - \Pi(t)e=\bar e\Pi(t). -\] -Thus $\Pi(t)\in\mathcal U\otimes_\Q\Omega$. Conversely, these are -exactly the blocks allowed by the second equation in -\eqref{eq:weil-U}. For $x,y\in V_\xi$, the form -$\psi_0(x,ty)$ depends only on $p_{\bar\xi}t p_\xi$. -It follows that projecting by \eqref{eq:weil-block-projection} -preserves all the same-embedding restrictions in question. - -Nondegeneracy on $V_{\bar\tau}$ is a polynomial open condition on -$\mathcal U\otimes_\Q\Omega$, and we have shown this open is -nonempty. The rational points of the rational vector space -$\mathcal U$ are Zariski dense after extension to $\Omega$. Hence -there is a rational $u\in\mathcal U$ for which -$\psi_0(\cdot,u\cdot)$ is nondegenerate on $V_{\bar\tau}$. - -To prove invertibility, we use the fact that a nonzero $E$-stable -abelian subvariety has a nonzero homology space at every embedding -of $E$. Suppose that $u$ is not invertible, and choose an integer -$N>0$ such that $Nu$ is an endomorphism. The reduced identity -component $C=((\ker(Nu))^0)_{\mathrm{red}}$ is a -positive-dimensional abelian subvariety. The relation $ue=\bar e u$ makes $C$ stable under -the rational action of $E$. This action is unital and therefore -faithful. Choose a primitive element $e$ of $E$ and an integer -$M>0$ such that $Me$ acts by an integral endomorphism of $C$. -The exterior algebra description of cohomology and normalized -top-degree pullback give, in every realization -\cite[Appendix, Proposition~(A.2)(a),(b)]{Milne1999}, -\[ - \det\bigl(n-e\mid H^1(C)\bigr) - =M^{-2\dim C}\deg\bigl([Mn]-Me\bigr) -\] -for every integer $n$ for which the map on the right is an -isogeny. All but finitely many integers have this property. -The right side is rational and independent of the realization. -Polynomial interpolation therefore gives one characteristic -polynomial in $\Q[T]$, identical also in crystalline cohomology. -Its roots lie among -the conjugates of $e$, and rationality forces their multiplicities to -be equal. Since $C\ne0$, every embedding of $E$ occurs with positive -multiplicity. The same statement holds on first homology, in the -chosen realization. Rational Poincar\'e reducibility -\cite[Proposition~12.1]{MilneAV} makes the inclusion -\(C\hookrightarrow A_0\) injective on first homology in every -realization. This inclusion is \(E\)-equivariant, and its -image is killed by \(u\). Thus the nonzero \(\bar\tau\)-eigenspace -of \(H_1(C)\) shows that the kernel of \(u\) meets -\(V_{\bar\tau}\) nontrivially. Such a vector lies in the radical of -$\psi_0(\cdot,u\cdot)|_{V_{\bar\tau}}$, a contradiction. - -A nonzero pairing would therefore give an invertible element of -$\mathcal U$. Taking the contrapositive proves the lemma. The -argument applies separately to every realization, so all the -pairings in this case have the common rational value zero. -\end{proof} - -\subsection{A divisor identity on a rational summand} - -For the rest of the section, suppose that $u\in\mathcal U$ is -invertible. It gives a rational divisor $D^u$ on $A_0$ with -alternating form -\begin{equation}\label{eq:weil-extra-divisor} - D^u(x,y)=\psi_0(x,uy). -\end{equation} -This form is nondegenerate and pairs each embedding of $E$ only with -itself: $u$ exchanges opposite eigenspaces, whereas $\psi_0$ pairs -opposite eigenspaces. - -Prescribed-filtration transport will replace $A_0$ by a power and -then by an isogenous special fiber $B_0$. We will retain the first -factor through a rational projector on $B_0$, and lift that projector, -the $E$-action on its image, and the transported divisor to $B$. -The next lemma explains the resulting divisor identity: a divisor -nondegenerate separately on those eigenspaces generates their -determinant classes. A rational projector here means an idempotent -in the rational endomorphism algebra. - -\begin{lemma}\label{lem:weil-divisor-span} -Let $B$ be a complex abelian variety, let $R\in\End^0(B)$ be a -rational projector, and suppose that there is an algebra action -$E\longrightarrow R\End^0(B)R$ whose unit maps to $R$. -Let $T=\operatorname{im}(R)$ in rational first homology, with -$\dim_E T=2s$; the indicated endomorphisms act by zero on -$\ker(R)$. Suppose there is a rational divisor class $D$ on -$B$ such that $R^*D=D$ and, after splitting $E$, its alternating form -is -\[ - D=\sum_\tau D_\tau, - \qquad D_\tau\in\bigwedge\nolimits^2 T_\tau^\vee\otimes\Q(1), -\] -with each $D_\tau$ nondegenerate. Then the rational subspace whose -scalar extension is -\[ - \bigoplus_\tau\bigwedge\nolimits^{2s}T_\tau^\vee\otimes\Q(s), -\] -extended by zero on the complementary summand, consists of rational -Lefschetz classes on $B$. -\end{lemma} - -\begin{proof} -For $b\in E$, let $\widetilde b\in\End^0(B)$ act by $b$ on $T$ and -by zero on the complement. Its pullback of a rational divisor is a -rational divisor: clear a denominator in $\widetilde b$, and use -that multiplication by $N$ acts as $N^2$ on $H^2$. Over a splitting -field $\Omega$, choosing Tate-line bases, we have -\[ - (\widetilde b^*D)^s - =\left(\sum_\tau\tau(b)^2D_\tau\right)^s. -\] -The coordinates $x_\tau=\tau(b)$ are independent coordinates on the -scalar extension of the rational vector group underlying $E$. -In the corresponding vector-valued polynomial -\begin{equation}\label{eq:weil-coefficient-polynomial} - P((x_\tau)_\tau)=\left(\sum_\tau x_\tau^2D_\tau\right)^s, -\end{equation} -the coefficient of $x_\tau^{2s}$ is $D_\tau^s$. Nondegeneracy makes -this coefficient nonzero, so it spans -$\bigwedge^{2s}T_\tau^\vee\otimes\Q(s)_\Omega$. - -Let $M$ be the rational span of the Lefschetz classes -$(\widetilde b^*D)^s$ for $b\in E$. The set $E$ of rational points -is Zariski dense in its vector group, also after extension to -$\Omega$. Any linear functional annihilating $M\otimes_\Q\Omega$ -annihilates all evaluations of \eqref{eq:weil-coefficient-polynomial} -on those points. It therefore annihilates the polynomial, and hence -each coefficient. Every $D_\tau^s$ belongs to -$M\otimes_\Q\Omega$. - -This proves the required inclusion after scalar extension. Both -subspaces are rational; applying faithful scalar extension to their -quotient proves the inclusion over $\Q$. In particular it gives -rational coefficients for every rational class in the asserted -subspace, without requiring an individual embedding line to be -defined over $\Q$. -\end{proof} - -\subsection{A lift preserving the divisor} - -We now construct the auxiliary variety to which -Lemma~\ref{lem:weil-divisor-span} applies. To transport the Weil class, -we use the $E$-linear polarization similitudes whose determinant has -the same transformation law as a codimension-$s$ Tate twist. -Choose a rational basis -for the Betti Tate line, let $c$ be the multiplier of -$\GSp(V,\psi)$, and set -\begin{equation}\label{eq:weil-group} - G=\{a\in\GSp(V,\psi)\cap\GL_E(V): - \det_E(a)=c(a)^s\}. -\end{equation} -On points over a $\Q$-algebra $R$, the determinant equality is in -$(E\otimes_\Q R)^\times$, with the right side embedded diagonally. - -\begin{lemma}\label{lem:weil-group} -The Hodge homomorphism of $A$ factors through $G$, and its conjugacy -class defines a Shimura datum of Hodge type through the displayed -embedding. Over an algebraically closed coefficient field, the orbit -of the corresponding filtration on $H^1$ consists exactly of the -$E$-stable Lagrangians for the dual polarization pairing whose -dimension at each embedding of $E$ is $s$. Using a basis of its Tate -line, every alternating form in \eqref{eq:weil-pure-space} transforms -by $c^s$. -\end{lemma} - -\begin{proof} -The polarization pairs $V_\tau$ with $V_{\bar\tau}$, and no other pair -of eigenspaces. Over an algebraically closed field, choose one -embedding in each conjugate pair. An element of $G$ is specified by -$c$ and matrices $a_\tau\in\GL_{2s}$ with -$\det(a_\tau)=c^s$; the matrices on the opposite spaces are then -determined by the polarization. In particular, -\[ - \ker(c:G\longrightarrow\mathbb G_m) - \simeq\prod_{\{\tau,\bar\tau\}}\SL_{2s}. -\] -The multiplier is surjective geometrically. Thus $G$ is connected -reductive, with the same adjoint group as the group of $E$-linear -polarization similitudes. - -The two Hodge characters occur $s$ times at every embedding. Their -product is the polarization multiplier, so the Hodge homomorphism -satisfies the determinant condition in \eqref{eq:weil-group}. Its -weight is rational and central. On the adjoint Lie algebra the Hodge -types are among $(-1,1),(0,0),(1,-1)$, as follows from the weight-one -representation. Conjugation by the image of $i$ is a Cartan -involution, by the polarization. At every real embedding of the -maximal totally real subfield of $E$, the adjoint group has unitary -signature $(s,s)$. Its Hodge homomorphism is nontrivial there, so no -rational simple adjoint factor has trivial Hodge homomorphism. -These are the Shimura-datum conditions for the given Hodge embedding. - -On $H^1$ the polarization still pairs opposite eigenspaces. An -$E$-stable Lagrangian is specified by an $s$-dimensional subspace at -one embedding of each pair; the subspace at the opposite embedding -is its annihilator. The factor $\SL_{2s}$ acts transitively on these -subspaces, as in the cohomological orbit calculation of -Theorem~\ref{thm:filtration}. Finally, an alternating determinant form -evaluated on $a_\tau v_1,\ldots,a_\tau v_{2s}$ is multiplied by -$\det(a_\tau)=c^s$. This proves the asserted transformation law. -\end{proof} - -\begin{proof}[Proof of Proposition~\ref{prop:weil}] -The vanishing alternative is Lemma~\ref{lem:weil-orthogonality}. -Suppose that $u$ is invertible, and use the divisor $D^u$ from -\eqref{eq:weil-extra-divisor}. - -\medskip\noindent -\emph{A filtration preserving the divisor.} - -Let $\mathcal D=H^1_{\mathrm{cris}}(A_0/W(\F))[1/p]$, with its -contravariant convention, and let $\beta$ be the dual polarization -pairing, taking values in the isocrystal line of slope one. The -actions on $\mathcal D$ satisfy -\[ - \beta(ux,y)=\beta(x,uy),\qquad - \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u. -\] -All these endomorphisms and the polarization have finite field -descent. Theorem~\ref{thm:filtration} supplies, over a finite -extension with finite local field descent, a weakly admissible -filtration $\Fil^1\mathcal D=L$ which is $E$- and $u$-stable, -is Lagrangian for $\beta$, and has dimension $s$ at every embedding -of $E$. By Lemma~\ref{lem:weil-group} it lies geometrically in the -required $G$-orbit. The -\hyperref[par:transport-tensor-descent]{finite-descent argument} in -Proposition~\ref{prop:transport} shows that its full finite family of -Hodge-type structure tensors descends from the original number-field -point by crystalline comparison. Taking a compositum with the local -field of definition of $L$ gives the common finite local-field descent -required by its prescribed-filtration clause. - -\medskip\noindent -\emph{Transport and the first rational summand.} -Apply Proposition~\ref{prop:transport} to $G$ and $\gamma$ with this -filtration. It gives a lift $B$, a -quasi-isogeny $a:A_0^n\longrightarrow B_0$, and a rational Hodge -class $\kappa$ on $B$, with -\[ - f_0=aj:A_0\longrightarrow B_0,\qquad - f_0^*\kappa_0=\gamma_0, -\] -simultaneously in the realizations under consideration. The -filtration on $B$, marked by $a$, is the direct sum of $n$ copies of -the chosen filtration. Here $j$ includes the first factor. The map -$f_0$ identifies $A_0$ with a rational direct summand of $B_0$. -Its retraction $q_0$ and the corresponding projector $R_0$ are -\begin{equation}\label{eq:weil-projector-identity} - q_0=\operatorname{pr}_1a^{-1},\qquad R_0=f_0q_0. -\end{equation} -Thus $q_0f_0=\id$ and $R_0^2=R_0$. Pulling back -$f_0^*\kappa_0=\gamma_0$ by $q_0$ gives -\[ - R_0^*\kappa_0=q_0^*\gamma_0. -\] -Act by $E$ on $\operatorname{im}(R_0)$ using $e\mapsto f_0e q_0$, -and by zero on the complement. Also put $D_0=q_0^*D^u$. - -\medskip\noindent -\emph{Lifting the projector, action, and divisor.} -The projector and this $E$-action preserve the repeated filtration. -By Corollary~\ref{cor:filtration-divisor}, $D^u$ satisfies the divisor -lifting condition for the chosen filtration on the first summand. -The crystalline map induced by $q_0=\operatorname{pr}_1a^{-1}$ -respects that filtration and the marked repeated filtration on $B_0$. -Hence $D_0=q_0^*D^u$ satisfies the same condition on $B$. -Apply Lemma~\ref{lem:lift-hom} to \(R_0\), to a finite -\(\Q\)-basis of the action of \(E\) on \(\operatorname{im}(R_0)\), and to the -symmetric homomorphism \(h_{D_0}:B_0\to B_0^\vee\) defined by \(D_0\). -Denote their lifts by \(R\), the corresponding basis of maps, and -\(h_D\). Injectivity of specialization preserves \(R^2=R\), the -rational multiplication table of that basis, and its unit \(R\). -It therefore gives an algebra action \(E\to R\End^0(B)R\). -The symmetric map \(h_D\) defines a rational divisor class \(D\). -Moreover, \(q_0R_0=q_0\), so -\[ - R_0^\vee h_{D_0}R_0=h_{D_0}. -\] -Injectivity lifts this identity to \(R^\vee h_DR=h_D\), which is -exactly \(R^*D=D\). Thus the required projector, action, and divisor -all exist on the one auxiliary variety $B$. - -\medskip\noindent -\emph{The rational identity and its specialization.} -Consider now rational Betti cohomology at a complex realization of -$B$. The summand $T=\operatorname{im}(R)$ has dimension $2s$ at -each embedding of $E$. The form of $D$ is supported on this summand -and pairs each embedding with itself nondegenerately. Also -$R^*\kappa$ is supported on the pure determinant spaces of this -summand. These assertions can be checked after extension to any -one prime-to-$p$ realization: for the divisor they follow from its -specified specialization, and for the Hodge class they follow from -the equality $R_0^*\kappa_0=q_0^*\gamma_0$ and the -pure support of $\gamma_0$. Compatibility of the transported -realizations then gives these support and nondegeneracy assertions -in Betti cohomology. - -Lemma~\ref{lem:weil-divisor-span} now expresses the rational Betti -class $R^*\kappa$ as a finite rational linear combination of products -of rational divisor classes on $B$. The realization and -specialization compatibilities in -Proposition~\ref{prop:transport}, together with those for divisors -and homomorphisms, specialize this identity with the same rational -coefficients in every component. Hence $R_0^*\kappa_0$ is a -simultaneous rational Lefschetz class. Finally, -\[ - \gamma_0=f_0^*R_0^*\kappa_0, -\] -because $R_0f_0=f_0$. Pullback by a rational homomorphism preserves -simultaneous rational Lefschetz classes, so the same is true of -$\gamma_0$. Its complementary traces are therefore one common -rational intersection number, proving the second alternative and -the proposition. -\end{proof} +\section{Balanced Weil classes}\label{sec:weil} + +Let $A/\Qbar$ have good reduction at the fixed place $w$, and let a CM +field $E$ act on $A$ up to isogeny. We write $e\mapsto\bar e$ for its +complex conjugation. Suppose that a polarization $\lambda$ induces this +involution on $E$. Put +\[ + V=H_1(A(\C),\Q),\qquad \dim_E V=2s,\qquad + g=\dim A=s[E:\Q], +\] +where $s\geq 1$. We assume that the Hodge multiplicities at every +embedding of $E$ are $(s,s)$. The polarization is an alternating form +$\psi:V\times V\longrightarrow\Q(1)$ satisfying +\[ + \psi(ex,y)=\psi(x,\bar e y). +\] + +\begin{definition}\label{def:weil-space} +The rational space of \emph{Weil classes} associated with $E$ is the +subspace +\[ + W_E^s(A)\subset + \bigwedge\nolimits_{\Q}^{2s} V^\vee\otimes\Q(s) + =H^{2s}(A(\C),\Q(s)) +\] +whose scalar extension to a splitting field $\Omega$ of $E$ is +\begin{equation}\label{eq:weil-pure-space} + W_E^s(A)\otimes_\Q\Omega + =\bigoplus_{\tau:E\hookrightarrow\Omega} + \bigwedge\nolimits_\Omega^{2s}V_\tau^\vee + \otimes\Q(s)_\Omega. +\end{equation} +Here each summand is extended by zero on the other eigenspaces. The +right side is Galois stable, so it defines the indicated rational +subspace. +\end{definition} + +The balanced multiplicities imply that every element of $W_E^s(A)$ is +a Hodge class: each determinant in \eqref{eq:weil-pure-space} has +untwisted type $(s,s)$. We use its compatible realizations as an +absolute Hodge class; see \cite{Deligne1982}. The same eigenspace +description of its support holds after specialization, since the +$E$-action specializes compatibly. + +Our goal is the Lefschetz pairing property for these specialized +classes. In fact, the reduction satisfies a dichotomy determined by +its rational endomorphisms. Let $\dagger$ be Rosati for the reduced +polarization and put +\begin{equation}\label{eq:weil-U} + \mathcal U= + \{u\in\End^0(A_0):u^\dagger=u,\quad + ue=\bar e u\ \text{for every }e\in E\}. +\end{equation} + +\begin{proposition}\label{prop:weil} +Every $\gamma\in W_E^s(A)$ satisfies the Lefschetz pairing property +at $w$. More precisely, if $\mathcal U$ in \eqref{eq:weil-U} has no +invertible element, all complementary pairings are zero. If it has +an invertible element, $\gamma_0$ is a simultaneous rational +Lefschetz class. +\end{proposition} + +We first prove the vanishing alternative. For the other alternative, +an invertible $u$ supplies a divisor whose form is nondegenerate on +each $E$-eigenspace separately. We will lift this divisor and use it +to obtain a rational identity for the determinant classes. + +\subsection{The orthogonality alternative} + +\begin{lemma}\label{lem:weil-orthogonality} +If $\mathcal U$ contains no invertible element, every specialization of +a class in $W_E^s(A)$ is orthogonal to all complementary Lefschetz +classes, in each cohomological realization. +\end{lemma} + +\begin{proof} +We prove the contrapositive. A nonzero pairing will produce a divisor +form nondegenerate on one $E$-eigenspace. We will then show that a +rational endomorphism in $\mathcal U$ with this property is invertible. + +Fix one realization and extend its coefficient field to an +algebraically closed field $\Omega$. At the crystalline component we +also include the coefficient extension used for the specialized +class. In this proof, $V_\tau$ also denotes the corresponding +eigenspace in the dual of first cohomology of $A_0$, identified by +specialization and comparison. Write $L^j_\Omega(A_0)$ for the $\Omega$-span of products of +$j$ divisor classes. Since $[E:\Q]\geq2$, we have $g-2s\geq0$. +Hard Lefschetz on the Lefschetz algebra gives +\begin{equation}\label{eq:weil-hard-lefschetz} + L^{g-s}_\Omega(A_0) + =\psi_0^{g-2s}L^s_\Omega(A_0). +\end{equation} +The restriction of hard Lefschetz to this algebra, including for +crystalline cohomology, follows from the Lefschetz correspondences +of \cite[Theorem~5.9]{Milne1999}. + +Suppose a Weil class pairs nontrivially with +$D_1\cdots D_s\psi_0^{g-2s}$, and choose a pure term, supported on +$V_\tau$, whose contribution is nonzero. This term already has +degree $2s$ on $V_\tau$, so no further factor in a nonzero top-degree +product can use that space. A factor from $\psi_0$ which uses +$V_{\bar\tau}$ also uses $V_\tau$; hence the polarization factors +contribute to neither space in this pair. All $2s$ degrees on +$V_{\bar\tau}$ must therefore come from the $s$ divisor factors. +Each has degree two, so each must contribute its restriction to +$V_{\bar\tau}$, and those restrictions have nonzero product. +The coefficient of $t_1\cdots t_s$ in +\[ + (t_1D_1+\cdots+t_sD_s)^s\big|_{V_{\bar\tau}} +\] +is $s!$ times that product. Some $\Omega$-linear combination of the +divisors therefore restricts to a nondegenerate alternating form on +$V_{\bar\tau}$. + +We identify which endomorphisms can give these restrictions. Let +$p_\xi\in E\otimes_\Q\Omega$ be the eigenspace projectors. Rosati +satisfies $p_\xi^\dagger=p_{\bar\xi}$. A divisor corresponds under +$\psi_0$ to a Rosati-symmetric endomorphism $t$. Define +\begin{equation}\label{eq:weil-block-projection} + \Pi(t)=\sum_\xi p_{\bar\xi}t p_\xi. +\end{equation} +This operation takes place inside +$\End^0(A_0)\otimes_\Q\Omega$. Moreover, +\[ + (p_{\bar\xi}t p_\xi)^\dagger + =p_{\bar\xi}t^\dagger p_\xi, + \qquad + \Pi(t)e=\bar e\Pi(t). +\] +Thus $\Pi(t)\in\mathcal U\otimes_\Q\Omega$. Conversely, these are +exactly the blocks allowed by the second equation in +\eqref{eq:weil-U}. For $x,y\in V_\xi$, the form +$\psi_0(x,ty)$ depends only on $p_{\bar\xi}t p_\xi$. +It follows that projecting by \eqref{eq:weil-block-projection} +preserves all the same-embedding restrictions in question. + +Nondegeneracy on $V_{\bar\tau}$ is a polynomial open condition on +$\mathcal U\otimes_\Q\Omega$, and we have shown this open is +nonempty. The rational points of the rational vector space +$\mathcal U$ are Zariski dense after extension to $\Omega$. Hence +there is a rational $u\in\mathcal U$ for which +$\psi_0(\cdot,u\cdot)$ is nondegenerate on $V_{\bar\tau}$. + +To prove invertibility, we use the fact that a nonzero $E$-stable +abelian subvariety has a nonzero homology space at every embedding +of $E$. Suppose that $u$ is not invertible, and choose an integer +$N>0$ such that $Nu$ is an endomorphism. The reduced identity +component $C=((\ker(Nu))^0)_{\mathrm{red}}$ is a +positive-dimensional abelian subvariety. The relation $ue=\bar e u$ makes $C$ stable under +the rational action of $E$. This action is unital and therefore +faithful. Choose a primitive element $e$ of $E$ and an integer +$M>0$ such that $Me$ acts by an integral endomorphism of $C$. +The exterior algebra description of cohomology and normalized +top-degree pullback give, in every realization +\cite[Appendix, Proposition~(A.2)(a),(b)]{Milne1999}, +\[ + \det\bigl(n-e\mid H^1(C)\bigr) + =M^{-2\dim C}\deg\bigl([Mn]-Me\bigr) +\] +for every integer $n$ for which the map on the right is an +isogeny. All but finitely many integers have this property. +The right side is rational and independent of the realization. +Polynomial interpolation therefore gives one characteristic +polynomial in $\Q[T]$, identical also in crystalline cohomology. +Its roots lie among +the conjugates of $e$, and rationality forces their multiplicities to +be equal. Since $C\ne0$, every embedding of $E$ occurs with positive +multiplicity. The same statement holds on first homology, in the +chosen realization. Rational Poincar\'e reducibility +\cite[Proposition~12.1]{MilneAV} makes the inclusion +\(C\hookrightarrow A_0\) injective on first homology in every +realization. This inclusion is \(E\)-equivariant, and its +image is killed by \(u\). Thus the nonzero \(\bar\tau\)-eigenspace +of \(H_1(C)\) shows that the kernel of \(u\) meets +\(V_{\bar\tau}\) nontrivially. Such a vector lies in the radical of +$\psi_0(\cdot,u\cdot)|_{V_{\bar\tau}}$, a contradiction. + +A nonzero pairing would therefore give an invertible element of +$\mathcal U$. Taking the contrapositive proves the lemma. The +argument applies separately to every realization, so all the +pairings in this case have the common rational value zero. +\end{proof} + +\subsection{A divisor identity on a rational summand} + +For the rest of the section, suppose that $u\in\mathcal U$ is +invertible. It gives a rational divisor $D^u$ on $A_0$ with +alternating form +\begin{equation}\label{eq:weil-extra-divisor} + D^u(x,y)=\psi_0(x,uy). +\end{equation} +This form is nondegenerate and pairs each embedding of $E$ only with +itself: $u$ exchanges opposite eigenspaces, whereas $\psi_0$ pairs +opposite eigenspaces. + +Prescribed-filtration transport will replace $A_0$ by a power and +then by an isogenous special fiber $B_0$. We will retain the first +factor through a rational projector on $B_0$, and lift that projector, +the $E$-action on its image, and the transported divisor to $B$. +The next lemma explains the resulting divisor identity: a divisor +nondegenerate separately on those eigenspaces generates their +determinant classes. A rational projector here means an idempotent +in the rational endomorphism algebra. + +\begin{lemma}\label{lem:weil-divisor-span} +Let $B$ be a complex abelian variety, let $R\in\End^0(B)$ be a +rational projector, and suppose that there is an algebra action +$E\longrightarrow R\End^0(B)R$ whose unit maps to $R$. +Let $T=\operatorname{im}(R)$ in rational first homology, with +$\dim_E T=2s$; the indicated endomorphisms act by zero on +$\ker(R)$. Suppose there is a rational divisor class $D$ on +$B$ such that $R^*D=D$ and, after splitting $E$, its alternating form +is +\[ + D=\sum_\tau D_\tau, + \qquad D_\tau\in\bigwedge\nolimits^2 T_\tau^\vee\otimes\Q(1), +\] +with each $D_\tau$ nondegenerate. Then the rational subspace whose +scalar extension is +\[ + \bigoplus_\tau\bigwedge\nolimits^{2s}T_\tau^\vee\otimes\Q(s), +\] +extended by zero on the complementary summand, consists of rational +Lefschetz classes on $B$. +\end{lemma} + +\begin{proof} +For $b\in E$, let $\widetilde b\in\End^0(B)$ act by $b$ on $T$ and +by zero on the complement. Its pullback of a rational divisor is a +rational divisor: clear a denominator in $\widetilde b$, and use +that multiplication by $N$ acts as $N^2$ on $H^2$. Over a splitting +field $\Omega$, choosing Tate-line bases, we have +\[ + (\widetilde b^*D)^s + =\left(\sum_\tau\tau(b)^2D_\tau\right)^s. +\] +The coordinates $x_\tau=\tau(b)$ are independent coordinates on the +scalar extension of the rational vector group underlying $E$. +In the corresponding vector-valued polynomial +\begin{equation}\label{eq:weil-coefficient-polynomial} + P((x_\tau)_\tau)=\left(\sum_\tau x_\tau^2D_\tau\right)^s, +\end{equation} +the coefficient of $x_\tau^{2s}$ is $D_\tau^s$. Nondegeneracy makes +this coefficient nonzero, so it spans +$\bigwedge^{2s}T_\tau^\vee\otimes\Q(s)_\Omega$. + +Let $M$ be the rational span of the Lefschetz classes +$(\widetilde b^*D)^s$ for $b\in E$. The set $E$ of rational points +is Zariski dense in its vector group, also after extension to +$\Omega$. Any linear functional annihilating $M\otimes_\Q\Omega$ +annihilates all evaluations of \eqref{eq:weil-coefficient-polynomial} +on those points. It therefore annihilates the polynomial, and hence +each coefficient. Every $D_\tau^s$ belongs to +$M\otimes_\Q\Omega$. + +This proves the required inclusion after scalar extension. Both +subspaces are rational; applying faithful scalar extension to their +quotient proves the inclusion over $\Q$. In particular it gives +rational coefficients for every rational class in the asserted +subspace, without requiring an individual embedding line to be +defined over $\Q$. +\end{proof} + +\subsection{A lift preserving the divisor} + +We now construct the auxiliary variety to which +Lemma~\ref{lem:weil-divisor-span} applies. To transport the Weil class, +we use the $E$-linear polarization similitudes whose determinant has +the same transformation law as a codimension-$s$ Tate twist. +Choose a rational basis +for the Betti Tate line, let $c$ be the multiplier of +$\GSp(V,\psi)$, and set +\begin{equation}\label{eq:weil-group} + G=\{a\in\GSp(V,\psi)\cap\GL_E(V): + \det_E(a)=c(a)^s\}. +\end{equation} +On points over a $\Q$-algebra $R$, the determinant equality is in +$(E\otimes_\Q R)^\times$, with the right side embedded diagonally. + +\begin{lemma}\label{lem:weil-group} +The Hodge homomorphism of $A$ factors through $G$, and its conjugacy +class defines a Shimura datum of Hodge type through the displayed +embedding. Over an algebraically closed coefficient field, the orbit +of the corresponding filtration on $H^1$ consists exactly of the +$E$-stable Lagrangians for the dual polarization pairing whose +dimension at each embedding of $E$ is $s$. Using a basis of its Tate +line, every alternating form in \eqref{eq:weil-pure-space} transforms +by $c^s$. +\end{lemma} + +\begin{proof} +The polarization pairs $V_\tau$ with $V_{\bar\tau}$, and no other pair +of eigenspaces. Over an algebraically closed field, choose one +embedding in each conjugate pair. An element of $G$ is specified by +$c$ and matrices $a_\tau\in\GL_{2s}$ with +$\det(a_\tau)=c^s$; the matrices on the opposite spaces are then +determined by the polarization. In particular, +\[ + \ker(c:G\longrightarrow\mathbb G_m) + \simeq\prod_{\{\tau,\bar\tau\}}\SL_{2s}. +\] +The multiplier is surjective geometrically. Thus $G$ is connected +reductive, with the same adjoint group as the group of $E$-linear +polarization similitudes. + +The two Hodge characters occur $s$ times at every embedding. Their +product is the polarization multiplier, so the Hodge homomorphism +satisfies the determinant condition in \eqref{eq:weil-group}. Its +weight is rational and central. On the adjoint Lie algebra the Hodge +types are among $(-1,1),(0,0),(1,-1)$, as follows from the weight-one +representation. Conjugation by the image of $i$ is a Cartan +involution, by the polarization. At every real embedding of the +maximal totally real subfield of $E$, the adjoint group has unitary +signature $(s,s)$. Its Hodge homomorphism is nontrivial there, so no +rational simple adjoint factor has trivial Hodge homomorphism. +These are the Shimura-datum conditions for the given Hodge embedding. + +On $H^1$ the polarization still pairs opposite eigenspaces. An +$E$-stable Lagrangian is specified by an $s$-dimensional subspace at +one embedding of each pair; the subspace at the opposite embedding +is its annihilator. The factor $\SL_{2s}$ acts transitively on these +subspaces, as in the cohomological orbit calculation of +Theorem~\ref{thm:filtration}. Finally, an alternating determinant form +evaluated on $a_\tau v_1,\ldots,a_\tau v_{2s}$ is multiplied by +$\det(a_\tau)=c^s$. This proves the asserted transformation law. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:weil}] +The vanishing alternative is Lemma~\ref{lem:weil-orthogonality}. +Suppose that $u$ is invertible, and use the divisor $D^u$ from +\eqref{eq:weil-extra-divisor}. + +\medskip\noindent +\emph{A filtration preserving the divisor.} + +Let $\mathcal D=H^1_{\mathrm{cris}}(A_0/W(\F))[1/p]$, with its +contravariant convention, and let $\beta$ be the dual polarization +pairing, taking values in the isocrystal line of slope one. The +actions on $\mathcal D$ satisfy +\[ + \beta(ux,y)=\beta(x,uy),\qquad + \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u. +\] +All these endomorphisms and the polarization have finite field +descent. Theorem~\ref{thm:filtration} supplies, over a finite +extension with finite local field descent, a weakly admissible +filtration $\Fil^1\mathcal D=L$ which is $E$- and $u$-stable, +is Lagrangian for $\beta$, and has dimension $s$ at every embedding +of $E$. By Lemma~\ref{lem:weil-group} it lies geometrically in the +required $G$-orbit. The +\hyperref[par:transport-tensor-descent]{finite-descent argument} in +Proposition~\ref{prop:transport} shows that its full finite family of +Hodge-type structure tensors descends from the original number-field +point by crystalline comparison. Taking a compositum with the local +field of definition of $L$ gives the common finite local-field descent +required by its prescribed-filtration clause. + +\medskip\noindent +\emph{Transport and the first rational summand.} +Apply Proposition~\ref{prop:transport} to $G$ and $\gamma$ with this +filtration. It gives a lift $B$, a +quasi-isogeny $a:A_0^n\longrightarrow B_0$, and a rational Hodge +class $\kappa$ on $B$, with +\[ + f_0=aj:A_0\longrightarrow B_0,\qquad + f_0^*\kappa_0=\gamma_0, +\] +simultaneously in the realizations under consideration. The +filtration on $B$, marked by $a$, is the direct sum of $n$ copies of +the chosen filtration. Here $j$ includes the first factor. The map +$f_0$ identifies $A_0$ with a rational direct summand of $B_0$. +Its retraction $q_0$ and the corresponding projector $R_0$ are +\begin{equation}\label{eq:weil-projector-identity} + q_0=\operatorname{pr}_1a^{-1},\qquad R_0=f_0q_0. +\end{equation} +Thus $q_0f_0=\id$ and $R_0^2=R_0$. Pulling back +$f_0^*\kappa_0=\gamma_0$ by $q_0$ gives +\[ + R_0^*\kappa_0=q_0^*\gamma_0. +\] +Act by $E$ on $\operatorname{im}(R_0)$ using $e\mapsto f_0e q_0$, +and by zero on the complement. Also put $D_0=q_0^*D^u$. + +\medskip\noindent +\emph{Lifting the projector, action, and divisor.} +The projector and this $E$-action preserve the repeated filtration. +By Corollary~\ref{cor:filtration-divisor}, $D^u$ satisfies the divisor +lifting condition for the chosen filtration on the first summand. +The crystalline map induced by $q_0=\operatorname{pr}_1a^{-1}$ +respects that filtration and the marked repeated filtration on $B_0$. +Hence $D_0=q_0^*D^u$ satisfies the same condition on $B$. +Apply Lemma~\ref{lem:lift-hom} to \(R_0\), to a finite +\(\Q\)-basis of the action of \(E\) on \(\operatorname{im}(R_0)\), and to the +symmetric homomorphism \(h_{D_0}:B_0\to B_0^\vee\) defined by \(D_0\). +Denote their lifts by \(R\), the corresponding basis of maps, and +\(h_D\). Injectivity of specialization preserves \(R^2=R\), the +rational multiplication table of that basis, and its unit \(R\). +It therefore gives an algebra action \(E\to R\End^0(B)R\). +The symmetric map \(h_D\) defines a rational divisor class \(D\). +Moreover, \(q_0R_0=q_0\), so +\[ + R_0^\vee h_{D_0}R_0=h_{D_0}. +\] +Injectivity lifts this identity to \(R^\vee h_DR=h_D\), which is +exactly \(R^*D=D\). Thus the required projector, action, and divisor +all exist on the one auxiliary variety $B$. + +\medskip\noindent +\emph{The rational identity and its specialization.} +Consider now rational Betti cohomology at a complex realization of +$B$. The summand $T=\operatorname{im}(R)$ has dimension $2s$ at +each embedding of $E$. The form of $D$ is supported on this summand +and pairs each embedding with itself nondegenerately. Also +$R^*\kappa$ is supported on the pure determinant spaces of this +summand. These assertions can be checked after extension to any +one prime-to-$p$ realization: for the divisor they follow from its +specified specialization, and for the Hodge class they follow from +the equality $R_0^*\kappa_0=q_0^*\gamma_0$ and the +pure support of $\gamma_0$. Compatibility of the transported +realizations then gives these support and nondegeneracy assertions +in Betti cohomology. + +Lemma~\ref{lem:weil-divisor-span} now expresses the rational Betti +class $R^*\kappa$ as a finite rational linear combination of products +of rational divisor classes on $B$. The realization and +specialization compatibilities in +Proposition~\ref{prop:transport}, together with those for divisors +and homomorphisms, specialize this identity with the same rational +coefficients in every component. Hence $R_0^*\kappa_0$ is a +simultaneous rational Lefschetz class. Finally, +\[ + \gamma_0=f_0^*R_0^*\kappa_0, +\] +because $R_0f_0=f_0$. Pullback by a rational homomorphism preserves +simultaneous rational Lefschetz classes, so the same is true of +$\gamma_0$. Its complementary traces are therefore one common +rational intersection number, proving the second alternative and +the proposition. +\end{proof} diff --git a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/05_reduction.tex b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/05_reduction.tex index 4249a3e196a9976f09db766ebce44b458bd663c0..fdf72c10b047fdc86f9192781880cfc791c3c9bb 100644 --- a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/05_reduction.tex +++ b/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/05_reduction.tex @@ -1,448 +1,448 @@ -\section{CM reduction and consequences} -\label{sec:reduction} - -The remaining step in the main proof is to express CM Hodge classes -as pullbacks of balanced Weil classes. Together with CM transport -and Proposition~\ref{prop:weil}, this will prove -Theorem~\ref{thm:main}. We then derive two specialization consequences. - -\subsection{CM classes as pullbacks of Weil classes} - -We give the CM reduction in the form needed here. The stronger -reduction to classes on abelian varieties of split Weil type is due to -Deligne and Andr\'e \cite{Andre1992}; see the reconstruction in -\cite[Theorem~1 and Section~4]{Milne2020}. Only balanced signatures -and a compatible polarization are needed for our application, and -we prove this form directly using rational idempotents on powers. - -\begin{lemma}[CM reduction] -\label{lem:cm-reduction} -Let $A/\Qbar$ be an abelian variety of CM type with good reduction at $w$, -and let $r>0$. For every rational Hodge class -$\gamma\in H^{2r}(A(\C),\Q(r))$, there are finitely many abelian varieties -$B_i/\Qbar$ with good reduction at $w$, rational homomorphisms -$f_i\colon A\to B_i$, and rational Weil classes -$\eta_i\in H^{2r}(B_i(\C),\Q(r))$ such that -\begin{equation} -\label{eq:cm-pullback-decomposition} - \gamma=\sum_i f_i^*\eta_i. -\end{equation} -Each $B_i$ has an action of a CM field $K$ for which $H_1(B_i(\C),\Q)$ -has rank $2r$ over $K$, the Hodge signatures are $(r,r)$ at every -embedding of $K$, and a polarization induces complex conjugation on $K$. -The equality specializes simultaneously in all the realizations used in -Theorem~\ref{thm:main}. -\end{lemma} - -\begin{proof} -We first identify the eigenspace components that can support a rational -Hodge class. For each such component we construct a balanced abelian -factor of a power of $A$, whose Weil classes pull back onto that -component. Taking the sum of these rational pullback spaces will then -give the decomposition over $\Q$. - -\medskip\noindent -\emph{Balanced eigenspace components.} -Set $V=H_1(A(\C),\Q)$. We work throughout up to isogeny. There is a -CM algebra $E'$, a product of CM fields, acting on $A$ such that $V$ is -free of rank one over $E'$. To obtain it, decompose $A$ up to isogeny -into simple CM factors and take the product of their CM fields, with a -separate label for each repeated factor. Transporting this action back -along an isogeny gives the asserted action on $A$ itself. - -Choose a finite Galois CM field $K\subset\Qbar$ containing the images of -all the factors of $E'$, and put $S=\Hom_{\Q\text{-alg}}(E',K)$. Then -\[ - V_K:=V\otimes_{\Q}K=\bigoplus_{\rho\in S}V_\rho, - \qquad \dim_K V_\rho=1. -\] -The Hodge decomposition is diagonal for this decomposition after -extension to $\C$. We suppress the common Tate twist in the following -linear-algebra calculations. Alternating forms of degree $2r$ decompose -as -\[ - \bigwedge^{2r}V_K^* - =\bigoplus_{\substack{\Delta\subset S\\ |\Delta|=2r}}L_\Delta, - \qquad - L_\Delta= - \bigwedge^{2r}\left(\bigoplus_{\rho\in\Delta}V_\rho\right)^*, -\] -where each form on the right is extended by zero on the other summands. -Every $L_\Delta$ is one-dimensional and has a definite Hodge type. - -Suppose the component of $\gamma$ in $L_\Delta$ is nonzero. Since -$\gamma$ is rational, its component in $L_{\sigma\Delta}$ is the -semilinear conjugate of that component and is nonzero for every -$\sigma\in\Gal(K/\Q)$. The Hodge condition therefore implies that -every $\sigma\Delta$ contains $r$ lines of each of the two Hodge types. -We call such a subset balanced under all conjugates. - -\medskip\noindent -\emph{An abelian factor with balanced signatures.} -Fix one of these subsets $\Delta$. Form the abelian variety -$A\otimes_{\Q}K$ in the category up to isogeny: it is a power of $A$ -whose index space is the underlying rational vector space of $K$. -The regular action of $K$ on this index space is an action by rational -endomorphisms of the power. Its rational homology is -\[ - T=V\otimes_{\Q}K,\qquad - \dim_\Q T=[K:\Q]\dim_\Q V. -\] -In contrast with the coefficient space $V_K$, we regard $T$ here as a -$\Q$-vector space, with $K$ acting on its second factor. The abelian -variety remains over $\Qbar$. The actions on the two factors give -an action of the rational algebra -\[ - E'\otimes_{\Q}K\simeq\prod_{\rho\in S}K, - \qquad e\otimes k\longmapsto\bigl(\rho(e)k\bigr)_{\rho\in S}. -\] -Let $e_\Delta$ be the idempotent with coordinates $1$ on $\Delta$ and -$0$ elsewhere. It is a rational endomorphism of $A\otimes_{\Q}K$, -so its image is an abelian variety $B_\Delta$ up to isogeny. In -particular no Galois invariance of $\Delta$ is required for this -construction. Write -\[ - U=H_1(B_\Delta(\C),\Q)=e_\Delta T. -\] -The index-field action of $K$ preserves $U$. - -Now extend the coefficients of $U$ to $K$, and use the fixed inclusion -$K\subset\C$ to identify the embeddings of $K$ with $\Gal(K/\Q)$. -Evaluation of the index factor at $\tau$ identifies its eigenspace in -$T\otimes_\Q K$ with $V_K$. In that eigenspace the idempotent retains -exactly the summands indexed by $\tau\Delta$. Thus -\begin{equation} -\label{eq:cm-embedding-summand} - U_\tau=\bigoplus_{\rho\in\Delta}V_{\tau\rho}, -\end{equation} -and the same description holds after extension to $\C$. Explicitly, -evaluation of the idempotent at the pair of embeddings $(\xi,\tau)$ is $1$ exactly -when $\xi=\tau\rho$ for some $\rho\in\Delta$. Thus -$\dim_K U_\tau=2r$ for every $\tau$. Since $K$ is Galois and -$\Delta$ is balanced under all conjugates, each $U_\tau$ has $r$ lines -of each Hodge type. Consequently $U$ has rank $2r$ over $K$, -$\dim B_\Delta=r[K:\Q]$, and the required Hodge signatures are -$(r,r)$. - -\medskip\noindent -\emph{A compatible polarization.} -Let $\psi$ be a -rational polarization form on $V$, and equip the index space $K$ with -the positive definite rational symmetric form -\[ - t(k,l)=\Tr_{K/\Q}(k\bar l). -\] -The form $\Psi=\psi\otimes t$ is a rational polarization form on the -power $A\otimes_{\Q}K$: its alternating and Hodge properties follow -from those of $\psi$, and its positivity follows by tensoring the -positive symmetric form associated to $\psi$ with $t$. A positive -integral multiple gives an actual polarization. For $a\in K$, -\[ - t(ak,l)=t(k,\bar a l), -\] -so the Rosati involution of $\Psi$ induces conjugation on the -index-field action. Restrict $\Psi$ to $B_\Delta$. This is again a -polarization, and because $U$ is $K$-stable its adjoint identity still -induces conjugation on $K$. This restriction does not require -$e_\Delta$ to be self-adjoint. - -\medskip\noindent -\emph{Pullback and descent of the rational span.} -The factor $B_\Delta$ now has the field action, balanced signatures, -and polarization needed to define its Weil classes. The rational -homomorphism $f_\Delta\colon A\to B_\Delta$ defined on -homology by -\[ - v\longmapsto e_\Delta(v\otimes 1) -\] -has the needed effect on Weil classes. After splitting coefficients, -its component at the identity embedding of $K$ is the projection -\[ - \bigoplus_{\rho\in S}V_\rho - \longrightarrow - \bigoplus_{\rho\in\Delta}V_\rho. -\] -Thus pullback maps $\bigwedge^{2r}U_{\id}^*$ isomorphically onto -$L_\Delta$. More generally the component indexed by $\tau$ pulls back -onto $L_{\tau\Delta}$. The rational Weil-class space $W_K^r(B_\Delta)$ -is characterized, with the twist restored, by -\[ - W_K^r(B_\Delta)\otimes_{\Q}K - =\bigoplus_{\tau:K\hookrightarrow K} - \bigwedge^{2r}U_\tau^*(r). -\] -In particular $f_\Delta^*W_K^r(B_\Delta)$ is a rational subspace whose -extension to $K$ contains $L_\Delta(r)$. - -Take the sum of these rational pullback spaces over the finitely many -subsets supporting $\gamma$. Its scalar extension contains every -component of $\gamma$, and therefore contains $\gamma$ itself. Since -extension of scalars is injective on the quotient by a rational -subspace, the sum already contains $\gamma$ over $\Q$. This proves -\eqref{eq:cm-pullback-decomposition} with rational Weil classes. Notice -that an individual embedding component need not descend to $\Q$; -only the entire Weil-class space and its pullback are used in this -descent argument. - -\medskip\noindent -\emph{Good reduction and specialization.} -All the constructions use finitely many rational endomorphisms of -powers of $A$ and their images. They therefore exist over $\Qbar$ and -descend, together with the maps, to a common number field after a finite -extension. Choose that extension so that $A$ has good reduction at the -place specified by $w$. For any $\ell\ne p$, the rational Tate module -of $B_\Delta$ is a Galois-stable direct summand of the rational Tate -module of the power of $A$. It is unramified at this place. The -N\'eron--Ogg--Shafarevich criterion gives good reduction of -$B_\Delta$ there; it also shows that the isogenies used in the -construction do not affect this conclusion -\cite[Chapter~IV, Theorem~3.5]{MilneAVNotes}. Rational homomorphisms between -the resulting good-reduction varieties specialize after clearing -denominators. The equality of rational Hodge classes -\eqref{eq:cm-pullback-decomposition} is respected by their compatible -realizations and by specialization, with the same rational coefficients -in every component. This proves the last assertion. -\end{proof} - -\begin{corollary} -\label{cor:cm-pairing} -The Lefschetz pairing property holds for the specialization of every -rational Hodge class on a CM abelian variety over $\Qbar$ with good -reduction at $w$. -\end{corollary} - -\begin{proof} -In codimension zero the class is a rational multiple of the unit, so -the assertion follows from the compatibility of divisor intersection -numbers with the normalized traces. In positive codimension use -Lemma~\ref{lem:cm-reduction}. Proposition~\ref{prop:weil} gives the -Lefschetz pairing property for the specialization of each $\eta_i$. -Lemma~\ref{lem:pairing-functorial} preserves this property under -rational pullback and rational linear combinations. Applying it to -\eqref{eq:cm-pullback-decomposition} proves the assertion. -\end{proof} - -\subsection{Proof of the main theorem} - -\begin{proof}[Proof of Theorem~\ref{thm:main}] -The case \(r=0\) follows from divisor intersection numbers as in the -preceding corollary. This includes \(\dim A=0\), where the abelian -variety is a point. If \(r=\dim A\), a rational top class is a -rational multiple of the point class; its normalized trace is -preserved by specialization, and the complementary product is \(1\). -We may therefore assume \(00$, take $q$ times the -point class, where $q\in\Q$ is the normalized trace of $\gamma$. -The point class belongs to $\mathcal R^d(A_0)$ -\cite[\S2.3]{Milne2009}, and its normalized trace is one. Compatibility -of traces with specialization gives the required components in these -cases. - -Suppose now that $00\) such that \(h=Nb\) is an actual -homomorphism \(A_0\to B_0\). Pullback by \(h\) preserves the theory -by axiom \textup{(R1)} of \cite[Definition~2.1]{Milne2009}. Since -\(\mathcal R^r(A_0)\) is a \(\Q\)-vector space, we may set -\[ - \rho_\gamma=N^{-2r}h^*\rho_\kappa\in\mathcal R^r(A_0). -\] -Functoriality of the component maps and -\eqref{eq:rational-pullback} give -\begin{align*} - (\rho_\gamma)_\ell - &=N^{-2r}h^*\kappa_{0,\ell} - =b^*\kappa_{0,\ell}=\gamma_{0,\ell} - &&(\ell\ne p),\\ - (\rho_\gamma)_p\otimes1 - &=N^{-2r}h^*\kappa_{0,p} - =b^*\kappa_{0,p}=\gamma_{0,p}. -\end{align*} -The second line is taken after extension from $\Kzero$ to $\Cw$. -Pullback in \textup{(R1)} applies to any homomorphism $h$, and rational -scalar multiplication allows any denominator $N$. Thus the argument -requires neither an isogeny nor separability, and remains valid when -$p\mid N$, including at $p=2$. - -Finally, two classes with the required components have the same -\(\ell\)-adic component for any one \(\ell\ne p\). Injectivity of -that component map by \textup{(R4)} makes them equal. Thus the class -is unique in the fixed theory, independently of the auxiliary choices. -\end{proof} - -This cohomological membership statement alone gives no algebraic cycle -and proves neither the Hodge nor the Tate conjecture. The construction -takes place at the fixed -reduction place: its CM auxiliary may have dimension $n\dim A$, and -the map $b$ is between special fibers. It supplies no same-dimensional -CM lift or map between the generic fibers. - -\subsection{Algebraicity after good reduction} - -The CM Hodge theorem \cite[Theorem~1.1]{OpenAICMHodge2026}, combined -with our CM transport, strengthens the preceding specialization -statement. It is an additional input to the following consequence, -not to the proof of Theorem~\ref{thm:main}. - -\begin{corollary}[Algebraic specialization of rational Hodge classes] -\label{cor:algebraic-specialization} -Fix the prime and place as in Theorem~\ref{thm:main}. Let -\(A/\Qbar\) be an abelian variety of dimension \(d\) with good -reduction \(A_0/\F\) at \(w\), and let -\(\gamma\in H^{2r}(A(\C),\Q(r))\) be a rational Hodge class, -where \(0\leq r\leq d\). There is one rational algebraic cycle -\(z_\gamma\in\operatorname{CH}^r(A_0)_{\Q}\) such that -\[ - \operatorname{cl}_\ell(z_\gamma)=\gamma_{0,\ell} - \quad(\ell\ne p),\qquad - \operatorname{cl}_{\mathrm{cris}}(z_\gamma)\otimes1=\gamma_{0,p}. -\] -The crystalline equality is in -\[ - H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r) - \otimes_{\Kzero}\Cw. -\] -This holds for every prime \(p\), including \(p=2\). -Consequently, for every -\(y\in\operatorname{CH}^{d-r}(A_0)_{\Q}\), the traces of -\(\gamma_{0,\ell}\operatorname{cl}_\ell(y)\) for \(\ell\ne p\) -and of \(\gamma_{0,p}\operatorname{cl}_{\mathrm{cris}}(y)\) -are all the same rational number \(\deg(z_\gamma\cdot y)\). -\end{corollary} - -\begin{proof} -In codimension zero or top codimension, use the corresponding rational -multiple of the unit or the point class, as in the preceding proof. -This includes dimension zero. For $00\) with \(h=Nb\) an actual homomorphism -\(A_0\to B_0\), and put -\[ - z_\gamma=N^{-2r}h^*Z_0\in\operatorname{CH}^r(A_0)_{\Q}. -\] -Here \(h^*\) is the Chow pullback between smooth varieties, defined -by the graph and refined intersection; it requires neither flatness -nor separability of \(h\). Functoriality of cycle classes and -\eqref{eq:rational-pullback} give the required equalities. Rational -coefficients permit this denominator even if \(p\mid N\); the -crystalline realization has already inverted \(p\). Finally, -compatibility of cycle classes with products and normalized traces -gives the asserted intersection number for every \(y\). -\end{proof} - -This conclusion concerns only the special fiber at the fixed place. -It neither asserts the Hodge conjecture for arbitrary abelian varieties -in characteristic zero nor lifts \(z_\gamma\) to the original -generic fiber. The representing cycle is not asserted to be unique -modulo rational equivalence, and no comparison between different -reduction places is made. +\section{CM reduction and consequences} +\label{sec:reduction} + +The remaining step in the main proof is to express CM Hodge classes +as pullbacks of balanced Weil classes. Together with CM transport +and Proposition~\ref{prop:weil}, this will prove +Theorem~\ref{thm:main}. We then derive two specialization consequences. + +\subsection{CM classes as pullbacks of Weil classes} + +We give the CM reduction in the form needed here. The stronger +reduction to classes on abelian varieties of split Weil type is due to +Deligne and Andr\'e \cite{Andre1992}; see the reconstruction in +\cite[Theorem~1 and Section~4]{Milne2020}. Only balanced signatures +and a compatible polarization are needed for our application, and +we prove this form directly using rational idempotents on powers. + +\begin{lemma}[CM reduction] +\label{lem:cm-reduction} +Let $A/\Qbar$ be an abelian variety of CM type with good reduction at $w$, +and let $r>0$. For every rational Hodge class +$\gamma\in H^{2r}(A(\C),\Q(r))$, there are finitely many abelian varieties +$B_i/\Qbar$ with good reduction at $w$, rational homomorphisms +$f_i\colon A\to B_i$, and rational Weil classes +$\eta_i\in H^{2r}(B_i(\C),\Q(r))$ such that +\begin{equation} +\label{eq:cm-pullback-decomposition} + \gamma=\sum_i f_i^*\eta_i. +\end{equation} +Each $B_i$ has an action of a CM field $K$ for which $H_1(B_i(\C),\Q)$ +has rank $2r$ over $K$, the Hodge signatures are $(r,r)$ at every +embedding of $K$, and a polarization induces complex conjugation on $K$. +The equality specializes simultaneously in all the realizations used in +Theorem~\ref{thm:main}. +\end{lemma} + +\begin{proof} +We first identify the eigenspace components that can support a rational +Hodge class. For each such component we construct a balanced abelian +factor of a power of $A$, whose Weil classes pull back onto that +component. Taking the sum of these rational pullback spaces will then +give the decomposition over $\Q$. + +\medskip\noindent +\emph{Balanced eigenspace components.} +Set $V=H_1(A(\C),\Q)$. We work throughout up to isogeny. There is a +CM algebra $E'$, a product of CM fields, acting on $A$ such that $V$ is +free of rank one over $E'$. To obtain it, decompose $A$ up to isogeny +into simple CM factors and take the product of their CM fields, with a +separate label for each repeated factor. Transporting this action back +along an isogeny gives the asserted action on $A$ itself. + +Choose a finite Galois CM field $K\subset\Qbar$ containing the images of +all the factors of $E'$, and put $S=\Hom_{\Q\text{-alg}}(E',K)$. Then +\[ + V_K:=V\otimes_{\Q}K=\bigoplus_{\rho\in S}V_\rho, + \qquad \dim_K V_\rho=1. +\] +The Hodge decomposition is diagonal for this decomposition after +extension to $\C$. We suppress the common Tate twist in the following +linear-algebra calculations. Alternating forms of degree $2r$ decompose +as +\[ + \bigwedge^{2r}V_K^* + =\bigoplus_{\substack{\Delta\subset S\\ |\Delta|=2r}}L_\Delta, + \qquad + L_\Delta= + \bigwedge^{2r}\left(\bigoplus_{\rho\in\Delta}V_\rho\right)^*, +\] +where each form on the right is extended by zero on the other summands. +Every $L_\Delta$ is one-dimensional and has a definite Hodge type. + +Suppose the component of $\gamma$ in $L_\Delta$ is nonzero. Since +$\gamma$ is rational, its component in $L_{\sigma\Delta}$ is the +semilinear conjugate of that component and is nonzero for every +$\sigma\in\Gal(K/\Q)$. The Hodge condition therefore implies that +every $\sigma\Delta$ contains $r$ lines of each of the two Hodge types. +We call such a subset balanced under all conjugates. + +\medskip\noindent +\emph{An abelian factor with balanced signatures.} +Fix one of these subsets $\Delta$. Form the abelian variety +$A\otimes_{\Q}K$ in the category up to isogeny: it is a power of $A$ +whose index space is the underlying rational vector space of $K$. +The regular action of $K$ on this index space is an action by rational +endomorphisms of the power. Its rational homology is +\[ + T=V\otimes_{\Q}K,\qquad + \dim_\Q T=[K:\Q]\dim_\Q V. +\] +In contrast with the coefficient space $V_K$, we regard $T$ here as a +$\Q$-vector space, with $K$ acting on its second factor. The abelian +variety remains over $\Qbar$. The actions on the two factors give +an action of the rational algebra +\[ + E'\otimes_{\Q}K\simeq\prod_{\rho\in S}K, + \qquad e\otimes k\longmapsto\bigl(\rho(e)k\bigr)_{\rho\in S}. +\] +Let $e_\Delta$ be the idempotent with coordinates $1$ on $\Delta$ and +$0$ elsewhere. It is a rational endomorphism of $A\otimes_{\Q}K$, +so its image is an abelian variety $B_\Delta$ up to isogeny. In +particular no Galois invariance of $\Delta$ is required for this +construction. Write +\[ + U=H_1(B_\Delta(\C),\Q)=e_\Delta T. +\] +The index-field action of $K$ preserves $U$. + +Now extend the coefficients of $U$ to $K$, and use the fixed inclusion +$K\subset\C$ to identify the embeddings of $K$ with $\Gal(K/\Q)$. +Evaluation of the index factor at $\tau$ identifies its eigenspace in +$T\otimes_\Q K$ with $V_K$. In that eigenspace the idempotent retains +exactly the summands indexed by $\tau\Delta$. Thus +\begin{equation} +\label{eq:cm-embedding-summand} + U_\tau=\bigoplus_{\rho\in\Delta}V_{\tau\rho}, +\end{equation} +and the same description holds after extension to $\C$. Explicitly, +evaluation of the idempotent at the pair of embeddings $(\xi,\tau)$ is $1$ exactly +when $\xi=\tau\rho$ for some $\rho\in\Delta$. Thus +$\dim_K U_\tau=2r$ for every $\tau$. Since $K$ is Galois and +$\Delta$ is balanced under all conjugates, each $U_\tau$ has $r$ lines +of each Hodge type. Consequently $U$ has rank $2r$ over $K$, +$\dim B_\Delta=r[K:\Q]$, and the required Hodge signatures are +$(r,r)$. + +\medskip\noindent +\emph{A compatible polarization.} +Let $\psi$ be a +rational polarization form on $V$, and equip the index space $K$ with +the positive definite rational symmetric form +\[ + t(k,l)=\Tr_{K/\Q}(k\bar l). +\] +The form $\Psi=\psi\otimes t$ is a rational polarization form on the +power $A\otimes_{\Q}K$: its alternating and Hodge properties follow +from those of $\psi$, and its positivity follows by tensoring the +positive symmetric form associated to $\psi$ with $t$. A positive +integral multiple gives an actual polarization. For $a\in K$, +\[ + t(ak,l)=t(k,\bar a l), +\] +so the Rosati involution of $\Psi$ induces conjugation on the +index-field action. Restrict $\Psi$ to $B_\Delta$. This is again a +polarization, and because $U$ is $K$-stable its adjoint identity still +induces conjugation on $K$. This restriction does not require +$e_\Delta$ to be self-adjoint. + +\medskip\noindent +\emph{Pullback and descent of the rational span.} +The factor $B_\Delta$ now has the field action, balanced signatures, +and polarization needed to define its Weil classes. The rational +homomorphism $f_\Delta\colon A\to B_\Delta$ defined on +homology by +\[ + v\longmapsto e_\Delta(v\otimes 1) +\] +has the needed effect on Weil classes. After splitting coefficients, +its component at the identity embedding of $K$ is the projection +\[ + \bigoplus_{\rho\in S}V_\rho + \longrightarrow + \bigoplus_{\rho\in\Delta}V_\rho. +\] +Thus pullback maps $\bigwedge^{2r}U_{\id}^*$ isomorphically onto +$L_\Delta$. More generally the component indexed by $\tau$ pulls back +onto $L_{\tau\Delta}$. The rational Weil-class space $W_K^r(B_\Delta)$ +is characterized, with the twist restored, by +\[ + W_K^r(B_\Delta)\otimes_{\Q}K + =\bigoplus_{\tau:K\hookrightarrow K} + \bigwedge^{2r}U_\tau^*(r). +\] +In particular $f_\Delta^*W_K^r(B_\Delta)$ is a rational subspace whose +extension to $K$ contains $L_\Delta(r)$. + +Take the sum of these rational pullback spaces over the finitely many +subsets supporting $\gamma$. Its scalar extension contains every +component of $\gamma$, and therefore contains $\gamma$ itself. Since +extension of scalars is injective on the quotient by a rational +subspace, the sum already contains $\gamma$ over $\Q$. This proves +\eqref{eq:cm-pullback-decomposition} with rational Weil classes. Notice +that an individual embedding component need not descend to $\Q$; +only the entire Weil-class space and its pullback are used in this +descent argument. + +\medskip\noindent +\emph{Good reduction and specialization.} +All the constructions use finitely many rational endomorphisms of +powers of $A$ and their images. They therefore exist over $\Qbar$ and +descend, together with the maps, to a common number field after a finite +extension. Choose that extension so that $A$ has good reduction at the +place specified by $w$. For any $\ell\ne p$, the rational Tate module +of $B_\Delta$ is a Galois-stable direct summand of the rational Tate +module of the power of $A$. It is unramified at this place. The +N\'eron--Ogg--Shafarevich criterion gives good reduction of +$B_\Delta$ there; it also shows that the isogenies used in the +construction do not affect this conclusion +\cite[Chapter~IV, Theorem~3.5]{MilneAVNotes}. Rational homomorphisms between +the resulting good-reduction varieties specialize after clearing +denominators. The equality of rational Hodge classes +\eqref{eq:cm-pullback-decomposition} is respected by their compatible +realizations and by specialization, with the same rational coefficients +in every component. This proves the last assertion. +\end{proof} + +\begin{corollary} +\label{cor:cm-pairing} +The Lefschetz pairing property holds for the specialization of every +rational Hodge class on a CM abelian variety over $\Qbar$ with good +reduction at $w$. +\end{corollary} + +\begin{proof} +In codimension zero the class is a rational multiple of the unit, so +the assertion follows from the compatibility of divisor intersection +numbers with the normalized traces. In positive codimension use +Lemma~\ref{lem:cm-reduction}. Proposition~\ref{prop:weil} gives the +Lefschetz pairing property for the specialization of each $\eta_i$. +Lemma~\ref{lem:pairing-functorial} preserves this property under +rational pullback and rational linear combinations. Applying it to +\eqref{eq:cm-pullback-decomposition} proves the assertion. +\end{proof} + +\subsection{Proof of the main theorem} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +The case \(r=0\) follows from divisor intersection numbers as in the +preceding corollary. This includes \(\dim A=0\), where the abelian +variety is a point. If \(r=\dim A\), a rational top class is a +rational multiple of the point class; its normalized trace is +preserved by specialization, and the complementary product is \(1\). +We may therefore assume \(00$, take $q$ times the +point class, where $q\in\Q$ is the normalized trace of $\gamma$. +The point class belongs to $\mathcal R^d(A_0)$ +\cite[\S2.3]{Milne2009}, and its normalized trace is one. Compatibility +of traces with specialization gives the required components in these +cases. + +Suppose now that $00\) such that \(h=Nb\) is an actual +homomorphism \(A_0\to B_0\). Pullback by \(h\) preserves the theory +by axiom \textup{(R1)} of \cite[Definition~2.1]{Milne2009}. Since +\(\mathcal R^r(A_0)\) is a \(\Q\)-vector space, we may set +\[ + \rho_\gamma=N^{-2r}h^*\rho_\kappa\in\mathcal R^r(A_0). +\] +Functoriality of the component maps and +\eqref{eq:rational-pullback} give +\begin{align*} + (\rho_\gamma)_\ell + &=N^{-2r}h^*\kappa_{0,\ell} + =b^*\kappa_{0,\ell}=\gamma_{0,\ell} + &&(\ell\ne p),\\ + (\rho_\gamma)_p\otimes1 + &=N^{-2r}h^*\kappa_{0,p} + =b^*\kappa_{0,p}=\gamma_{0,p}. +\end{align*} +The second line is taken after extension from $\Kzero$ to $\Cw$. +Pullback in \textup{(R1)} applies to any homomorphism $h$, and rational +scalar multiplication allows any denominator $N$. Thus the argument +requires neither an isogeny nor separability, and remains valid when +$p\mid N$, including at $p=2$. + +Finally, two classes with the required components have the same +\(\ell\)-adic component for any one \(\ell\ne p\). Injectivity of +that component map by \textup{(R4)} makes them equal. Thus the class +is unique in the fixed theory, independently of the auxiliary choices. +\end{proof} + +This cohomological membership statement alone gives no algebraic cycle +and proves neither the Hodge nor the Tate conjecture. The construction +takes place at the fixed +reduction place: its CM auxiliary may have dimension $n\dim A$, and +the map $b$ is between special fibers. It supplies no same-dimensional +CM lift or map between the generic fibers. + +\subsection{Algebraicity after good reduction} + +The CM Hodge theorem \cite[Theorem~1.1]{OpenAICMHodge2026}, combined +with our CM transport, strengthens the preceding specialization +statement. It is an additional input to the following consequence, +not to the proof of Theorem~\ref{thm:main}. + +\begin{corollary}[Algebraic specialization of rational Hodge classes] +\label{cor:algebraic-specialization} +Fix the prime and place as in Theorem~\ref{thm:main}. Let +\(A/\Qbar\) be an abelian variety of dimension \(d\) with good +reduction \(A_0/\F\) at \(w\), and let +\(\gamma\in H^{2r}(A(\C),\Q(r))\) be a rational Hodge class, +where \(0\leq r\leq d\). There is one rational algebraic cycle +\(z_\gamma\in\operatorname{CH}^r(A_0)_{\Q}\) such that +\[ + \operatorname{cl}_\ell(z_\gamma)=\gamma_{0,\ell} + \quad(\ell\ne p),\qquad + \operatorname{cl}_{\mathrm{cris}}(z_\gamma)\otimes1=\gamma_{0,p}. +\] +The crystalline equality is in +\[ + H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r) + \otimes_{\Kzero}\Cw. +\] +This holds for every prime \(p\), including \(p=2\). +Consequently, for every +\(y\in\operatorname{CH}^{d-r}(A_0)_{\Q}\), the traces of +\(\gamma_{0,\ell}\operatorname{cl}_\ell(y)\) for \(\ell\ne p\) +and of \(\gamma_{0,p}\operatorname{cl}_{\mathrm{cris}}(y)\) +are all the same rational number \(\deg(z_\gamma\cdot y)\). +\end{corollary} + +\begin{proof} +In codimension zero or top codimension, use the corresponding rational +multiple of the unit or the point class, as in the preceding proof. +This includes dimension zero. For $00\) with \(h=Nb\) an actual homomorphism +\(A_0\to B_0\), and put +\[ + z_\gamma=N^{-2r}h^*Z_0\in\operatorname{CH}^r(A_0)_{\Q}. +\] +Here \(h^*\) is the Chow pullback between smooth varieties, defined +by the graph and refined intersection; it requires neither flatness +nor separability of \(h\). Functoriality of cycle classes and +\eqref{eq:rational-pullback} give the required equalities. Rational +coefficients permit this denominator even if \(p\mid N\); the +crystalline realization has already inverted \(p\). Finally, +compatibility of cycle classes with products and normalized traces +gives the asserted intersection number for every \(y\). +\end{proof} + +This conclusion concerns only the special fiber at the fixed place. +It neither asserts the Hodge conjecture for arbitrary abelian varieties +in characteristic zero nor lifts \(z_\gamma\) to the original +generic fiber. The representing cycle is not asserted to be unique +modulo rational equivalence, and no comparison between different +reduction places is made. diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/README.md b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/README.md index 5d0e2e1835ed39cf6d237491809779d68ceb1a8e..0ab8ec4ac41e2cf365a78a94a60ab02f9d406ca8 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/README.md +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/README.md @@ -1,21 +1,21 @@ -# [Numerical semiampleness of nef adjoint classes on compact Kähler manifolds](numerical-generalized-abundance.pdf) - -**Author:** OpenAI - -**Date:** October 6, 2026 - -## Citation - -```bibtex -@misc{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026, - author = {{OpenAI}}, - title = {{Numerical semiampleness of nef adjoint classes on compact K{\"a}hler manifolds}}, - howpublished = {OpenAI Math Release preprint - \href{https://github.com/openai/math/blob/main/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/numerical-generalized-abundance.pdf}{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026}}, - year = {2026} -} -``` - -## Version note - -This version states the Iitaka, model, and contraction inputs used in the divisorial-decomposition argument explicitly; see the [previous version](../Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026/numerical-generalized-abundance.pdf). +# [Numerical semiampleness of nef adjoint classes on compact Kähler manifolds](numerical-generalized-abundance.pdf) + +**Author:** OpenAI + +**Date:** October 6, 2026 + +## Citation + +```bibtex +@misc{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026, + author = {{OpenAI}}, + title = {{Numerical semiampleness of nef adjoint classes on compact K{\"a}hler manifolds}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/numerical-generalized-abundance.pdf}{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026}}, + year = {2026} +} +``` + +## Version note + +This version states the Iitaka, model, and contraction inputs used in the divisorial-decomposition argument explicitly; see the [previous version](../Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026/numerical-generalized-abundance.pdf). diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/companions.bib b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/companions.bib index f38c42fe951d69d96d6bed9b4309298d3ae03286..cf76c842ea698f853ff925031498a9dd38d9b0f5 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/companions.bib +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/companions.bib @@ -1,41 +1,41 @@ -@misc{K, - author = {{OpenAI}}, - title = {{Log abundance for compact K{\"a}hler spaces under logarithmic Iitaka subadditivity}}, - howpublished = {OpenAI Math Release preprint - \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf}{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026}}, - year = {2026}, - note = {Theorem 2.13, Lemma 3.6, Proposition 3.8, Proposition 3.30, and Theorem 3.55} -} - -@misc{LI, - author = {{OpenAI}}, - title = {{Orbifold and logarithmic Iitaka subadditivity}}, - howpublished = {OpenAI Math Release preprint - \href{https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf}{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026}}, - year = {2026}, - note = {Corollary 6.2; Lemmas 4.1 and 4.3 and Proposition 4.5} -} - -@misc{P, - author = {{OpenAI}}, - title = {{Numerical Semiampleness of Nef Adjoint Divisors}}, - howpublished = {OpenAI Math Release preprint - \href{https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf}{OAI:Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026}}, - year = {2026} -} - -@misc{MM, - author = {{OpenAI}}, - title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-pairs}}, - howpublished = {OpenAI Math Release preprint - \href{https://github.com/openai/math/blob/main/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf}{OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026}}, - year = {2026} -} - -@misc{LA, - author = {{OpenAI}}, - title = {{Log abundance in characteristic zero}}, - howpublished = {OpenAI Math Release preprint - \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf}{OAI:Log-abundance-in-characteristic-zero-September-24-2026}}, - year = {2026} -} +@misc{K, + author = {{OpenAI}}, + title = {{Log abundance for compact K{\"a}hler spaces under logarithmic Iitaka subadditivity}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf}{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026}}, + year = {2026}, + note = {Theorem 2.13, Lemma 3.6, Proposition 3.8, Proposition 3.30, and Theorem 3.55} +} + +@misc{LI, + author = {{OpenAI}}, + title = {{Orbifold and logarithmic Iitaka subadditivity}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf}{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026}}, + year = {2026}, + note = {Corollary 6.2; Lemmas 4.1 and 4.3 and Proposition 4.5} +} + +@misc{P, + author = {{OpenAI}}, + title = {{Numerical Semiampleness of Nef Adjoint Divisors}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf}{OAI:Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026}}, + year = {2026} +} + +@misc{MM, + author = {{OpenAI}}, + title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-pairs}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf}{OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026}}, + year = {2026} +} + +@misc{LA, + author = {{OpenAI}}, + title = {{Log abundance in characteristic zero}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf}{OAI:Log-abundance-in-characteristic-zero-September-24-2026}}, + year = {2026} +} diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/figures/zero-two-diagonals.tex b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/figures/zero-two-diagonals.tex index 92f8230f6cca83baff1b0e43af240aac8e9de4c7..4857702c33569a5b7eadbfdb0d2f0d4a1f7ab124 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/figures/zero-two-diagonals.tex +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/figures/zero-two-diagonals.tex @@ -1,42 +1,42 @@ -\begin{minipage}[c]{0.47\linewidth} -\centering -\small -\begin{tikzpicture}[>=stealth, baseline=(current bounding box.center)] -\node (v) at (0,1.5) {$V$}; -\node (b) at (2.7,1.5) {$B$}; -\node (v0) at (0,0) {$\mathcal V_0$}; -\node (zz) at (2.7,0) {$Z\times Z$}; -\draw[->] (v)--node[above] {$\subset$} (b); -\draw[->] (v0)--node[above] {$\subset$} (zz); -\draw[->] (v)--node[left] {$b|_V$} (v0); -\draw[->] (b)--node[right] {$b$} (zz); -\end{tikzpicture} -\[ - \begin{gathered} - V_z=\operatorname{Bl}_xX,\\ - (V\cap E)_z=\mathbb P(T_z^*D_\lambda) - \subset E_z=\mathbb P(T_z^*Z). - \end{gathered} -\] -\end{minipage} -\hfill -\begin{minipage}[c]{0.50\linewidth} -\centering -\small -\begin{tikzpicture}[>=stealth, baseline=(current bounding box.center)] -\node (ep) at (0,1.5) {$E^+$}; -\node (e) at (3,1.5) {$E=\mathbb P_Z(\Omega_Z)$}; -\node (zp) at (0,0) {$Z^+$}; -\node (z) at (3,0) {$Z$}; -\draw[->] (ep)--(e); -\draw[->] (ep)--(zp); -\draw[->] (e)--(z); -\draw[->] (zp)--node[below] {$p_+=\operatorname{Bl}_{U_0}$} (z); -\end{tikzpicture} -\[ - \begin{gathered} - J_+=p_+^{-1}(U_0),\qquad U_0=X\times\mathbb P^1,\\ - \mathcal J=(\mathcal I_V\cdot\cO_E)\cdot\cO_{E^+}. - \end{gathered} -\] -\end{minipage} +\begin{minipage}[c]{0.47\linewidth} +\centering +\small +\begin{tikzpicture}[>=stealth, baseline=(current bounding box.center)] +\node (v) at (0,1.5) {$V$}; +\node (b) at (2.7,1.5) {$B$}; +\node (v0) at (0,0) {$\mathcal V_0$}; +\node (zz) at (2.7,0) {$Z\times Z$}; +\draw[->] (v)--node[above] {$\subset$} (b); +\draw[->] (v0)--node[above] {$\subset$} (zz); +\draw[->] (v)--node[left] {$b|_V$} (v0); +\draw[->] (b)--node[right] {$b$} (zz); +\end{tikzpicture} +\[ + \begin{gathered} + V_z=\operatorname{Bl}_xX,\\ + (V\cap E)_z=\mathbb P(T_z^*D_\lambda) + \subset E_z=\mathbb P(T_z^*Z). + \end{gathered} +\] +\end{minipage} +\hfill +\begin{minipage}[c]{0.50\linewidth} +\centering +\small +\begin{tikzpicture}[>=stealth, baseline=(current bounding box.center)] +\node (ep) at (0,1.5) {$E^+$}; +\node (e) at (3,1.5) {$E=\mathbb P_Z(\Omega_Z)$}; +\node (zp) at (0,0) {$Z^+$}; +\node (z) at (3,0) {$Z$}; +\draw[->] (ep)--(e); +\draw[->] (ep)--(zp); +\draw[->] (e)--(z); +\draw[->] (zp)--node[below] {$p_+=\operatorname{Bl}_{U_0}$} (z); +\end{tikzpicture} +\[ + \begin{gathered} + J_+=p_+^{-1}(U_0),\qquad U_0=X\times\mathbb P^1,\\ + \mathcal J=(\mathcal I_V\cdot\cO_E)\cdot\cO_{E^+}. + \end{gathered} +\] +\end{minipage} diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/literature.bib b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/literature.bib index d86e044fc6b3a77fd40ae5bfe0ffc510f588860a..94591fd6bd8ee8b53dabf619536d09275abeb1f1 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/literature.bib +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/literature.bib @@ -1,536 +1,536 @@ - 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Y{\'a}{\~n}ez, Jos{\'e} Ignacio}, + title = {{MMP} for Generalized Pairs on {K{\"a}hler} $3$-folds}, + year = {2026}, + note = {arXiv:2305.00524v3; to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze}, + doi = {10.2422/2036-2145.202506_004}, + eprint = {2305.00524}, archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2305.00524v3} +} + +@article{Ambro, + author = {Ambro, Florin}, + title = {The moduli $b$-divisor of an lc-trivial fibration}, + journal = {Compositio Mathematica}, + volume = {141}, number = {2}, year = {2005}, pages = {385--403}, + doi = {10.1112/S0010437X04001071}, + eprint = {math/0308143}, archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/0308143} +} + +@article{Schmid, + author = {Schmid, Wilfried}, + title = {Variation of {Hodge} structure: the singularities of the period mapping}, + journal = {Inventiones Mathematicae}, + volume = {22}, year = {1973}, pages = {211--319}, + doi = {10.1007/BF01389674} +} + +@article{CKS, + 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+@misc{BBYv2, + author = {Matsumura, Shin-ichi and Wang, Juanyong and Wu, Xiaojun and Zhang, Qimin}, + title = {{Beauville--Bogomolov--Yau} decomposition for {K{\"a}hler} generalized pairs}, + year = {2026}, + note = {arXiv:2506.23218v2, September 9, 2026}, + eprint = {2506.23218}, archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2506.23218v2} +} + +@article{COP, + author = {Campana, Fr{\'e}d{\'e}ric and Oguiso, Keiji and Peternell, Thomas}, + title = {Non-algebraic hyperk{\"a}hler manifolds}, + journal = {Journal of Differential Geometry}, + volume = {85}, number = {3}, year = {2010}, pages = {397--424}, + doi = {10.4310/jdg/1292940689}, + eprint = {0804.1682}, archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/0804.1682} +} + +@book{SGA1, + author = {Grothendieck, Alexander and others}, + title = {Rev{\^e}tements {\'e}tales et groupe fondamental ({SGA} 1)}, + series = {Documents Math{\'e}matiques}, volume = {3}, + publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France}, + address = {Paris}, year = {2003}, + note = {Updated edition of Lecture Notes in Mathematics 224 (1971); arXiv:math/0206203v2}, + eprint = {math/0206203}, archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/0206203v2} +} + +@book{Ueno, + author = {Ueno, Kenji}, + title = {Classification Theory of Algebraic Varieties and Compact Complex Spaces}, + series = {Lecture Notes in Mathematics}, volume = {439}, + publisher = {Springer}, address = {Berlin}, year = {1975}, + doi = {10.1007/BFb0070570} +} + +@misc{DemaillyCADG, + author = {Demailly, Jean-Pierre}, + title = {Complex Analytic and Differential Geometry}, + year = {2012}, note = {Version of June 21, 2012}, + url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf} +} + +@article{PPS, + author = {Pareschi, Giuseppe and Popa, Mihnea and Schnell, Christian}, + title = {{Hodge} modules on complex tori and generic vanishing for compact {K{\"a}hler} manifolds}, + journal = {Geometry {\&} Topology}, + volume = {21}, number = {4}, year = {2017}, pages = {2419--2460}, + doi = {10.2140/gt.2017.21.2419}, + url = {https://msp.org/gt/2017/21-4/p12.xhtml} +} + + +@incollection{Barlet, + author={Barlet, Daniel}, + title={Espace analytique r\'eduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie}, + booktitle={Fonctions de plusieurs variables complexes, II}, + editor={Norguet, Fran\c{c}ois}, + series={Lecture Notes in Mathematics}, volume={482}, + publisher={Springer}, address={Berlin}, year={1975}, pages={1--158}, + doi={10.1007/BFb0080243} +} + +@incollection{BarletSurvey, + author={Barlet, Daniel}, + title={How to use the cycle space in complex geometry}, + booktitle={Several Complex Variables}, + series={Mathematical Sciences Research Institute Publications}, volume={37}, + publisher={Cambridge University Press}, address={Cambridge}, + year={1999}, pages={25--42}, + url={https://library.slmath.org/books/Book37/files/barlet.pdf} +} + +@article{Grauert, + author={Grauert, Hans}, + title={Ein {Theorem} der analytischen {Garbentheorie} und die {Modulr\"aume} komplexer {Strukturen}}, + journal={Publ. Math. Inst. Hautes \'Etudes Sci.}, + volume={5}, year={1960}, pages={5--64}, + doi={10.1007/BF02684746}, + note={Correction: ibid. 16 (1963), 35--36, doi:10.1007/BF02684291} +} + +@article{HironakaFlattening, + author = {Hironaka, Heisuke}, + title = {Flattening Theorem in Complex-Analytic Geometry}, + journal = {American Journal of Mathematics}, + volume = {97}, + number = {2}, + pages = {503--547}, + year = {1975}, + doi = {10.2307/2373721} +} + +@article{BierstoneMilman2008, + author = {Bierstone, Edward and Milman, Pierre D.}, + title = {Functoriality in Resolution of Singularities}, + journal = {Publications of the Research Institute for Mathematical Sciences}, + volume = {44}, + number = {2}, + pages = {609--639}, + year = {2008}, + doi = {10.2977/PRIMS/1210167338}, + eprint = {math/0702375}, + archivePrefix = {arXiv} +} + +@misc{FujinoAnalyticBCHM, + author = {Fujino, Osamu}, + title = {Minimal model program for projective morphisms between complex analytic spaces}, + year = {2022}, + eprint = {2201.11315}, + archivePrefix = {arXiv}, + primaryClass = {math.AG}, + note = {arXiv:2201.11315v1}, + doi = {10.48550/arXiv.2201.11315}, + url = {https://arxiv.org/abs/2201.11315v1} +} + +@article{BCHM2010, + author = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James}, + title = {Existence of minimal models for varieties of log general type}, + journal = {Journal of the American Mathematical Society}, + volume = {23}, + number = {2}, + year = {2010}, + pages = {405--468}, + doi = {10.1090/S0894-0347-09-00649-3}, + eprint = {math/0610203}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/0610203} +} + +@article{GrauertCorrection, + author = {Grauert, Hans}, + title = {Berichtigung: {Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen}}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + volume = {16}, year = {1963}, pages = {35--36}, + doi = {10.1007/BF02684291}, + url = {https://www.numdam.org/item/PMIHES_1963__16__35_0/} +} + +@article{Campana1981, + author = {Campana, Fr{\'e}d{\'e}ric}, + title = {Cor{\'e}duction alg{\'e}brique d'un espace analytique faiblement {K{\"a}hl{\'e}rien} compact}, + journal = {Inventiones Mathematicae}, + volume = {63}, number = {2}, year = {1981}, pages = {187--223}, + doi = {10.1007/BF01393876} +} diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/main.tex b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/main.tex index a230ddda2765721e1f2265d3b53df6e4e722389d..a2b366385f1124f6bf8fa341b030bd0ede2334a1 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/main.tex +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/main.tex @@ -1,30 +1,30 @@ -\documentclass[11pt]{article} -\input{preamble} -\title{Numerical semiampleness of nef adjoint classes\\on compact K\"ahler manifolds} -\author{OpenAI} -\date{October 6, 2026} -\begin{document} -\maketitle -\begin{abstract} -Let \(X\) be a smooth connected compact K\"ahler manifold, let \(B\) be an -effective rational simple normal crossing divisor with coefficients less -than one, and let \(M\) be a nef rational holomorphic line bundle on \(X\). -If \(K_X+B\) is pseudo-effective and \(K_X+B+M\) is nef, we prove that its -first Chern class in real Bott--Chern cohomology is represented by a semiample -rational line bundle. This numerical statement allows a flat change of line -bundle; it does not assert semiampleness of the original adjoint. -\end{abstract} -\tableofcontents -\input{sections/01-introduction} -\input{sections/02-preliminaries} -\input{sections/03-descent} -\input{sections/04-adjoint} -\input{sections/05-zero} -\input{sections/06-cycles} -\input{sections/07-completion} -\begingroup -\raggedright -\bibliographystyle{plain} -\bibliography{literature,companions} -\endgroup -\end{document} +\documentclass[11pt]{article} +\input{preamble} +\title{Numerical semiampleness of nef adjoint classes\\on compact K\"ahler manifolds} +\author{OpenAI} +\date{October 6, 2026} +\begin{document} +\maketitle +\begin{abstract} +Let \(X\) be a smooth connected compact K\"ahler manifold, let \(B\) be an +effective rational simple normal crossing divisor with coefficients less +than one, and let \(M\) be a nef rational holomorphic line bundle on \(X\). +If \(K_X+B\) is pseudo-effective and \(K_X+B+M\) is nef, we prove that its +first Chern class in real Bott--Chern cohomology is represented by a semiample +rational line bundle. This numerical statement allows a flat change of line +bundle; it does not assert semiampleness of the original adjoint. +\end{abstract} +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-preliminaries} +\input{sections/03-descent} +\input{sections/04-adjoint} +\input{sections/05-zero} +\input{sections/06-cycles} +\input{sections/07-completion} +\begingroup +\raggedright +\bibliographystyle{plain} +\bibliography{literature,companions} +\endgroup +\end{document} diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/preamble.tex b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/preamble.tex index 5bcb38d7532962e7d5613d9c781855a5dfa132f9..cb37e04a151339ad4bdbd5f12c4e61154030a90f 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/preamble.tex +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/preamble.tex @@ -1,46 +1,46 @@ -\usepackage[T1]{fontenc} -\usepackage[utf8]{inputenc} -\usepackage{lmodern} -\usepackage{amsmath,amssymb,amsthm,mathtools} -\usepackage{mathrsfs} -\usepackage{microtype} -\usepackage[margin=1.05in]{geometry} -\usepackage{enumitem} -\usepackage{tikz} -\usetikzlibrary{arrows.meta,positioning,calc} -\usepackage{xcolor} -\usepackage{hyperref} -\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black, - urlcolor=blue!45!black,pdftitle={Numerical semiampleness of nef adjoint classes on compact Kahler manifolds}, - pdfauthor={OpenAI}} -\newcommand{\Q}{\mathbb Q} -\newcommand{\R}{\mathbb R} -\newcommand{\C}{\mathbb C} -\newcommand{\Z}{\mathbb Z} -\newcommand{\N}{\mathbb N} -\newcommand{\PP}{\mathbb P} -\newcommand{\cO}{\mathcal O} -\newcommand{\num}{\equiv} -\newcommand{\qsim}{\sim_{\Q}} -\DeclareMathOperator{\Pic}{Pic} -\DeclareMathOperator{\Supp}{Supp} -\DeclareMathOperator{\codim}{codim} -\DeclareMathOperator{\rank}{rank} -\DeclareMathOperator{\vol}{vol} -\DeclareMathOperator{\im}{im} -\DeclareMathOperator{\tr}{tr} -\DeclareMathOperator{\Alb}{Alb} -\newtheorem{theorem}{Theorem}[section] -\newtheorem{proposition}[theorem]{Proposition} -\newtheorem{lemma}[theorem]{Lemma} -\newtheorem{corollary}[theorem]{Corollary} -\newtheorem{claim}[theorem]{Claim} -\theoremstyle{definition} -\newtheorem{definition}[theorem]{Definition} -\newtheorem{convention}[theorem]{Convention} -\theoremstyle{remark} -\newtheorem{remark}[theorem]{Remark} -\numberwithin{equation}{section} -\setlist[enumerate]{leftmargin=*,itemsep=3pt} -\setlist[itemize]{leftmargin=*,itemsep=3pt} -\emergencystretch=1.2em +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{mathrsfs} +\usepackage{microtype} +\usepackage[margin=1.05in]{geometry} +\usepackage{enumitem} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc} +\usepackage{xcolor} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black, + urlcolor=blue!45!black,pdftitle={Numerical semiampleness of nef adjoint classes on compact Kahler manifolds}, + pdfauthor={OpenAI}} +\newcommand{\Q}{\mathbb Q} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\PP}{\mathbb P} +\newcommand{\cO}{\mathcal O} +\newcommand{\num}{\equiv} +\newcommand{\qsim}{\sim_{\Q}} +\DeclareMathOperator{\Pic}{Pic} +\DeclareMathOperator{\Supp}{Supp} +\DeclareMathOperator{\codim}{codim} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Alb}{Alb} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{claim}[theorem]{Claim} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{convention}[theorem]{Convention} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\setlist[enumerate]{leftmargin=*,itemsep=3pt} +\setlist[itemize]{leftmargin=*,itemsep=3pt} +\emergencystretch=1.2em diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/01-introduction.tex b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/01-introduction.tex index 25ebe108740bef203223d1f80067759dee06bf46..03bc09bc7b92fdb4417d265cc95093bc28e1fa53 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/01-introduction.tex +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/01-introduction.tex @@ -1,160 +1,160 @@ -\section{Introduction}\label{sec:introduction} - -A central aim of abundance theory is to recover a holomorphic map from -the positivity of an adjoint class. For an ordinary log canonical line, -the expected conclusion is semiampleness of that line itself. -Adding an arbitrary nef line introduces a different phenomenon: its -flat part need not be torsion. The natural conclusion is then numerical -semiampleness, namely the existence of a semiample line with the same -real Chern class. - -We prove this statement for smooth compact K\"ahler manifolds with a -klt boundary. A \emph{rational line bundle} is an element of -\(\Pic(X)\otimes_{\Z}\Q\). It is nef when its first Chern class belongs -to the closure of the K\"ahler cone, and pseudo-effective when that -class contains a closed positive \((1,1)\)-current. It is semiample -when a positive integral multiple is a holomorphic line bundle -generated everywhere by global sections. - -\begin{theorem}\label{thm:main} -Let \(X\) be a smooth connected compact K\"ahler manifold, let -\(B\geq0\) be a rational simple normal crossing divisor whose -coefficients are strictly less than one, and let -\(M\in\Pic(X)\otimes_{\Z}\Q\). -Assume that \(K_X+B\) is pseudo-effective, \(M\) is nef, and -\(D=K_X+B+M\) is nef. Then there is a semiample rational line bundle -\(L\) on \(X\) such that -\[ - c_1(L)=c_1(D)\quad\text{in }H^{1,1}_{\mathrm{BC}}(X,\R). -\] -\end{theorem} - -The numerical formulation is essential even on an elliptic curve: -a nontorsion degree-zero line \(M\) is nef but has no section in any -positive power, whereas its Chern class is represented by the trivial -line. Theorem~\ref{thm:main} allows exactly this change of flat part. -It keeps the nef line \(M\) on the original manifold; it is not a -statement about an arbitrary nef b-divisor on a higher model. - -\subsection{Context and the inputs to the proof} - -Lazi\'c and Peternell formulated generalized abundance for projective -klt pairs as precisely this numerical semiampleness problem -\cite[Generalised Abundance Conjecture]{LP1}. Their work relates it to ordinary abundance -and to positivity on varieties with numerically trivial canonical -class, using nef reduction and the positive part of the divisorial -Zariski decomposition \cite{LP1,LP2}. They proved the surface case and -the threefold case in which the ordinary adjoint has positive numerical -dimension \cite[Corollaries~C and~D]{LP1}. -The present proof uses the projective positive-part theorem of -\cite[Proposition~8.1]{P} as an input. Our task is to pass from that -projective statement to the analytic setting, where the nef summand -need not be represented by a divisor and a manifold may have no -nonconstant meromorphic functions. - -The compact K\"ahler minimal model program has developed alongside -these abundance questions. H\"oring and Peternell constructed minimal -models for compact K\"ahler threefolds \cite{HoeringPeternell2016}. -The threefold abundance theorems -\cite{CHP2016,CHP2023} and the log abundance theorem of Das and Ou -\cite{DasOu2024,DasOu2026} establish important cases of the ordinary problem. -Hacon and Xie's analytic cone theorem \cite{HX} supplies the rational -curve length bound. The restricted K\"ahler model constructions and -ordinary scaling in \cite[Section~3]{K} supply the detected projective -steps used here. For ordinary adjoints in arbitrary dimension, we use -that paper's divisorial decomposition theorem, with its precise -logarithmic Iitaka hypothesis supplied by \cite[Corollary~6.2]{LI}. - -These are substantial inputs. Section~\ref{sec:preliminaries} states -their precise forms, retaining the distinction between actual -rational line identities and equalities of real classes. -The proof below supplies the further arguments needed for the nef -summand. It uses Boucksom's analytic divisorial decomposition -\cite{Boucksom2004}, the compactness theory of cycle spaces -\cite{Barlet,LiebermanCycles,Fujiki}, and the geometry of compact K\"ahler -spaces of algebraic dimension zero -\cite{COP,BBYv2,Ou}. The adjoint construction on a projective base -follows the discriminant and moduli-line strategy of Ambro -\cite[Theorem~0.2]{Ambro}, using the period results of \cite{LI} -to accommodate a real polarization. - -\subsection{The stronger statement and the proof} - -For a pseudo-effective real \((1,1)\)-class \(\alpha\), denote by -\(N(\alpha)\) its divisorial negative part: the effective real divisor -whose coefficient along each prime is the least generic multiplicity -forced in positive representatives, with arbitrarily small K\"ahler -perturbations. A nef class has zero negative part. -We prove a stronger statement in which the adjoint sum need not be nef. - -The algebraic dimension \(a(T)\) is the transcendence degree of the -field of meromorphic functions on \(T\). Its algebraic reduction, -after modification, is a morphism \(h:T\to W\) to a smooth projective -variety of dimension \(a(T)\) whose meromorphic functions account for -all those on \(T\). - -\begin{theorem}\label{thm:decomposition} -Let \(T\) be a smooth connected compact K\"ahler manifold, \(B\geq0\) -a rational simple normal crossing divisor with coefficients less than -one, and \(M\) a nef rational line bundle. Suppose -\(J=K_T+B\) is pseudo-effective. There are a smooth compact K\"ahler -modification \(\mu:U\to T\), a semiample rational line \(P\) on \(U\), -and an effective rational divisor \(R\) on \(U\), such that -\[ - \mu^*(J+M)\num P+R,\qquad - R=N\bigl(c_1(\mu^*(J+M))\bigr). -\] -Here \(\num\) denotes equality of real Chern classes. -Moreover, if \(a(T)=0\), then \(c_1(M)=0\). -\end{theorem} - -We prove both assertions of Theorem~\ref{thm:decomposition} -simultaneously by induction on dimension. Projective manifolds are -covered by \cite{P}. For \(a(T)=0\), the ordinary decomposition and -the decomposition theorem for K\"ahler klt pairs of Calabi--Yau type -\cite[Corollary~1.4]{BBYv2} reduce the problem to tori and simple -spaces carrying a generically symplectic form. The simple case requires -a meromorphic nonvanishing statement for \(K_T+M\). -Section~\ref{sec:zero} develops the required extension of the -two-diagonal argument in \cite[Section~6]{K}. The new geometric input -excludes a family of correspondences sweeping \(T\times T\): such a -family would produce a nonzero holomorphic one-form on a fixed finite -cover. - -When \(00$ and $d=\dim X-\dim Y>0$. - -Flatten $f$ after a smooth modification $q:Y'\to Y$ -\cite{HironakaFlattening}. -Let $Z$ be the normalization of the main component of $X\times_Y Y'$, -and resolve it by a projective modification $r:X'\to Z$. -Write $g_0:Z\to Y'$ and $g=g_0r$. The map $g_0$ is equidimensional; -it and $g$ have connected fibers by Stein factorization and normality -of $Y'$. All these spaces admit the asserted K\"ahler models. -Write $p_0:Z\to X$ for the map to the original source, so that -$p=p_0r$ is a modification. If $Y$ is projective, the base -modifications may be chosen projective. Figure~\ref{desc:diagram} -separates the equidimensional map from the resolution above it. - -\begin{figure}[ht] -\centering -\begin{tikzpicture}[>=Stealth, every node/.style={inner sep=3pt}] -\node (Xp) at (0,1.35) {$X'$}; -\node (Z) at (2.6,1.35) {$Z$}; -\node (X) at (5.2,1.35) {$X$}; -\node (Yp) at (2.6,0) {$Y'$}; -\node (Y) at (5.2,0) {$Y$}; -\draw[->] (Xp) -- node[above] {$r$} (Z); -\draw[->] (Z) -- node[above] {$p_0$} (X); -\draw[->] (Xp) -- node[below left] {$g$} (Yp); -\draw[->] (Z) -- node[right] {$g_0$} (Yp); -\draw[->] (X) -- node[right] {$f$} (Y); -\draw[->] (Yp) -- node[below] {$q$} (Y); -\end{tikzpicture} -\caption{The flattened main component $Z$ is equidimensional over -$Y'$. A prime divisor on the resolution $X'$ whose image in $Y'$ -has codimension at least two must therefore be exceptional for $r$.} -\label{desc:diagram} -\end{figure} - -Put $\alpha=c_1(p^*M)$, choose a K\"ahler form $\omega$ on $X'$, and -choose a closed positive current $T$ representing $\alpha$. -The volume -\[ - c=\int_{X'_y}\omega^d -\] -is positive and constant on the smooth-fibration locus. Define the -closed positive $(1,1)$-current -\begin{equation}\label{desc:base-current} - S=c^{-1}g_*(T\wedge\omega^d) -\end{equation} -on $Y'$. We claim that $T=g^*S$ over that locus. -Indeed $g_*(T\wedge\omega^{d-1})$ is a closed positive current of -degree zero, hence a nonnegative constant. Its cohomology class is -the fiber intersection of $\alpha$ with $\omega^{d-1}$, which vanishes. -Thus this current is zero. In local product coordinates this says -that the vertical block of the positive matrix of measures defining -$T$ vanishes. Positivity also forces its mixed block to vanish. -Closedness then makes the horizontal coefficients independent of the -fiber coordinates. Connectedness of the fibers makes these local -currents descend, and Equation~\eqref{desc:base-current} identifies -their common descent as $S$. - -The current $S$ has local plurisubharmonic potentials, so its pullback -by the dominant map $g$ is defined on all of $X'$. -The closed order-zero current $T-g^*S$ is supported on the inverse -image of the complement of the smooth-fibration locus in $Y'$. -The support theorem for currents -\cite[Chapter~III, Corollaries~2.11 and~2.14]{DemaillyCADG} therefore gives -\begin{equation}\label{desc:vertical-difference} - T-g^*S=[D],\qquad D=\sum_E x_E E, -\end{equation} -where $D$ is a signed real divisor supported on vertical primes. -We next show that its components dominating base divisors come in -whole pullbacks. - -Fix a prime divisor $Q\subset Y'$ occurring among their images, and -let $E_1,\ldots,E_s$ be the primes of $X'$ dominating $Q$. -Let $a_i>0$ be their multiplicities in $g^*Q$, and let $x_i$ be their -coefficients in $D$. For a K\"ahler form $\eta$ on $Y'$, set -\begin{equation}\label{desc:intersection-matrix} - C_{ij}=\int_{X'}\{E_i\}\{E_j\}\,(g^*\eta)^{k-1}\omega^{d-1}. -\end{equation} -For $i\ne j$, these numbers are nonnegative. -The inverse image $g^{-1}(Q)$ is an effective Cartier divisor, hence -has pure codimension one. After a generic choice of the point in -$Q$, its fiber components all have pure dimension $d$. Distinct -primes $E_i,E_j$ have pure codimension-two intersection; if they -meet over a general point of $Q$, that intersection dominates $Q$ -and has fibers of dimension $d-1$. It therefore contributes -positively to $C_{ij}$. These dimension statements follow by -generic flatness over $Q$, excluding the smaller base images. -The graph having an edge when $C_{ij}>0$ is consequently connected. Indeed, over a general point -of $Q$ the irreducible components of the connected fiber have a -connected intersection graph. Grouping those components according -to the global prime $E_i$ containing them gives the graph above as -a quotient, which is still connected. Moreover $Ca=0$, where -$a=(a_i)$. -To see this, pair $E_i$ with $g^*Q$ and $(g^*\eta)^{k-1}\omega^{d-1}$. -Its image lies in $Q$, so the $k$ classes from the base give zero. -Components of $g^*Q$ whose images have codimension at least two give -zero separately in this pairing. - -Let $\beta$ be the Bott--Chern class of $S$. Equation~\eqref{desc:vertical-difference} gives -$\{D\}=\alpha-g^*\beta$. Pairing this identity with each -$E_i(g^*\eta)^{k-1}\omega^{d-1}$ shows that -\[ - Cx\geq0. -\] -Here the $\alpha$ term is nonnegative by nefness, and the base term -vanishes by dimension. Components of $D$ above other base divisors, -or above sets of codimension at least two, also contribute zero. -Since $a$ has strictly positive entries and $a^{\mathsf t}Cx=0$, -we obtain $Cx=0$. The kernel is precisely $\R a$: indeed -\[ - x^{\mathsf t}Cx - =-\sum_{i0$ is the multiplicity of that prime in $f^*Q$. -The disk lies in $X$, so the character of $F$ is trivial on its -boundary. Hence $\chi$ has finite order dividing $a_Q$ on the base -meridian. There are finitely many such $Q$. A common power of -$\chi$ kills all their meridians and therefore factors through -$\pi_1(Y)$; subsets of complex codimension at least two add no -obstruction. The resulting unitary flat line on $Y$ has zero real -Chern class, and another power removes any torsion in its integral -Chern class. It then belongs to $\Pic^0(Y)$. - -The resulting characters agree after pullback on $X^\circ$. -Since $\pi_1(X^\circ)\to\pi_1(X)$ is surjective, they agree on $X$ -as well. Dividing by the powers taken in the argument proves the -asserted identity in $\Pic(X)\otimes\Q$. -\end{proof} +\section{Descent of nef and flat lines}\label{sec:descent} + +A nef class that vanishes on the fibers of a fibration should come from +its base. Singular fibers make this assertion more delicate than +cohomological descent on a smooth family: the discrepancy can contain +vertical divisors. We first remove those divisors, and then distinguish +numerical descent from descent of an actual rational line. Throughout +this section a \emph{fibration} is a surjective holomorphic map with +connected fibers. + +\begin{proposition}[Numerical descent of a nef line]\label{desc:nef} +Let $f:X\to Y$ be a fibration of smooth connected compact K\"ahler +manifolds, and let $M\in\Pic(X)\otimes\Q$ be analytically nef. +Suppose that $c_1(M|_{X_y})=0$ on a general smooth fiber. +There are smooth compact K\"ahler modifications $p:X'\to X$ and +$q:Y'\to Y$, a fibration $g:X'\to Y'$ satisfying $fp=qg$, and a +rational line $N$ on $Y'$ such that +\begin{equation}\label{desc:numerical-identity} + p^*M\num g^*N. +\end{equation} +If $Y$ is projective, $Y'$ can be chosen projective and $N$ is nef. +\end{proposition} + +\begin{proof} +We descend a positive current over the smooth fibers, remove the +vertical divisor discrepancy over the remaining fibers, and finally +recover the rationality of the descended class. +If $Y$ is a point, the hypothesis says that $c_1(M)=0$. +If the relative dimension is zero, $f$ is bimeromorphic; take a common +smooth model as both $X'$ and $Y'$. We may therefore assume +$k=\dim Y>0$ and $d=\dim X-\dim Y>0$. + +Flatten $f$ after a smooth modification $q:Y'\to Y$ +\cite{HironakaFlattening}. +Let $Z$ be the normalization of the main component of $X\times_Y Y'$, +and resolve it by a projective modification $r:X'\to Z$. +Write $g_0:Z\to Y'$ and $g=g_0r$. The map $g_0$ is equidimensional; +it and $g$ have connected fibers by Stein factorization and normality +of $Y'$. All these spaces admit the asserted K\"ahler models. +Write $p_0:Z\to X$ for the map to the original source, so that +$p=p_0r$ is a modification. If $Y$ is projective, the base +modifications may be chosen projective. Figure~\ref{desc:diagram} +separates the equidimensional map from the resolution above it. + +\begin{figure}[ht] +\centering +\begin{tikzpicture}[>=Stealth, every node/.style={inner sep=3pt}] +\node (Xp) at (0,1.35) {$X'$}; +\node (Z) at (2.6,1.35) {$Z$}; +\node (X) at (5.2,1.35) {$X$}; +\node (Yp) at (2.6,0) {$Y'$}; +\node (Y) at (5.2,0) {$Y$}; +\draw[->] (Xp) -- node[above] {$r$} (Z); +\draw[->] (Z) -- node[above] {$p_0$} (X); +\draw[->] (Xp) -- node[below left] {$g$} (Yp); +\draw[->] (Z) -- node[right] {$g_0$} (Yp); +\draw[->] (X) -- node[right] {$f$} (Y); +\draw[->] (Yp) -- node[below] {$q$} (Y); +\end{tikzpicture} +\caption{The flattened main component $Z$ is equidimensional over +$Y'$. A prime divisor on the resolution $X'$ whose image in $Y'$ +has codimension at least two must therefore be exceptional for $r$.} +\label{desc:diagram} +\end{figure} + +Put $\alpha=c_1(p^*M)$, choose a K\"ahler form $\omega$ on $X'$, and +choose a closed positive current $T$ representing $\alpha$. +The volume +\[ + c=\int_{X'_y}\omega^d +\] +is positive and constant on the smooth-fibration locus. Define the +closed positive $(1,1)$-current +\begin{equation}\label{desc:base-current} + S=c^{-1}g_*(T\wedge\omega^d) +\end{equation} +on $Y'$. We claim that $T=g^*S$ over that locus. +Indeed $g_*(T\wedge\omega^{d-1})$ is a closed positive current of +degree zero, hence a nonnegative constant. Its cohomology class is +the fiber intersection of $\alpha$ with $\omega^{d-1}$, which vanishes. +Thus this current is zero. In local product coordinates this says +that the vertical block of the positive matrix of measures defining +$T$ vanishes. Positivity also forces its mixed block to vanish. +Closedness then makes the horizontal coefficients independent of the +fiber coordinates. Connectedness of the fibers makes these local +currents descend, and Equation~\eqref{desc:base-current} identifies +their common descent as $S$. + +The current $S$ has local plurisubharmonic potentials, so its pullback +by the dominant map $g$ is defined on all of $X'$. +The closed order-zero current $T-g^*S$ is supported on the inverse +image of the complement of the smooth-fibration locus in $Y'$. +The support theorem for currents +\cite[Chapter~III, Corollaries~2.11 and~2.14]{DemaillyCADG} therefore gives +\begin{equation}\label{desc:vertical-difference} + T-g^*S=[D],\qquad D=\sum_E x_E E, +\end{equation} +where $D$ is a signed real divisor supported on vertical primes. +We next show that its components dominating base divisors come in +whole pullbacks. + +Fix a prime divisor $Q\subset Y'$ occurring among their images, and +let $E_1,\ldots,E_s$ be the primes of $X'$ dominating $Q$. +Let $a_i>0$ be their multiplicities in $g^*Q$, and let $x_i$ be their +coefficients in $D$. For a K\"ahler form $\eta$ on $Y'$, set +\begin{equation}\label{desc:intersection-matrix} + C_{ij}=\int_{X'}\{E_i\}\{E_j\}\,(g^*\eta)^{k-1}\omega^{d-1}. +\end{equation} +For $i\ne j$, these numbers are nonnegative. +The inverse image $g^{-1}(Q)$ is an effective Cartier divisor, hence +has pure codimension one. After a generic choice of the point in +$Q$, its fiber components all have pure dimension $d$. Distinct +primes $E_i,E_j$ have pure codimension-two intersection; if they +meet over a general point of $Q$, that intersection dominates $Q$ +and has fibers of dimension $d-1$. It therefore contributes +positively to $C_{ij}$. These dimension statements follow by +generic flatness over $Q$, excluding the smaller base images. +The graph having an edge when $C_{ij}>0$ is consequently connected. Indeed, over a general point +of $Q$ the irreducible components of the connected fiber have a +connected intersection graph. Grouping those components according +to the global prime $E_i$ containing them gives the graph above as +a quotient, which is still connected. Moreover $Ca=0$, where +$a=(a_i)$. +To see this, pair $E_i$ with $g^*Q$ and $(g^*\eta)^{k-1}\omega^{d-1}$. +Its image lies in $Q$, so the $k$ classes from the base give zero. +Components of $g^*Q$ whose images have codimension at least two give +zero separately in this pairing. + +Let $\beta$ be the Bott--Chern class of $S$. Equation~\eqref{desc:vertical-difference} gives +$\{D\}=\alpha-g^*\beta$. Pairing this identity with each +$E_i(g^*\eta)^{k-1}\omega^{d-1}$ shows that +\[ + Cx\geq0. +\] +Here the $\alpha$ term is nonnegative by nefness, and the base term +vanishes by dimension. Components of $D$ above other base divisors, +or above sets of codimension at least two, also contribute zero. +Since $a$ has strictly positive entries and $a^{\mathsf t}Cx=0$, +we obtain $Cx=0$. The kernel is precisely $\R a$: indeed +\[ + x^{\mathsf t}Cx + =-\sum_{i0$ is the multiplicity of that prime in $f^*Q$. +The disk lies in $X$, so the character of $F$ is trivial on its +boundary. Hence $\chi$ has finite order dividing $a_Q$ on the base +meridian. There are finitely many such $Q$. A common power of +$\chi$ kills all their meridians and therefore factors through +$\pi_1(Y)$; subsets of complex codimension at least two add no +obstruction. The resulting unitary flat line on $Y$ has zero real +Chern class, and another power removes any torsion in its integral +Chern class. It then belongs to $\Pic^0(Y)$. + +The resulting characters agree after pullback on $X^\circ$. +Since $\pi_1(X^\circ)\to\pi_1(X)$ is surjective, they agree on $X$ +as well. Dividing by the powers taken in the argument proves the +asserted identity in $\Pic(X)\otimes\Q$. +\end{proof} diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/04-adjoint.tex b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/04-adjoint.tex index 2132e8e0960e552fd39c6b16bb4410decdc0a5d4..db2d5e289a4c3cd8f09098dbbb07054dc5bfc9c8 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/04-adjoint.tex +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/04-adjoint.tex @@ -1,459 +1,459 @@ -\section{An ordinary adjoint on the projective base} -\label{sec:adjoint} - -The induction will produce a morphism to a projective variety for which an -ordinary adjoint is pulled back from the base. To apply the projective -numerical-semiampleness theorem, we must realize the line downstairs as an -ordinary klt adjoint. The following proposition provides this realization. -The total space need not be projective. - -\begin{proposition}[An adjoint on the base] -\label{adj:base} -Let $(Y,D)$ be an effective klt pair with rational boundary on a normal -compact K\"ahler space. Let $f:Y\to Z$ be a surjective morphism with connected -fibers to a normal projective variety, and suppose that -$\dim Y>\dim Z$. If $H$ is a rational Cartier line bundle on $Z$ and -\begin{equation} -\label{adj:initial} - K_Y+D\qsim f^*H, -\end{equation} -then there is an effective rational divisor $D_Z$ such that $(Z,D_Z)$ is klt -and -\[ - H\qsim K_Z+D_Z. -\] -\end{proposition} - -Ambro proves the corresponding statement for projective total spaces -\cite[Theorem~0.2]{Ambro}. We follow the same separation into a discriminant -and a Hodge-theoretic moduli line. The two points requiring attention here -are the real, possibly irrational polarization of a K\"ahler family and the -construction of its cyclic cover without a meromorphic frame of $K_Y$. -The period results recalled next address the first point; we give the cover -and the required infinitesimal calculation explicitly. - -\subsection{The period input and the discriminant} - -For a pure variation of Hodge structures on a smooth open set, the -\emph{line period map} of a rank-one highest Hodge piece is the map to the -projective space of the flat vector space that records this line in a local -flat trivialization. Its generic differential rank is independent of the -trivialization. The full period map records the entire Hodge filtration. -For quasi-unipotent boundary monodromy, the \emph{parabolic extension} of the -line is the rational line bundle obtained by making the monodromy unipotent -on local finite covers, extending the highest Hodge piece, and descending -with its rational boundary weights. - -We use the following precise consequence of the period results in -\cite[Lemmas~4.1 and~4.3 and Proposition~4.5]{LI}. - -\begin{lemma}[Period quotient input] -\label{adj:period-input} -Let $S$ be a smooth projective variety and let $S^\circ\subset S$ have simple -normal crossing complement. Let $\mathbb V$ be a real-polarizable pure -variation on $S^\circ$ with an integral lattice and quasi-unipotent local -monodromy. Suppose that a complex direct summand of $\mathbb V_{\C}$, as a -variation of Hodge structures, has a rank-one highest Hodge piece, with parabolic extension $L$. Then $L$ is nef, -and its numerical dimension is the generic rank of its line period map. - -After modification $\tau:S'\to S$, there are a smooth projective variety -$Q$, a surjective morphism $p:S'\to Q$ with connected fibers, and a nef -rational line bundle $P$ on $Q$ such that -\begin{equation} -\label{adj:period-descent} - \tau^*L\qsim p^*P. -\end{equation} -Moreover, $\dim Q$ is at most the generic rank of the full period map of -$\mathbb V$. If $Q$ is a point, $\tau^*L$ is rationally trivial. -\end{lemma} - -Here the identity is in $\Pic(S')\otimes\Q$, including the boundary. -For clarity, the dimension assertion follows because the descended, -generically immersive period map on $Q$ belongs to an adjoint variation -obtained by tensor constructions from $\mathbb V$. -The rational adjoint reduction in \cite[Lemma~4.3]{LI} retains entire -rational simple factors, so its monodromy is discrete even when the -original real polarization is irrational. It also kills only finite -scalar monodromy after taking a power. Thus -\eqref{adj:period-descent} does not discard an arbitrary flat twist. -We shall need exactly this actual line identity. - -Choose a projective resolution $b:S\to Z$, and normalize the main -component of $Y\times_Z S$. Choose its log resolution functorially -with respect to local isomorphisms preserving the marked boundary; -over the open where $b$ is an isomorphism this is a functorial log -resolution of $(Y,D)$. Denote the resulting smooth compact K\"ahler -space by $X$, with maps $\pi:X\to Y$ and $g:X\to S$. -Such a resolution is available in the complex analytic category; -see \cite[Theorem~1.1 and the preceding analytic remark]{BierstoneMilman2008}. -The functorial choice will matter when we lift local flows. -Additional resolutions used to compute thresholds and fiber integrals -are auxiliary and do not replace the family whose cohomology we use. -Define the crepant rational divisor $D_X$ by -\[ - K_X+D_X=\pi^*(K_Y+D). -\] -Its exceptional coefficients may be negative; all its coefficients are -strictly less than one. For a prime divisor $Q_0$ on $S$, let -\[ - t_{Q_0}=\sup\{c\in\R:(X,D_X+c g^*Q_0) - \text{ is log canonical over the general point of }Q_0\}. -\] -The definition is unchanged on a higher crepant resolution. It gives a -positive rational number, computed by a log resolution as -\begin{equation} -\label{adj:threshold} - t_{Q_0}=\min_i\frac{1-u_i}{a_i}, -\end{equation} -where $g^*Q_0=\sum_i a_i E_i$ over its general point and $u_i$ is the -coefficient of $E_i$ in the crepant boundary there. Define -\begin{equation} -\label{adj:discriminant} - \Delta_S=\sum_{Q_0}(1-t_{Q_0})Q_0, - \qquad L_S=b^*H-K_S-\Delta_S. -\end{equation} -Only finitely many terms of $\Delta_S$ are nonzero: off a suitable proper -analytic subset the resolved pair is relatively simple normal crossing, -the map is smooth, and the threshold of a base prime is one. We may choose -$S$ with a simple normal crossing divisor containing this exceptional set -and the support of $\Delta_S$. - -The coefficients of $\Delta_S$ are strictly less than one. To prove -Proposition~\ref{adj:base}, we will show that $L_S$ has an effective rational -representative whose addition to $\Delta_S$ is sub-klt. The relevant -positivity comes from the following root construction. - -\subsection{The root cover and its highest Hodge line} - -\begin{lemma}[The root eigenline] -\label{adj:root} -In the setting of Proposition~\ref{adj:base}, there is a dense Zariski open -$S^\circ\subset S$ and a smooth proper family $h:V^\circ\to S^\circ$ obtained -from a cyclic cover and resolution, such that $R^l h_*\R$, where -$l=\dim Y-\dim Z$, is real-polarizable and has an integral lattice. -One character summand of its complexification has a rank-one highest -Hodge piece. On $S^\circ$, this line agrees, as a rational line bundle, -with $b^*H-K_S$. -\end{lemma} - -\begin{proof} -Choose an integer $m>0$ clearing all divisors and rational line identities. -A meromorphic section of the line $m b^*H$ exists because $S$ is projective. -Its pullback, under the identity -\[ - m(K_X+D_X)\simeq g^*(m b^*H), -\] -and division by the canonical meromorphic section of $mD_X$ give a -meromorphic section $\sigma$ of $K_X^{\otimes m}$. -On the open where $\sigma$ has neither zeros nor poles, take its $m$th -roots in the fibers of $K_X$. In local canonical frames this is the -equation $z^m=\sigma$; on overlaps the roots transform by the transition -functions of $K_X$. The local covers therefore glue without a -meromorphic trivialization of $K_X$, and without choosing a root of $H$. - -The Grauert--Remmert extension theorem for finite analytic covers -\cite[Exposé~XII, Proposition~5.3 and Theorem~5.4]{SGA1} extends it -normally across the divisor of $\sigma$, giving a finite cyclic cover -of $X$, possibly disconnected. It is compact K\"ahler. -Take a resolution functorial for the cyclic action and for local -isomorphisms of the pair, as in \cite[Section~1.1 and Lemma~1.1]{Ambro}. -Thus both resolutions are functorial. A local flow preserving the -original pair lifts through them once its lift to the root cover has -been chosen; we will construct that lift explicitly below. -The resulting compact manifold $V$ is K\"ahler. After deleting from $S$ the zeros and -poles of the chosen base section and the degeneration locus, the map -$h:V^\circ\to S^\circ$ is smooth and proper. We take $S^\circ$ -inside the isomorphism locus of $b$. Shrinking it further, the original -fibers of $Y\to Z$ are normal effective klt pairs and the chosen -resolutions restrict to resolutions of those pairs. - -Average a global K\"ahler class on $V$ under the cyclic group. Its -restriction is a flat real section of $R^2h_*\R$ and polarizes the -primitive pieces. Their Lefschetz decomposition supplies a flat real -polarization of the full degree-$l$ cohomology. We retain the full -cohomology local system, with lattice $R^lh_*\Z$ modulo torsion; an -irrational primitive summand is not required to carry its own lattice. -The finite group action preserves this variation and its polarization. - -Locally over the base, the tautological root, divided by a local base -volume form, gives a relative meromorphic top form $\tau$ on the cover. -It belongs to a fixed character of the cyclic group. The klt condition -makes its squared volume locally integrable, including after resolution, -so it extends holomorphically: a meromorphic top form with a divisorial -pole cannot be locally square integrable. -To see the orders directly, consider a boundary prime on the original -normal fiber with coefficient $u=p/e\in(0,1)$ in lowest terms. At a -general point the cover has local coordinate $x=y^e$, and the pulled-back -root form has order -\begin{equation} -\label{adj:root-order} - e-1-eu=e-1-p\in\{0,\ldots,e-2\}. -\end{equation} -Away from the boundary its order at a nonexceptional prime is zero. -The corresponding calculation on a crepant resolution uses coefficients -less than one and proves integrability at exceptional divisors as well. - -Suppose that $\tau'$ is another holomorphic top form of the same -character on a fiber. The ratio $\tau'/\tau$ is an invariant meromorphic -function and descends to the original normal connected fiber. -A pole along a boundary prime downstairs would pull back with order at -least $e$, whereas \eqref{adj:root-order} permits only $e-2$ zeros of -$\tau$. No such pole is possible. Away from the boundary, $\tau$ has -no divisorial zeros downstairs. Normality extends the ratio across -codimension two, and compactness makes it constant. This also handles -a disconnected cover, since the full root group acts transitively on -its components over a connected fiber. The highest Hodge piece of the -chosen character is therefore one-dimensional. - -Finally, changing a local frame of $m b^*H$ changes the root by an -$m$th root of the corresponding base factor; changing the base volume -form contributes the inverse canonical transition. Taking the $m$th tensor power removes -the finite root ambiguities. These are precisely the transitions of -$m(b^*H-K_S)$, proving the asserted identity on $S^\circ$. -\end{proof} - -\begin{lemma}[Extension across the discriminant] -\label{adj:extension} -The parabolic extension of the eigenline in Lemma~\ref{adj:root} is $L_S$ -of \eqref{adj:discriminant}. This identification persists on higher -smooth base models with simple normal crossing complement. -\end{lemma} - -\begin{proof} -Fix a general point of a prime $Q_0=(t=0)$ on $S$. We normalize the -root form by a nonvanishing local frame of $b^*H-K_S$. More explicitly, -if the meromorphic section of $m b^*H$ chosen in Lemma~\ref{adj:root} -is $a$ times a nonvanishing local frame, divide its root by $a^{1/m}$ -and by the local base volume form. This is an ordinary frame after a -finite power substitution; the finite ambiguity is precisely part of -the parabolic convention. Thus zeros or poles of the chosen meromorphic -base section do not enter the following exponent. -Tangential base coordinates may be treated as parameters. On a log resolution, -\[ - t=\prod_{i=1}^k x_i^{a_i} -\] -up to a nowhere-zero factor, and the squared absolute root form has -vertical density factors $|x_i|^{-2u_i}$. Horizontal boundary factors -are integrable because their coefficients are less than one. -The squared Hodge norm of the relative form is its fiber integral. -Writing $s_i=-\log|x_i|$, the integral is computed on slices -\[ - \sum_i a_i s_i=-\log|t| -\] -with exponential weight $\exp(-2\sum_i(1-u_i)s_i)$. -Conversion to the ordinary base area element contributes $|t|^{-2}$. -The minimum in \eqref{adj:threshold} thus gives constants $C,N>0$ such -that, on a sufficiently small punctured disk, -\begin{equation} -\label{adj:norm-power} - C^{-1}|t|^{2(t_{Q_0}-1)}(-\log|t|)^{-N} - \leq \|\tau\|^2 - \leq C|t|^{2(t_{Q_0}-1)}(-\log|t|)^N. -\end{equation} -Indeed, the slice has at most polynomial volume in $-\log|t|$, which -proves the upper estimate after factoring out the least exponential -decay. For the lower estimate take a chart along a component attaining -the minimum, away from all other vertical components, and integrate -over a fixed compact set in the remaining directions. Compactness -allows finitely many charts; the integrable horizontal factors do not -alter the power of $|t|$. - -After a local power substitution making monodromy unipotent, extending -Hodge frames and their duals have norms bounded by powers of -$-\log|t|$. These are the nilpotent-orbit estimates of -Schmid and Cattani--Kaplan--Schmid \cite{Schmid,CKS}, in the form used in -\cite[Section~4.1]{LI}. Consequently the comparison between a -frame of $b^*H-K_S$ and an extending Hodge frame has rational order -$t_{Q_0}-1$. Multiplication by $t^{1-t_{Q_0}}$, on a cover where this -power is integral, removes exactly that order. Both the corrected -comparison and its inverse have at most logarithmic growth. A -holomorphic function on a punctured disk with such growth has no pole; -applying this to the inverse excludes a zero. The same argument with -tangential parameters extends the identification across each divisor, -and normal extension treats codimension two. Thus the extending line is -\[ - b^*H-K_S+\sum_{Q_0}(t_{Q_0}-1)Q_0=L_S. -\] -For the last assertion, first make the boundary monodromies -unipotent. In canonical extending frames a local monomial pullback -$t_i=\text{unit}\cdot\prod_j s_j^{a_{ij}}$ replaces the commuting -nilpotent residues $N_i$ by $\sum_i a_{ij}N_i$, which are again -nilpotent. Both the canonical flat extension and its extending Hodge -filtration therefore pull back without an additional divisor. -Descending with rational weights incorporates the finite-monodromy -parts and gives the same conclusion for the parabolic line, including -exceptional base divisors; see \cite[Section~4]{LI}. -\end{proof} - -We have now identified the moduli line with an extreme Hodge line, so -Lemma~\ref{adj:period-input} makes it nef and gives an actual line -descent. To obtain a big line on the quotient, we must compare the -variation seen by this one line with that seen by the whole family. -Effectivity of the original boundary is decisive in this comparison. - -\subsection{The line detects the full period rank} - -\begin{lemma}[Equality of period ranks] -\label{adj:rank} -For the family of Lemma~\ref{adj:root}, the line period map of its root -eigenline and the full period map of $R^lh_*\R$ have the same generic -differential rank. -\end{lemma} - -\begin{proof} -Work in an open patch on which the ranks are constant, and let $\xi$ -be a holomorphic base vector field in the kernel of the line period -map. Choose a local frame $\tau$ of the eigenline, viewed as a -relative top form on $V^\circ$. The infinitesimal period formula says -that contraction with $\tau$ sends the Kodaira--Spencer class of $\xi$ -to zero in the fiber cohomology $H^1(\Omega^{l-1})$. - -Here is the corresponding lifting construction. Pull back the tangent -sequence to the line generated by $\xi$ and push it out along -\[ - T_{V^\circ/S^\circ}\longrightarrow - T_{V^\circ/S^\circ}\otimes K_{V^\circ/S^\circ} - \simeq\Omega^{l-1}_{V^\circ/S^\circ}, - \qquad v\longmapsto\iota_v\tau. -\] -Write $U$ for a sufficiently small Stein base patch and put -$\mathcal F=\Omega^{l-1}_{V_U/U}$. The pushed-out extension is -\[ - 0\longrightarrow\mathcal F - \longrightarrow\mathcal E - \longrightarrow\cO_{V_U}\longrightarrow0, -\] -with class in $H^1(V_U,\mathcal F)$. Shrink within the generic locus -so that $R^1h_*\mathcal F$ is locally free and coherent base change -holds. The contraction formula makes the image of this class in -$H^0(U,R^1h_*\mathcal F)$ zero. The Leray exact sequence and the -vanishing $H^1(U,h_*\mathcal F)=0$ show that -$H^1(V_U,\mathcal F)\to H^0(U,R^1h_*\mathcal F)$ is injective. -The extension class is therefore zero, and a holomorphic splitting -exists on the entire $V_U$. -Where $\tau$ is nonzero, divide the vertical part of the splitting by -$\tau$. This gives a meromorphic vector field $\widetilde\xi$ on the -resolved cover projecting to $\xi$, with poles bounded by the zero -divisor of $\tau$. Average it under the root group. Its projection -remains $\xi$, and the averaged field is invariant. - -We check this field at codimension-one points of the original normal -total space. At a boundary prime use the ramification coordinate -$x=y^e$ from \eqref{adj:root-order}. A tangential coefficient of an -invariant vector field has Laurent exponents congruent to zero modulo -$e$. Its permitted pole order is at most $e-2$, so every exponent is -nonnegative. The coefficient of $\partial/\partial y$ has exponents -congruent to one modulo $e$. Its first possible negative exponent is -$1-e$, which is again excluded. The normal coefficient is therefore -divisible by $y$. Thus the descended vector field is holomorphic and -tangent to the boundary. At a prime outside the boundary the root -form has no zero, and the conclusion is immediate. - -The resolution is an isomorphism at the codimension-one points just -used. Remaining exceptional divisors map into codimension at least -two in the original normal space. Holomorphic derivations extend -across that subset, so the descended field is holomorphic everywhere -on the family over our base patch and still projects to $\xi$. -Its local flow preserves the boundary components, their coefficients, -and the singular locus. By the functorial choice made above, the flow -first lifts to $X$ and preserves its crepant boundary $D_X$. - -This flow then lifts holomorphically to the root cover. Indeed, on the good -base patch the defining pluriform has divisor exactly the negative -multiple of the boundary. Pullback by a pair-preserving flow gives a -pluriform with the same divisor. Their ratio is a holomorphic unit on -the normal space; its $m$th root can be chosen starting at one in the -flow parameter. The root normalized at flow time zero is unique as a germ. Thus the -local cover lifts agree on overlaps of the unbranched open and glue -by normality. Functorial -resolution then lifts the flow to $V^\circ$. -Consequently the Kodaira--Spencer class of the resolved family vanishes -on $\xi$, and so does its full period differential. -The reverse kernel inclusion holds because the line is part of the -full Hodge filtration. The two kernels, and therefore the generic -ranks, are equal. -\end{proof} - -The argument is the analytic counterpart of Ambro's infinitesimal -comparison \cite[Proposition~2.1]{Ambro}. Notice exactly where it uses -$D\geq0$: the strict pole bound in \eqref{adj:root-order} is imposed at -primes of the original pair. Negative crepant exceptional coefficients -are allowed because their images have codimension at least two. - -\subsection{Replacing the moduli line by a klt boundary} - -\begin{proof}[Proof of Proposition~\ref{adj:base}] -Apply Lemma~\ref{adj:period-input} to the variation of -Lemma~\ref{adj:root}. Its monodromy is quasi-unipotent by the monodromy -theorem for this proper K\"ahler family \cite{Schmid}. Lemma~\ref{adj:extension} -identifies its parabolic line with $L_S$. Passing to the indicated -higher smooth projective model, and keeping the notation $S$, we have -\[ - L_S\qsim p^*P -\] -for a nef rational line $P$ on a smooth projective quotient $Q$. -Let $r$ be the generic rank of the line period map. Numerical dimension -of a nef line is unchanged by a surjective pullback, so -\[ - r=\nu(L_S)=\nu(P)\leq\dim Q - \leq\operatorname{rank}(d\Phi)=r, -\] -where $\Phi$ is the full period map and the last equality is -Lemma~\ref{adj:rank}. Thus $P$ is big if $\dim Q>0$. If $Q$ is a point, -$L_S$ is rationally trivial. - -Suppose first that $\dim Q>0$. By Kodaira's lemma write -$P\qsim A+E$, with $A$ ample rational and $E\geq0$ rational. For every -small positive rational $\varepsilon$, -\begin{equation} -\label{adj:small-fixed-part} - P\qsim A_\varepsilon+\varepsilon E, - \qquad A_\varepsilon=(1-\varepsilon)P+\varepsilon A - \quad\text{ample}. -\end{equation} -The pair $(S,\Delta_S)$ is sub-klt. On a fixed log resolution of -$\Delta_S+p^*E$, choose $\varepsilon$ sufficiently small that -$(S,\Delta_S+\varepsilon p^*E)$ remains sub-klt. Choose a sufficiently -large divisible integer $N$ and a general member -$G\in|N A_\varepsilon|$. The system $|Np^*A_\varepsilon|$ is -basepoint-free; on that resolution its general member meets the fixed -boundary transversely. Taking $N$ large makes its coefficient $1/N$ -less than one. Bertini therefore gives an effective rational divisor -\[ - E_S=\varepsilon p^*E+\frac1N p^*G - \qsim L_S -\] -with $(S,\Delta_S+E_S)$ sub-klt. In the point case take $E_S=0$. -In either case, -\begin{equation} -\label{adj:upstairs-presentation} - K_S+\Delta_S+E_S\qsim b^*H. -\end{equation} - -Set $D_Z=b_*(\Delta_S+E_S)$. This divisor is effective. Indeed, for -each prime of the original normal base, some component of its pullback -on $Y$ dominates it. That component has multiplicity $a\geq1$ and -boundary coefficient $u\geq0$, so its threshold is at most -$(1-u)/a\leq1$. Thus the coefficient of the discriminant at every -nonexceptional base prime is nonnegative, and $E_S$ is effective. - -Choose compatible canonical divisors and a rational Cartier divisor -representing $H$. The line identity -\eqref{adj:upstairs-presentation} gives an integer $m>0$ and a rational -function $\varphi$ in the common function field of $S$ and $Z$ such that -\[ - K_S+\Delta_S+E_S-b^*H=\frac1m\operatorname{div}_S(\varphi). -\] -Pushing down gives -\[ - K_Z+D_Z-H=\frac1m\operatorname{div}_Z(\varphi). -\] -In particular, $K_Z+D_Z$ is rational Cartier and rationally linearly -equivalent to $H$. Pulling the latter divisor equality back and -comparing with the former proves the crepant identity -\[ - K_S+\Delta_S+E_S=b^*(K_Z+D_Z). -\] -Since the pair upstairs is sub-klt, the effective pair $(Z,D_Z)$ is klt. -This proves the proposition. -\end{proof} +\section{An ordinary adjoint on the projective base} +\label{sec:adjoint} + +The induction will produce a morphism to a projective variety for which an +ordinary adjoint is pulled back from the base. To apply the projective +numerical-semiampleness theorem, we must realize the line downstairs as an +ordinary klt adjoint. The following proposition provides this realization. +The total space need not be projective. + +\begin{proposition}[An adjoint on the base] +\label{adj:base} +Let $(Y,D)$ be an effective klt pair with rational boundary on a normal +compact K\"ahler space. Let $f:Y\to Z$ be a surjective morphism with connected +fibers to a normal projective variety, and suppose that +$\dim Y>\dim Z$. If $H$ is a rational Cartier line bundle on $Z$ and +\begin{equation} +\label{adj:initial} + K_Y+D\qsim f^*H, +\end{equation} +then there is an effective rational divisor $D_Z$ such that $(Z,D_Z)$ is klt +and +\[ + H\qsim K_Z+D_Z. +\] +\end{proposition} + +Ambro proves the corresponding statement for projective total spaces +\cite[Theorem~0.2]{Ambro}. We follow the same separation into a discriminant +and a Hodge-theoretic moduli line. The two points requiring attention here +are the real, possibly irrational polarization of a K\"ahler family and the +construction of its cyclic cover without a meromorphic frame of $K_Y$. +The period results recalled next address the first point; we give the cover +and the required infinitesimal calculation explicitly. + +\subsection{The period input and the discriminant} + +For a pure variation of Hodge structures on a smooth open set, the +\emph{line period map} of a rank-one highest Hodge piece is the map to the +projective space of the flat vector space that records this line in a local +flat trivialization. Its generic differential rank is independent of the +trivialization. The full period map records the entire Hodge filtration. +For quasi-unipotent boundary monodromy, the \emph{parabolic extension} of the +line is the rational line bundle obtained by making the monodromy unipotent +on local finite covers, extending the highest Hodge piece, and descending +with its rational boundary weights. + +We use the following precise consequence of the period results in +\cite[Lemmas~4.1 and~4.3 and Proposition~4.5]{LI}. + +\begin{lemma}[Period quotient input] +\label{adj:period-input} +Let $S$ be a smooth projective variety and let $S^\circ\subset S$ have simple +normal crossing complement. Let $\mathbb V$ be a real-polarizable pure +variation on $S^\circ$ with an integral lattice and quasi-unipotent local +monodromy. Suppose that a complex direct summand of $\mathbb V_{\C}$, as a +variation of Hodge structures, has a rank-one highest Hodge piece, with parabolic extension $L$. Then $L$ is nef, +and its numerical dimension is the generic rank of its line period map. + +After modification $\tau:S'\to S$, there are a smooth projective variety +$Q$, a surjective morphism $p:S'\to Q$ with connected fibers, and a nef +rational line bundle $P$ on $Q$ such that +\begin{equation} +\label{adj:period-descent} + \tau^*L\qsim p^*P. +\end{equation} +Moreover, $\dim Q$ is at most the generic rank of the full period map of +$\mathbb V$. If $Q$ is a point, $\tau^*L$ is rationally trivial. +\end{lemma} + +Here the identity is in $\Pic(S')\otimes\Q$, including the boundary. +For clarity, the dimension assertion follows because the descended, +generically immersive period map on $Q$ belongs to an adjoint variation +obtained by tensor constructions from $\mathbb V$. +The rational adjoint reduction in \cite[Lemma~4.3]{LI} retains entire +rational simple factors, so its monodromy is discrete even when the +original real polarization is irrational. It also kills only finite +scalar monodromy after taking a power. Thus +\eqref{adj:period-descent} does not discard an arbitrary flat twist. +We shall need exactly this actual line identity. + +Choose a projective resolution $b:S\to Z$, and normalize the main +component of $Y\times_Z S$. Choose its log resolution functorially +with respect to local isomorphisms preserving the marked boundary; +over the open where $b$ is an isomorphism this is a functorial log +resolution of $(Y,D)$. Denote the resulting smooth compact K\"ahler +space by $X$, with maps $\pi:X\to Y$ and $g:X\to S$. +Such a resolution is available in the complex analytic category; +see \cite[Theorem~1.1 and the preceding analytic remark]{BierstoneMilman2008}. +The functorial choice will matter when we lift local flows. +Additional resolutions used to compute thresholds and fiber integrals +are auxiliary and do not replace the family whose cohomology we use. +Define the crepant rational divisor $D_X$ by +\[ + K_X+D_X=\pi^*(K_Y+D). +\] +Its exceptional coefficients may be negative; all its coefficients are +strictly less than one. For a prime divisor $Q_0$ on $S$, let +\[ + t_{Q_0}=\sup\{c\in\R:(X,D_X+c g^*Q_0) + \text{ is log canonical over the general point of }Q_0\}. +\] +The definition is unchanged on a higher crepant resolution. It gives a +positive rational number, computed by a log resolution as +\begin{equation} +\label{adj:threshold} + t_{Q_0}=\min_i\frac{1-u_i}{a_i}, +\end{equation} +where $g^*Q_0=\sum_i a_i E_i$ over its general point and $u_i$ is the +coefficient of $E_i$ in the crepant boundary there. Define +\begin{equation} +\label{adj:discriminant} + \Delta_S=\sum_{Q_0}(1-t_{Q_0})Q_0, + \qquad L_S=b^*H-K_S-\Delta_S. +\end{equation} +Only finitely many terms of $\Delta_S$ are nonzero: off a suitable proper +analytic subset the resolved pair is relatively simple normal crossing, +the map is smooth, and the threshold of a base prime is one. We may choose +$S$ with a simple normal crossing divisor containing this exceptional set +and the support of $\Delta_S$. + +The coefficients of $\Delta_S$ are strictly less than one. To prove +Proposition~\ref{adj:base}, we will show that $L_S$ has an effective rational +representative whose addition to $\Delta_S$ is sub-klt. The relevant +positivity comes from the following root construction. + +\subsection{The root cover and its highest Hodge line} + +\begin{lemma}[The root eigenline] +\label{adj:root} +In the setting of Proposition~\ref{adj:base}, there is a dense Zariski open +$S^\circ\subset S$ and a smooth proper family $h:V^\circ\to S^\circ$ obtained +from a cyclic cover and resolution, such that $R^l h_*\R$, where +$l=\dim Y-\dim Z$, is real-polarizable and has an integral lattice. +One character summand of its complexification has a rank-one highest +Hodge piece. On $S^\circ$, this line agrees, as a rational line bundle, +with $b^*H-K_S$. +\end{lemma} + +\begin{proof} +Choose an integer $m>0$ clearing all divisors and rational line identities. +A meromorphic section of the line $m b^*H$ exists because $S$ is projective. +Its pullback, under the identity +\[ + m(K_X+D_X)\simeq g^*(m b^*H), +\] +and division by the canonical meromorphic section of $mD_X$ give a +meromorphic section $\sigma$ of $K_X^{\otimes m}$. +On the open where $\sigma$ has neither zeros nor poles, take its $m$th +roots in the fibers of $K_X$. In local canonical frames this is the +equation $z^m=\sigma$; on overlaps the roots transform by the transition +functions of $K_X$. The local covers therefore glue without a +meromorphic trivialization of $K_X$, and without choosing a root of $H$. + +The Grauert--Remmert extension theorem for finite analytic covers +\cite[Exposé~XII, Proposition~5.3 and Theorem~5.4]{SGA1} extends it +normally across the divisor of $\sigma$, giving a finite cyclic cover +of $X$, possibly disconnected. It is compact K\"ahler. +Take a resolution functorial for the cyclic action and for local +isomorphisms of the pair, as in \cite[Section~1.1 and Lemma~1.1]{Ambro}. +Thus both resolutions are functorial. A local flow preserving the +original pair lifts through them once its lift to the root cover has +been chosen; we will construct that lift explicitly below. +The resulting compact manifold $V$ is K\"ahler. After deleting from $S$ the zeros and +poles of the chosen base section and the degeneration locus, the map +$h:V^\circ\to S^\circ$ is smooth and proper. We take $S^\circ$ +inside the isomorphism locus of $b$. Shrinking it further, the original +fibers of $Y\to Z$ are normal effective klt pairs and the chosen +resolutions restrict to resolutions of those pairs. + +Average a global K\"ahler class on $V$ under the cyclic group. Its +restriction is a flat real section of $R^2h_*\R$ and polarizes the +primitive pieces. Their Lefschetz decomposition supplies a flat real +polarization of the full degree-$l$ cohomology. We retain the full +cohomology local system, with lattice $R^lh_*\Z$ modulo torsion; an +irrational primitive summand is not required to carry its own lattice. +The finite group action preserves this variation and its polarization. + +Locally over the base, the tautological root, divided by a local base +volume form, gives a relative meromorphic top form $\tau$ on the cover. +It belongs to a fixed character of the cyclic group. The klt condition +makes its squared volume locally integrable, including after resolution, +so it extends holomorphically: a meromorphic top form with a divisorial +pole cannot be locally square integrable. +To see the orders directly, consider a boundary prime on the original +normal fiber with coefficient $u=p/e\in(0,1)$ in lowest terms. At a +general point the cover has local coordinate $x=y^e$, and the pulled-back +root form has order +\begin{equation} +\label{adj:root-order} + e-1-eu=e-1-p\in\{0,\ldots,e-2\}. +\end{equation} +Away from the boundary its order at a nonexceptional prime is zero. +The corresponding calculation on a crepant resolution uses coefficients +less than one and proves integrability at exceptional divisors as well. + +Suppose that $\tau'$ is another holomorphic top form of the same +character on a fiber. The ratio $\tau'/\tau$ is an invariant meromorphic +function and descends to the original normal connected fiber. +A pole along a boundary prime downstairs would pull back with order at +least $e$, whereas \eqref{adj:root-order} permits only $e-2$ zeros of +$\tau$. No such pole is possible. Away from the boundary, $\tau$ has +no divisorial zeros downstairs. Normality extends the ratio across +codimension two, and compactness makes it constant. This also handles +a disconnected cover, since the full root group acts transitively on +its components over a connected fiber. The highest Hodge piece of the +chosen character is therefore one-dimensional. + +Finally, changing a local frame of $m b^*H$ changes the root by an +$m$th root of the corresponding base factor; changing the base volume +form contributes the inverse canonical transition. Taking the $m$th tensor power removes +the finite root ambiguities. These are precisely the transitions of +$m(b^*H-K_S)$, proving the asserted identity on $S^\circ$. +\end{proof} + +\begin{lemma}[Extension across the discriminant] +\label{adj:extension} +The parabolic extension of the eigenline in Lemma~\ref{adj:root} is $L_S$ +of \eqref{adj:discriminant}. This identification persists on higher +smooth base models with simple normal crossing complement. +\end{lemma} + +\begin{proof} +Fix a general point of a prime $Q_0=(t=0)$ on $S$. We normalize the +root form by a nonvanishing local frame of $b^*H-K_S$. More explicitly, +if the meromorphic section of $m b^*H$ chosen in Lemma~\ref{adj:root} +is $a$ times a nonvanishing local frame, divide its root by $a^{1/m}$ +and by the local base volume form. This is an ordinary frame after a +finite power substitution; the finite ambiguity is precisely part of +the parabolic convention. Thus zeros or poles of the chosen meromorphic +base section do not enter the following exponent. +Tangential base coordinates may be treated as parameters. On a log resolution, +\[ + t=\prod_{i=1}^k x_i^{a_i} +\] +up to a nowhere-zero factor, and the squared absolute root form has +vertical density factors $|x_i|^{-2u_i}$. Horizontal boundary factors +are integrable because their coefficients are less than one. +The squared Hodge norm of the relative form is its fiber integral. +Writing $s_i=-\log|x_i|$, the integral is computed on slices +\[ + \sum_i a_i s_i=-\log|t| +\] +with exponential weight $\exp(-2\sum_i(1-u_i)s_i)$. +Conversion to the ordinary base area element contributes $|t|^{-2}$. +The minimum in \eqref{adj:threshold} thus gives constants $C,N>0$ such +that, on a sufficiently small punctured disk, +\begin{equation} +\label{adj:norm-power} + C^{-1}|t|^{2(t_{Q_0}-1)}(-\log|t|)^{-N} + \leq \|\tau\|^2 + \leq C|t|^{2(t_{Q_0}-1)}(-\log|t|)^N. +\end{equation} +Indeed, the slice has at most polynomial volume in $-\log|t|$, which +proves the upper estimate after factoring out the least exponential +decay. For the lower estimate take a chart along a component attaining +the minimum, away from all other vertical components, and integrate +over a fixed compact set in the remaining directions. Compactness +allows finitely many charts; the integrable horizontal factors do not +alter the power of $|t|$. + +After a local power substitution making monodromy unipotent, extending +Hodge frames and their duals have norms bounded by powers of +$-\log|t|$. These are the nilpotent-orbit estimates of +Schmid and Cattani--Kaplan--Schmid \cite{Schmid,CKS}, in the form used in +\cite[Section~4.1]{LI}. Consequently the comparison between a +frame of $b^*H-K_S$ and an extending Hodge frame has rational order +$t_{Q_0}-1$. Multiplication by $t^{1-t_{Q_0}}$, on a cover where this +power is integral, removes exactly that order. Both the corrected +comparison and its inverse have at most logarithmic growth. A +holomorphic function on a punctured disk with such growth has no pole; +applying this to the inverse excludes a zero. The same argument with +tangential parameters extends the identification across each divisor, +and normal extension treats codimension two. Thus the extending line is +\[ + b^*H-K_S+\sum_{Q_0}(t_{Q_0}-1)Q_0=L_S. +\] +For the last assertion, first make the boundary monodromies +unipotent. In canonical extending frames a local monomial pullback +$t_i=\text{unit}\cdot\prod_j s_j^{a_{ij}}$ replaces the commuting +nilpotent residues $N_i$ by $\sum_i a_{ij}N_i$, which are again +nilpotent. Both the canonical flat extension and its extending Hodge +filtration therefore pull back without an additional divisor. +Descending with rational weights incorporates the finite-monodromy +parts and gives the same conclusion for the parabolic line, including +exceptional base divisors; see \cite[Section~4]{LI}. +\end{proof} + +We have now identified the moduli line with an extreme Hodge line, so +Lemma~\ref{adj:period-input} makes it nef and gives an actual line +descent. To obtain a big line on the quotient, we must compare the +variation seen by this one line with that seen by the whole family. +Effectivity of the original boundary is decisive in this comparison. + +\subsection{The line detects the full period rank} + +\begin{lemma}[Equality of period ranks] +\label{adj:rank} +For the family of Lemma~\ref{adj:root}, the line period map of its root +eigenline and the full period map of $R^lh_*\R$ have the same generic +differential rank. +\end{lemma} + +\begin{proof} +Work in an open patch on which the ranks are constant, and let $\xi$ +be a holomorphic base vector field in the kernel of the line period +map. Choose a local frame $\tau$ of the eigenline, viewed as a +relative top form on $V^\circ$. The infinitesimal period formula says +that contraction with $\tau$ sends the Kodaira--Spencer class of $\xi$ +to zero in the fiber cohomology $H^1(\Omega^{l-1})$. + +Here is the corresponding lifting construction. Pull back the tangent +sequence to the line generated by $\xi$ and push it out along +\[ + T_{V^\circ/S^\circ}\longrightarrow + T_{V^\circ/S^\circ}\otimes K_{V^\circ/S^\circ} + \simeq\Omega^{l-1}_{V^\circ/S^\circ}, + \qquad v\longmapsto\iota_v\tau. +\] +Write $U$ for a sufficiently small Stein base patch and put +$\mathcal F=\Omega^{l-1}_{V_U/U}$. The pushed-out extension is +\[ + 0\longrightarrow\mathcal F + \longrightarrow\mathcal E + \longrightarrow\cO_{V_U}\longrightarrow0, +\] +with class in $H^1(V_U,\mathcal F)$. Shrink within the generic locus +so that $R^1h_*\mathcal F$ is locally free and coherent base change +holds. The contraction formula makes the image of this class in +$H^0(U,R^1h_*\mathcal F)$ zero. The Leray exact sequence and the +vanishing $H^1(U,h_*\mathcal F)=0$ show that +$H^1(V_U,\mathcal F)\to H^0(U,R^1h_*\mathcal F)$ is injective. +The extension class is therefore zero, and a holomorphic splitting +exists on the entire $V_U$. +Where $\tau$ is nonzero, divide the vertical part of the splitting by +$\tau$. This gives a meromorphic vector field $\widetilde\xi$ on the +resolved cover projecting to $\xi$, with poles bounded by the zero +divisor of $\tau$. Average it under the root group. Its projection +remains $\xi$, and the averaged field is invariant. + +We check this field at codimension-one points of the original normal +total space. At a boundary prime use the ramification coordinate +$x=y^e$ from \eqref{adj:root-order}. A tangential coefficient of an +invariant vector field has Laurent exponents congruent to zero modulo +$e$. Its permitted pole order is at most $e-2$, so every exponent is +nonnegative. The coefficient of $\partial/\partial y$ has exponents +congruent to one modulo $e$. Its first possible negative exponent is +$1-e$, which is again excluded. The normal coefficient is therefore +divisible by $y$. Thus the descended vector field is holomorphic and +tangent to the boundary. At a prime outside the boundary the root +form has no zero, and the conclusion is immediate. + +The resolution is an isomorphism at the codimension-one points just +used. Remaining exceptional divisors map into codimension at least +two in the original normal space. Holomorphic derivations extend +across that subset, so the descended field is holomorphic everywhere +on the family over our base patch and still projects to $\xi$. +Its local flow preserves the boundary components, their coefficients, +and the singular locus. By the functorial choice made above, the flow +first lifts to $X$ and preserves its crepant boundary $D_X$. + +This flow then lifts holomorphically to the root cover. Indeed, on the good +base patch the defining pluriform has divisor exactly the negative +multiple of the boundary. Pullback by a pair-preserving flow gives a +pluriform with the same divisor. Their ratio is a holomorphic unit on +the normal space; its $m$th root can be chosen starting at one in the +flow parameter. The root normalized at flow time zero is unique as a germ. Thus the +local cover lifts agree on overlaps of the unbranched open and glue +by normality. Functorial +resolution then lifts the flow to $V^\circ$. +Consequently the Kodaira--Spencer class of the resolved family vanishes +on $\xi$, and so does its full period differential. +The reverse kernel inclusion holds because the line is part of the +full Hodge filtration. The two kernels, and therefore the generic +ranks, are equal. +\end{proof} + +The argument is the analytic counterpart of Ambro's infinitesimal +comparison \cite[Proposition~2.1]{Ambro}. Notice exactly where it uses +$D\geq0$: the strict pole bound in \eqref{adj:root-order} is imposed at +primes of the original pair. Negative crepant exceptional coefficients +are allowed because their images have codimension at least two. + +\subsection{Replacing the moduli line by a klt boundary} + +\begin{proof}[Proof of Proposition~\ref{adj:base}] +Apply Lemma~\ref{adj:period-input} to the variation of +Lemma~\ref{adj:root}. Its monodromy is quasi-unipotent by the monodromy +theorem for this proper K\"ahler family \cite{Schmid}. Lemma~\ref{adj:extension} +identifies its parabolic line with $L_S$. Passing to the indicated +higher smooth projective model, and keeping the notation $S$, we have +\[ + L_S\qsim p^*P +\] +for a nef rational line $P$ on a smooth projective quotient $Q$. +Let $r$ be the generic rank of the line period map. Numerical dimension +of a nef line is unchanged by a surjective pullback, so +\[ + r=\nu(L_S)=\nu(P)\leq\dim Q + \leq\operatorname{rank}(d\Phi)=r, +\] +where $\Phi$ is the full period map and the last equality is +Lemma~\ref{adj:rank}. Thus $P$ is big if $\dim Q>0$. If $Q$ is a point, +$L_S$ is rationally trivial. + +Suppose first that $\dim Q>0$. By Kodaira's lemma write +$P\qsim A+E$, with $A$ ample rational and $E\geq0$ rational. For every +small positive rational $\varepsilon$, +\begin{equation} +\label{adj:small-fixed-part} + P\qsim A_\varepsilon+\varepsilon E, + \qquad A_\varepsilon=(1-\varepsilon)P+\varepsilon A + \quad\text{ample}. +\end{equation} +The pair $(S,\Delta_S)$ is sub-klt. On a fixed log resolution of +$\Delta_S+p^*E$, choose $\varepsilon$ sufficiently small that +$(S,\Delta_S+\varepsilon p^*E)$ remains sub-klt. Choose a sufficiently +large divisible integer $N$ and a general member +$G\in|N A_\varepsilon|$. The system $|Np^*A_\varepsilon|$ is +basepoint-free; on that resolution its general member meets the fixed +boundary transversely. Taking $N$ large makes its coefficient $1/N$ +less than one. Bertini therefore gives an effective rational divisor +\[ + E_S=\varepsilon p^*E+\frac1N p^*G + \qsim L_S +\] +with $(S,\Delta_S+E_S)$ sub-klt. In the point case take $E_S=0$. +In either case, +\begin{equation} +\label{adj:upstairs-presentation} + K_S+\Delta_S+E_S\qsim b^*H. +\end{equation} + +Set $D_Z=b_*(\Delta_S+E_S)$. This divisor is effective. Indeed, for +each prime of the original normal base, some component of its pullback +on $Y$ dominates it. That component has multiplicity $a\geq1$ and +boundary coefficient $u\geq0$, so its threshold is at most +$(1-u)/a\leq1$. Thus the coefficient of the discriminant at every +nonexceptional base prime is nonnegative, and $E_S$ is effective. + +Choose compatible canonical divisors and a rational Cartier divisor +representing $H$. The line identity +\eqref{adj:upstairs-presentation} gives an integer $m>0$ and a rational +function $\varphi$ in the common function field of $S$ and $Z$ such that +\[ + K_S+\Delta_S+E_S-b^*H=\frac1m\operatorname{div}_S(\varphi). +\] +Pushing down gives +\[ + K_Z+D_Z-H=\frac1m\operatorname{div}_Z(\varphi). +\] +In particular, $K_Z+D_Z$ is rational Cartier and rationally linearly +equivalent to $H$. Pulling the latter divisor equality back and +comparing with the former proves the crepant identity +\[ + K_S+\Delta_S+E_S=b^*(K_Z+D_Z). +\] +Since the pair upstairs is sub-klt, the effective pair $(Z,D_Z)$ is klt. +This proves the proposition. +\end{proof} diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/05-zero.tex b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/05-zero.tex index 4e408e205d58aa47a04f3cfbfa0bcd95ac0404cb..e8f00dd0229d3fa98f9852d09602e83b61f385e9 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/05-zero.tex +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/05-zero.tex @@ -1,1745 +1,1745 @@ -\section{Nef classes in algebraic dimension zero}\label{sec:zero} - -A semiample line bundle on a space of algebraic dimension zero is -rationally trivial. The induction therefore requires a stronger -conclusion in this case: the additional nef class must itself vanish. -The ordinary adjoint decomposition reduces this assertion to a question -about nef line bundles on simple manifolds. We first carry out that -geometric reduction. The remaining argument adapts the meromorphic -nonvanishing construction of \cite[Section~6]{K} to a line bundle that -need not be canonical. - -\begin{proposition}\label{zero:vanishing} -Let \(n\ge1\). Assume that the algebraic-dimension-zero assertion of -Theorem~\ref{thm:decomposition} holds in every dimension less than \(n\). -Let \(T\) be a smooth connected compact Kähler \(n\)-fold with -\(a(T)=0\), and let \(B\) be an effective rational simple normal crossing -boundary with coefficients less than one. If \(K_T+B\) is -pseudo-effective, then every nef rational holomorphic line bundle \(M\) -on \(T\) satisfies \(c_1(M)=0\). -\end{proposition} - -For intersections in this Section, a line bundle and its first Chern -class are distinguished when necessary by braces: \(\{H\}=c_1(H)\). -An inequality between real \((1,1)\)-classes is in pseudo-effective -order. A compact manifold is \emph{simple} if no positive-dimensional -proper compact analytic subvariety passes through a very general point. -Here, as usual, ``very general'' permits deletion of a countable union -of proper analytic subsets. - -\subsection{Reduction to a simple symplectic manifold} - -The following elementary intersection test lets us use finite covers -without requiring the nef line bundle to descend through them. - -\begin{lemma}\label{zero:mass-test} -Let \(V\) be a smooth compact Kähler \(n\)-fold, where \(n\ge1\), -and let \(\gamma\) be a nef -class, and let \(C\) be a nef and big class. If -\(\gamma\cdot C^{n-1}=0\), then \(\gamma=0\). -In particular, if \(V\) admits a generically finite surjective morphism -to a complex torus of algebraic dimension zero, every nef rational line -class on \(V\) is zero. -\end{lemma} - -\begin{proof} -Choose a Kähler class \(\omega\) and \(\varepsilon>0\) such that -\(C-\varepsilon\omega\) is pseudo-effective. When \(n\ge2\), the -identity -\[ - C^{n-1}-(\varepsilon\omega)^{n-1} - =(C-\varepsilon\omega) - \sum_{j=0}^{n-2}C^{n-2-j}(\varepsilon\omega)^j -\] -and nonnegativity of intersections of a pseudo-effective class with -nef classes give -\[ - 0=\gamma\cdot C^{n-1} - \ge \varepsilon^{n-1}\gamma\cdot\omega^{n-1}\ge0. -\] -The same conclusion is immediate when \(n=1\). A positive current in -\(\gamma\) thus has zero mass, so it vanishes. - -For the last assertion, write \(f:V\to A\) for the morphism. The -pushforward of the nef rational line class is a rational -pseudo-effective \((1,1)\)-class on \(A\). Averaging a positive current -under translations represents it by a constant semipositive form. The -kernel of a rational such form is the tangent space of a subtorus; the -induced positive integral form, after clearing denominators, polarizes -the quotient. Since \(a(A)=0\), this quotient is a point, and hence -\(f_*\gamma=0\). For a Kähler class \(\eta\) on \(A\), the class -\(C=f^*\eta\) is nef and big and -\[ - \gamma\cdot C^{n-1}=(f_*\gamma)\cdot\eta^{n-1}=0. -\] -Apply the first assertion. -\end{proof} - -\begin{lemma}\label{zero:geometric-reduction} -Under the induction hypothesis of Proposition~\ref{zero:vanishing}, -it suffices to prove the following assertion: every nef rational line -class on a smooth simple compact Kähler manifold of algebraic dimension -zero carrying a generically nondegenerate holomorphic two-form is zero. -\end{lemma} - -\begin{proof} -Apply the ordinary good divisorial decomposition to \(K_T+B\). On a -smooth modification its semiample part is rationally trivial, because -\(a(T)=0\). After the klt modification convention of -Section~\ref{sec:preliminaries}, the resulting ordinary adjoint still -equals its rational negative divisor as an actual rational line bundle. -The contraction of a known negative part -\cite[Theorem~2.13 and Proposition~3.8]{K}, with nef part zero, produces -an ordinary compact Kähler klt pair \((T_0,B_0)\) such that -\(K_{T_0}+B_0\qsim0\). - -The decomposition theorem of Matsumura--Wang--Wu--Zhang -\cite[Corollary~1.4]{BBYv2} applies to this pair: its nef b-part is zero. -A finite quasi-étale cover of \(T_0\) is a product of a rationally -connected factor, strict Calabi--Yau factors, irreducible holomorphic -symplectic factors, and a torus. Algebraic dimension is unchanged by -proper generically finite maps and by modifications. No -positive-dimensional rationally connected factor can occur, since a -Kähler resolution of such a factor is projective by algebraic -connectedness \cite{Campana1981}. A strict Calabi--Yau factor of -dimension at least three has no holomorphic two-forms. Extension of -forms for klt spaces \cite{KebekusSchnell2021} gives the same vanishing on a resolution, which is -projective by Kodaira's criterion. Such factors are also excluded. -Dimension-two factors of Calabi--Yau type are counted among the -symplectic factors. A resolution of an irreducible symplectic factor -has \(h^{2,0}=1\), generated by a generically nondegenerate form. - -Resolve the cover and the maps to \(T\) and to smooth models of all its -factors. We obtain a smooth compact Kähler manifold \(V\), with the -pulled-back nef line class \(\gamma\), mapping generically finitely to -\(T\) and to a product \(X_1\times\cdots\times X_r\). If \(r\ge2\), -let \(C\) be the pullback of a sum of Kähler classes from the factors. -For the projection omitting \(X_i\), every component of a smooth general -fiber maps generically finitely to \(X_i\). This fiber has algebraic -dimension zero and a nonzero top form, obtained by pulling back either -a torus volume form or a power of a symplectic form. Its canonical class -is therefore pseudo-effective. Its dimension is less than \(n\), so -the induction hypothesis, with zero boundary, makes the restriction of -\(\gamma\) zero. - -Expand \(C^{n-1}\). A nonzero term has top powers from all factors -except one, where precisely one power is missing. Integration over the -fibers just considered shows that its pairing with \(\gamma\) is zero. -Thus \(\gamma\cdot C^{n-1}=0\), and Lemma~\ref{zero:mass-test} gives -\(\gamma=0\). Vanishing descends to \(T\) by push--pull for a -generically finite map. A single torus factor is covered by the last -assertion of that lemma. - -It remains to consider one symplectic factor. On a smooth model \(X\) -we have \(a(X)=0\) and \(H^0(X,\Omega_X^2)=\C\sigma\), with -\(\sigma\) generically nondegenerate. There is no dominant meromorphic -fibration from \(X\) to an intermediate-dimensional compact base. -Indeed, on smooth Kähler models such a base has algebraic dimension -zero, hence is nonprojective and has a nonzero holomorphic two-form by -Kodaira's criterion. Its pullback would be a nonzero degenerate -holomorphic two-form on \(X\), which is impossible. - -The minimal-fibration theorem of Campana, in the form -\cite[Theorem~2.3(3) and Corollary~2.5(2)]{COP}, now says that \(X\) -is isotypically semi-simple. This means that \(X\) and a power -\(S^r\) of a simple manifold have a common generically finite cover; -we may take smooth Kähler models. If \(r\ge2\), repeat the preceding -product intersection argument. The smaller-dimensional fibers are -simple: simplicity is preserved by generically finite maps between -spaces of algebraic dimension zero. Indeed, a covering family of -proper subvarieties upstairs or downstairs gives such a family on the -other space by images or inverse images at points where the map is -finite; the countability of compact cycle components gives the very -general formulation. They are not uniruled, and hence -their canonical bundles are pseudo-effective by \cite[Theorem~1.1]{Ou}. -The induction hypothesis again applies. If \(r=1\), the common smooth -cover is simple and carries the pullback of \(\sigma\), a generically -nondegenerate holomorphic two-form. This is precisely the assertion -stated in the lemma. -\end{proof} - -\subsection{Finite correspondences and cotangent slopes} - -We next isolate the two geometric properties needed for the -nonvanishing argument. The first controls subvarieties of a product; -the second controls line subsheaves of cotangent tensors. - -\begin{lemma}\label{zero:no-correspondences} -Let \(X\) be a smooth simple compact Kähler manifold of algebraic -dimension zero, carrying a generically nondegenerate holomorphic -two-form. Suppose that every smooth connected compact Kähler manifold -mapping generically finitely onto \(X\) has irregularity zero. -Through a very general point of \(X^2\), the only positive-dimensional -proper irreducible compact analytic subvarieties are the two factor -slices. -\end{lemma} - -\begin{proof} -Fix a generically nondegenerate holomorphic two-form \(\sigma\) on -\(X\). A moving correspondence would produce a holomorphic one-form -on a fixed finite cover of \(X\). To see this, we identify its first projection -with a fixed finite cover over a parameter ball, and then contract the -pulled-back two-form in a direction varying the second projection. - -For a subvariety through a pair whose coordinates are very general, -simplicity makes each projection image either a point or all of \(X\). -If both projections dominate, their fibers through general points are -either zero-dimensional or the full other factor. A proper such -subvariety must therefore be a generically finite correspondence. - -Suppose these correspondences sweep \(X^2\). The compact cycle spaces -of a compact Kähler manifold have countably many compact irreducible -components \cite{LiebermanCycles}. One component must have incidence -image equal to \(X^2\). Resolve that component and the relevant -incidence component, obtaining a compact Kähler parameter space \(S\), -a smooth incidence space \(Y\), and maps -\[ - h:Y\longrightarrow S,\qquad F_1,F_2:Y\longrightarrow X. -\] -For a general parameter, the fiber is smooth and irreducible, and each -\(F_i\) restricts to a generically finite map. The map -\((F_1,h):Y\to X\times S\) is generically finite. Its normal finite -Stein factor is a finite cover of the smooth product. - -By purity, the branch locus of this cover is divisorial. Every branch -component dominating \(S\) has a proper image in \(X\): otherwise its -divisorial fibers over \(S\) would sweep \(X\), contradicting -simplicity. Compactness of \(S\) makes these images closed analytic -subsets. Remove the images in \(S\) of the remaining branch components -and take a small ball \(B\) about a general parameter. There is a fixed -proper analytic subset \(A\subset X\) such that the cover is étale -over \((X\setminus A)\times B\). Since \(B\) is simply connected, -this étale cover is pulled back from a fixed cover of \(X\setminus A\). -Normalize \(X\) in that cover. Uniqueness of normal finite extension -then identifies the cover over \(X\times B\) with the product of this -fixed normal cover and \(B\); see -\cite[Exposé~XII, Proposition~5.3 and Theorem~5.4]{SGA1}. - -Let \(\widetilde X\) be a smooth Kähler resolution of the fixed cover. -The second projection gives a meromorphic map -\(G:\widetilde X\times B\dashrightarrow X\). The pullback \(G^*\sigma\) -is a holomorphic two-form: resolve the meromorphic map and descend -holomorphic forms across a modification of a smooth space. Dominance -of \((F_1,F_2)\) implies that, at suitable general points, the -derivatives of \(G\) in the parameter directions span \(T_X\), while -the derivative along \(\widetilde X\) is an isomorphism. Choose a -constant tangent vector on \(B\) for which this parameter derivative -is nonzero at such a point. Contracting \(G^*\sigma\) with that vector -and restricting to \(\widetilde X\times\{t\}\) yields a nonzero global -holomorphic one-form on \(\widetilde X\). This contradicts the -hypothesis on irregularity. - -Each component parametrizing generically finite correspondences thus -has proper incidence image. Their countable union misses a very -general pair, proving the assertion. -\end{proof} - -\begin{lemma}\label{zero:cotangent-slope} -Let \(X\) be a smooth simple compact Kähler manifold of dimension -\(n\ge2\), algebraic dimension zero and irregularity zero, and suppose \(K_X\) is -pseudo-effective. Let \(\mathscr L\) be a holomorphic line bundle such -that \(L=c_1(\mathscr L)\ge c_1(K_X)\) in pseudo-effective order. -For every nonzero map \(\mathscr H\to\Omega_X^{\otimes k}\), where -\(\mathscr H\) is a line bundle and \(k\ge0\), -\begin{equation}\label{zero:slope-X} - c_1(\mathscr H)\le kL. -\end{equation} -For every very general \(x\in X\), write -\(a:Y=\operatorname{Bl}_xX\to X\). Every nonzero line map -\(\mathscr H_Y\to(a^*\Omega_X)^{\otimes k}\) satisfies -\begin{equation}\label{zero:slope-blowup} - c_1(\mathscr H_Y)\le ka^*L. -\end{equation} -The exceptional set of points can be chosen simultaneously for all -\(\mathscr H_Y\) and \(k\). -\end{lemma} - -\begin{proof} -We give the slope argument from \cite[Lemma~6.2]{K}, indicating why it -does not require any hypothesis about canonical sections. Let -\(\gamma\) belong to the full dual of the pseudo-effective cone. -Ou's convention calls precisely these classes movable, and -Harder--Narasimhan filtrations exist for them -\cite[Definition~4.2 and Lemma~4.3]{Ou}. If the minimum -\(\gamma\)-slope of \(\Omega_X\) were negative, the first -Harder--Narasimhan subsheaf \(\mathscr T\subset T_X\) would have -positive slope and be semistable. Tensor slope inequalities make its -bracket into \(T_X/\mathscr T\) zero, so it is a foliation. Its dual -has negative maximum slope and is non-pseudo-effective. By -\cite[Theorem~1.4, Lemmas~4.3--4.4 and Proposition~4.5]{Ou}, its leaves have -compact closures given by a meromorphic fibration. The rank of -\(\mathscr T\) is strictly less than \(\dim X\), since -\(c_1(T_X)\cdot\gamma\le0\). A general leaf closure contradicts -simplicity. - -All Harder--Narasimhan quotient slopes of \(\Omega_X\) are consequently -nonnegative, and its maximum slope is at most -\(c_1(K_X)\cdot\gamma\). Tensor slopes give -\[ - c_1(\mathscr H)\cdot\gamma - \le k c_1(K_X)\cdot\gamma\le kL\cdot\gamma. -\] -Separation by the dual cone proves \eqref{zero:slope-X}. - -Irregularity zero embeds \(\Pic(X)\) in the countable group -\(H^2(X,\Z)\). For any vector bundle on \(X\), linearly independent -global sections are generically pointwise independent: the coefficients -expressing one section in a maximal pointwise independent subfamily -are meromorphic functions, hence constants because \(a(X)=0\). -Evaluation is therefore injective outside a proper analytic subset. -Apply this simultaneously to -\(\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) for all lines -\(\mathscr H\) and all \(k\). - -For a point outside the resulting countable exceptional union, write -\(\mathscr H_Y=a^*\mathscr H\otimes\cO_Y(mF)\), where \(F\) is the -exceptional divisor. A nonzero map as in the statement extends away -from \(F\) to a section on \(X\) by Hartogs. If \(m>0\), that section -vanishes at \(x\), because \(a_*\cO_Y(-mF)=\mathcal I_x^m\), -contrary to injectivity of evaluation. Thus \(m\le0\), and -\eqref{zero:slope-X} proves \eqref{zero:slope-blowup}. -\end{proof} - -\subsection{Meromorphic nonvanishing from cotangent slope bounds} - -The next proposition extracts the line-bundle content of the -construction in \cite[Section~6]{K}. Its hypotheses deliberately separate -cotangent slopes from the identity of the chosen line bundle. This is -what permits the application to the sum of the canonical bundle and a -nef line bundle. - -\begin{proposition}\label{zero:line-nonvanishing} -Let \(X\) be a smooth simple compact Kähler manifold of dimension -\(n\ge2\), with \(a(X)=0\) and irregularity zero. Assume that the only -positive-dimensional proper compact irreducible analytic subvarieties -through a very general point of \(X^2\) are its factor slices. -Let \(\mathscr L\) be a holomorphic line bundle with pseudo-effective -class \(L\). Assume the following line-subsheaf bounds: -\begin{enumerate} -\item For every line bundle \(\mathscr H\), every integer \(k\ge0\), -and every nonzero map \(\mathscr H\to\Omega_X^{\otimes k}\), one has -\(c_1(\mathscr H)\le kL\). -\item Outside a countable union of proper analytic subsets of \(X\), -every point \(x\) has this property: if -\(a:Y=\operatorname{Bl}_xX\to X\), then for every line bundle -\(\mathscr H_Y\), every \(k\ge0\), and every nonzero map -\(\mathscr H_Y\to(a^*\Omega_X)^{\otimes k}\), one has -\(c_1(\mathscr H_Y)\le ka^*L\). -\end{enumerate} -Then some positive power of \(\mathscr L\) has a nonzero meromorphic -section. -\end{proposition} - -We prove the proposition by comparing point poles with vanishing along -a diagonal. Assume, throughout the proof, that no positive power of -\(\mathscr L\) has a nonzero meromorphic section. A normalized big class -on \(X\) has bounded pole order at a very general point. On the rank-two projective bundle over \(X^2\) formed from the two -pullbacks of \(\mathscr L\), a class of growing volume produces a -large family of symmetric cotangent tensors. -The diagonal in \(X^2\) forces a large common vanishing order in -their determinant. The slope inequalities then turn that vanishing into an -impossible point pole on \(X\). - -The proof below follows the volume and two-diagonal construction of -\cite[results~6.3--6.10]{K}. We retain its analytic estimates and give -their proofs along with the determinant calculations, so that the -chosen line bundle enters only through the hypotheses stated above. - -\subsubsection{Point pole thresholds} - -We will use several standard facts about volumes of real \((1,1)\)-classes. -Our normalization is \(\operatorname{vol}(\alpha)=\int\alpha^e\) for a nef class on an -\(e\)-fold. Volume is continuous, homogeneous, monotone in -pseudo-effective order, invariant under modification, and its \(e\)-th -root is concave on the big cone. Analytic Fujita approximation computes -it by Kähler parts on smooth projective modifications. Here a -\emph{smooth projective modification} means a projective modification -whose source is smooth; the sources used here are compact Kähler -manifolds obtained by resolving coherent analytic ideals. For a smooth -irreducible divisor \(D\) and a big class \(\alpha\), write -\(\operatorname{vol}_{\,|D}(\alpha)\) for the numerical restricted volume. -It is zero when \(D\) is contained in the non-Kähler locus -\cite[Theorem~1.1]{Vu2023}; otherwise it is -the supremum of the masses on \(D\) of restrictions of Kähler currents -with analytic singularities that are not generically singular on \(D\) -\cite[Lemma~2.7]{CollinsTosatti2018}. The divisorial derivative and its -continuity on the big cone are -\begin{equation}\label{zero:volume-derivative} - \frac{\mathrm{d}}{\mathrm{d} u}\operatorname{vol}(\alpha-u\{D\}) - =-e\,\operatorname{vol}_{\,|D}(\alpha-u\{D\}); -\end{equation} -see \cite[Theorem~1.1]{Vu2023}. We apply this formula only on smooth -manifolds while the varying class is big. - -Here is a useful precise form of the approximation in the restricted -volume formula. Resolve the log-ideal singularities of a current -restricted to \(D\). Its pullback is an effective real divisor plus a -positive residual current with locally bounded potentials. The latter -dominates a positive multiple of the pulled-back Kähler form. A current -with locally bounded potentials has zero Lelong numbers, so Demailly -regularization makes its class, after subtracting that multiple, nef -\cite[Theorem~1.1]{DemaillyRegularization1992}. -On a projective modification there is an effective exceptional divisor -whose negative is relatively ample. Subtracting a sufficiently small -multiple of this divisor from the residual class therefore makes that -class Kähler; add the same multiple to the divisor part. Round all -divisor coefficients slightly upwards to rational numbers. Openness of -the Kähler cone preserves the Kähler property. These changes can be -arbitrarily small in top intersections, which compute the original mass -by the bounded-potential product formula. Thus we may approximate a -restricted mass by -\begin{equation}\label{zero:rational-fujita} - h^*(\alpha|_D)=\beta+\{D'\},\qquad - \beta\ \text{Kähler},\quad D'\ge0\ \text{a rational divisor}. -\end{equation} -Moreover, \(D'\) has at least the log-ideal orders of the original -restriction on every further resolution. We will use this last -property to turn poles into vanishing conditions. Only \(D'\) is made -rational; the class \(\beta\) and the horizontal part of \(\alpha\) remain -real. - -\begin{lemma}[A point pole bound]\label{zero:pole-threshold} -Let \(W\) be a smooth compact Kähler manifold of dimension \(e\ge2\), -\(\alpha\) a big real \((1,1)\)-class, and \(z\in W\). Suppose a smooth -projective modification \(\mu:W'\to W\), which is an isomorphism near -\(z\), admits a decomposition -\[ - \mu^*\alpha=K+\{D\},\qquad K\ \text{Kähler},\quad D\ge0, -\] -where \(D\) misses the point \(z'\) over \(z\). If the ordinary analytic -Seshadri constant \(\epsilon(K,z')\) is at least \(\eta>0\), then, on the -blowup \(b:\widehat W=\operatorname{Bl}_zW\to W\) with exceptional divisor \(G\), -\begin{equation}\label{zero:pole-formula} - \sup\{u\ge0:b^*\alpha-u\{G\}\ge0\} - \le \frac{\eta}{2} - +2^{e-1}\operatorname{vol}(\alpha)\eta^{-(e-1)} . -\end{equation} -\end{lemma} - -\begin{proof} -Set \(u_0=\eta/2\). Blowing up \(z'\), the class -\(K-u_0\{G'\}\) is Kähler. The Fujita decomposition is unchanged near -\(G'\), so it supplies a Kähler current for -\(\alpha_{u_0}=b^*\alpha-u_0\{G\}\) that is smooth near \(G\). -Its restriction there has class \(u_0c_1(\cO_{\mathbb{P}^{e-1}}(1))\). -The restricted volume is consequently \(u_0^{e-1}\): the current gives -this lower bound, and the volume of the restricted class gives the -opposite bound. - -Let \(\tau\) denote the left side of Equation~\eqref{zero:pole-formula}. The classes -\(\alpha_u=b^*\alpha-u\{G\}\) are big for \(0\le u<\tau\), since -\(\alpha_0\) is big and the pseudo-effective cone is convex. The -concave function \(f(u)=\operatorname{vol}(\alpha_u)^{1/e}\) satisfies, by -Equation~\eqref{zero:volume-derivative}, -\[ - f'(u_0)=-\frac{u_0^{e-1}}{f(u_0)^{e-1}}. -\] -Its tangent line at \(u_0\) must remain positive up to \(\tau\). -Therefore -\[ - \tau\le u_0+\frac{f(u_0)^e}{u_0^{e-1}} - \le u_0+\frac{\operatorname{vol}(\alpha)}{u_0^{e-1}}, -\] -which is Equation~\eqref{zero:pole-formula}. -\end{proof} - -Fix a Kähler class \(\omega\) on \(X\). The class \(L\) is not big: -a big holomorphic line bundle would make \(X\) Moishezon, contrary to -\(a(X)=0\). For \(t>0\) define -\begin{equation}\label{zero:normalization} - P=r(L+t\omega),\qquad - r=\operatorname{vol}(L+t\omega)^{-1/n}. -\end{equation} -Then \(P\) is a big real class, \(\operatorname{vol}(P)=1\), \(L\le P/r\), and -\(r\to\infty\) as \(t\downarrow0\). Choose a smooth Fujita model -\(\mu:Y_P\to X\) with Kähler part \(P'\) satisfying -\begin{equation}\label{zero:P-prime} - \mu^*P=P'+\{D_P\},\qquad v:=\int_{Y_P}(P')^n>\frac12. -\end{equation} -At a very general point \(y\) of this model there is no -positive-dimensional proper subvariety: its image would be one through a -very general point of \(X\), and \(y\) avoids the exceptional locus. -The ordinary Seshadri formula for a Kähler class, -\[ - \epsilon(K,y) - =\inf_{\substack{W\ni y\\ \dim W>0}} - \left(\frac{\int_W K^{\dim W}} - {\operatorname{mult}_y W}\right)^{1/\dim W}, -\] -therefore gives \(\epsilon(P',y)=v^{1/n}\ge2^{-1/n}\). -This is the ordinary nef/Kähler formula -\cite[Theorem~2.8]{Tosatti2016}; see also -\cite[Equation~(1.5)]{CollinsTosatti2018} and the -Kähler cone criterion of \cite{DemaillyPaun2004}. We use this ordinary -formula in both applications below. Lemma~\ref{zero:pole-threshold}, with -\(\operatorname{vol}(P)=1\), now yields a constant \(C_n\) depending only on \(n\) -such that, for a very general \(x\), -\begin{equation}\label{zero:point-bound} - \tau(P,x):=\sup\{u\ge0:a^*P-u\{F\}\ge0\}\le C_n, - \qquad a:\operatorname{Bl}_xX\to X. -\end{equation} -Here and below positive constants denoted \(c_n,C_n\) may be decreased or -increased from one occurrence to the next. They are independent of all -parameters and choices of currents and modifications. - -\subsubsection{A projective bundle with controlled volume} - -We now produce a class whose volume grows, while its point pole threshold -grows much more slowly. On \(X^2\), let \(\mathscr L_i\) and \(L_i\) -denote the pullbacks of \(\mathscr L\) and \(L\) from the \(i\)-th factor. -Use the quotient convention and put -\[ - \pi:Z=\mathbb{P}_{X^2}(\mathscr L_1\oplus\mathscr L_2)\longrightarrow X^2, - \qquad \xi=c_1(\cO_Z(1)),\qquad d=\dim Z=2n+1 . -\] -Thus \(\pi_*\cO_Z(m)=\operatorname{Sym}^m(\mathscr L_1\oplus\mathscr L_2)\) for -\(m\ge0\). The zero divisor \(A_i\) of -\(\pi^*\mathscr L_i\to\cO_Z(1)\) has class \(\xi-L_i\); it is the -section on which \(\xi\) restricts to \(L_{3-i}\). In particular -\(\xi=\{A_i\}+L_i\ge0\). - -\begin{lemma}\label{zero:subvarieties} -For the projective bundle \(Z\) just defined, the only -positive-dimensional compact irreducible analytic subvarieties through -a very general point are a fiber of \(\pi\), the full inverse images of -the two factor slices, and \(Z\) itself. -\end{lemma} - -\begin{proof} -The image of such a subvariety in \(X^2\) is a point, a factor slice, -or all of \(X^2\), by the hypothesis of -Proposition~\ref{zero:line-nonvanishing}. Generic relative dimension -one gives the full inverse image. Otherwise, except for a point, the -subvariety is a multisection of the restricted projective line bundle. -On the slice or product \(S\), its divisor line is -\(\cO(k)\otimes\pi^*\mathscr H\), with \(k>0\). Its homogeneous -equation has coefficients in -\[ - H^0\bigl(S,\mathscr H\otimes - \mathscr L_1^{\otimes i}\otimes - \mathscr L_2^{\otimes(k-i)}\bigr),\qquad 0\le i\le k. -\] -A single nonzero monomial cuts out only axes and vertical divisors. -A non-axis multisection therefore has two nonzero coefficients. Their -quotient is a meromorphic section of a nonzero power of -\(\mathscr L_1\otimes\mathscr L_2^{-1}\). Restricting to a general -factor slice gives a meromorphic section of a nonzero power of -\(\mathscr L\); inversion makes the power positive if necessary. -This contradicts the standing contrary hypothesis. The axes -avoid a very general point of \(Z\), proving the lemma. -\end{proof} - -Let \(q\) tend to infinity through positive integers. For each sufficiently -large \(q\), choose \(t=t(q)\) in Equation~\eqref{zero:normalization} so that -\(r=r(q)\ge q^2\), and use the resulting normalized class \(P=P(q)\). -Define the real class and the scale -\begin{equation}\label{zero:M-definition} - M=P_1+P_2+q\xi,\qquad A_q=q^{1/d}. -\end{equation} - -The quantitative goal can now be stated. On the blowup -\(a:\operatorname{Bl}_xX\to X\) of a very general point, with exceptional -divisor \(F\), we will produce a positive rational scale \(s\) satisfying -\(c_nA_q\le s\le C_nA_q\) and -\[ - 2a^*P+(s+2q)a^*L-c_ns\{F\}\ge0. -\] -Since \(L\le P/r\), this would imply -\[ - \frac{c_ns}{2+(s+2q)/r}\le\tau(P,x)\le C_n. -\] -The denominator stays bounded because \(r\ge q^2\), while the numerator -tends to infinity. To obtain the class inequality, the first diagonal -will produce a subsheaf occupying a fixed positive fraction of a -symmetric cotangent power. An incidence construction will then force -its determinant to vanish to order proportional to \(s\). We begin -with the volume that supplies this rank fraction. - -\begin{lemma}\label{zero:two-slot-volume} -For these choices, \(M\) is big and -\begin{equation}\label{zero:M-bounds} - c_nq\le\operatorname{vol}(M)\le C_nq,\qquad - \tau(M,z):=\sup\{u\ge0:b_z^*M-u\{F_z\}\ge0\}\le C_nA_q -\end{equation} -at a very general point \(z\in Z\), where -\(b_z:\operatorname{Bl}_zZ\to Z\) has exceptional divisor \(F_z\). -\end{lemma} - -\begin{proof} -Pull \(Z\) to the Fujita model \(Y_P^2\) from -Equation~\eqref{zero:P-prime}, and put \(H=P'_1+P'_2\) on this pulled-back bundle. -There is a further smooth projective modification -\(\nu:\widehat Z\to\mathbb{P}_{Y_P^2}(\mu^*\mathscr L_1\oplus -\mu^*\mathscr L_2)\) and a Fujita decomposition of the pullback of \(M\) -whose Kähler part is exactly -\begin{equation}\label{zero:exact-K} - K=\frac12\nu^*H+\Theta , -\end{equation} -where \(\Theta\) is Kähler and has degree \(q\) on a general vertical -line. The modification and the divisor part miss that line. - -Here is the construction. It suffices to decompose \(H/2+q\xi\) as -\(\Theta\) plus an effective divisor, clean on a whole general vertical -line. Since \(\cO(1)\) is relatively ample, a small positive class of -the form \(\delta(\xi+cH)\) is Kähler for some \(c>0\). Choose -\(\delta>0\) so small that \(\delta0\) and \(C_s\) is big for -\(0\le s0 -\] -on a general fiber. For sufficiently large divisible integers \(j\), -let \(\mathcal F_j=f_*\cO_U(j\Lambda)\) and \(R_j=\operatorname{rk}\mathcal F_j\). -Here \(j\Lambda\) is an actual line bundle and -\(c_1(\mathcal F_j)\) means \(c_1(\det\mathcal F_j)\). Then -\begin{align} - R_j&=\frac{w}{e!}j^e+O(j^{e-1}),\label{zero:GRR-rank}\\ - B_0:=G+\lim_j\frac{c_1(\mathcal F_j)}{jR_j} - &=\frac{f_*\beta^{e+1}}{(e+1)w}\ge0,\label{zero:GRR-base}\\ - \int_U\beta^{b_0+e} - &\le (e+1)^{b_0}w\,\operatorname{vol}(B_0).\label{zero:mixed-bound} -\end{align} -The limit in Equation~\eqref{zero:GRR-base} is a limit of real Bott--Chern classes. -\end{lemma} - -\begin{proof} -The restriction of \(c_1(\Lambda)\) to each fiber is represented by the -restriction of the Kähler form \(\beta\). The fiberwise criterion for -relative ampleness makes \(\Lambda\) relatively ample after clearing -denominators. Relative Serre vanishing then kills the higher direct -images for all sufficiently large divisible \(j\). Analytic -Grothendieck--Riemann--Roch \cite{LevyGRR}, in degrees zero and two, gives -\[ - R_j=\frac{f_*c_1(\Lambda)^e}{e!}j^e+O(j^{e-1}),\qquad - c_1(\mathcal F_j) - =\frac{f_*c_1(\Lambda)^{e+1}}{(e+1)!}j^{e+1}+O(j^e). -\] -The degree-zero pushforward is \(w\). Expanding -\(\beta=f^*G+c_1(\Lambda)\) shows that -\[ - f_*\beta^{e+1} - =f_*c_1(\Lambda)^{e+1}+(e+1)wG, -\] -because terms with at least two horizontal factors have negative -fiber degree after pushforward. This proves the equality in -Equation~\eqref{zero:GRR-base}; the cohomological GRR equality is an equality in -Bott--Chern cohomology by the \(\partial\bar\partial\)-lemma on compact -Kähler manifolds. Pushforward of the positive form -\(\beta^{e+1}\) is a positive closed \((1,1)\)-current, so \(B_0\) is -pseudo-effective. - -Apply Lemma~\ref{zero:nef-pushforward} to choose \(p:Z'_0\to Z_0\), -\(q:U'\to U\), and \(f':U'\to Z'_0\), and put \(\beta'=q^*\beta\). -The resulting class \(B'_0\) is nef and satisfies -Equation~\eqref{zero:flattened-volume}. -Choose a Kähler class \(\omega'\) on \(Z'_0\) and put -\(C=B'_0+\varepsilon\omega'\). For \(0\le k\le b_0\), set -\[ - I_k=\int_{U'}(\beta')^{e+k}(f'^*C)^{b_0-k}. -\] -These are mixed intersections of nef classes. The first two satisfy -\[ - I_0=w\int_{Z'_0}C^{b_0},\qquad - I_1=(e+1)w\int_{Z'_0}B'_0C^{b_0-1} - \le(e+1)w\int_{Z'_0}C^{b_0}. -\] -The mixed nef inequalities make the sequence \(I_k\) log-concave. -Every \(I_k\) is positive: on the dense open where \(q\) is a local -biholomorphism and \(f'\) is a submersion, \(\beta'\) is positive -definite and \(f'^*C\) has rank \(b_0\), so the defining top form -is strictly positive. The successive ratios are therefore at most -\(I_1/I_0\le e+1\). Thus -\[ - \int_U\beta^{b_0+e}=I_{b_0} - \le(e+1)^{b_0}w\int_{Z'_0}C^{b_0}. -\] -Letting \(\varepsilon\downarrow0\) and using -Equation~\eqref{zero:flattened-volume} proves Equation~\eqref{zero:mixed-bound}. -\end{proof} - -We will also use the determinant formula without its volume bound. - -\begin{corollary}\label{zero:slope-descent} -Let \(Y\) be a smooth compact Kähler manifold and \(\mathscr E\) a -holomorphic vector bundle of rank at least two. Write -\(\pi_{\mathscr E}:\mathbb{P}_Y(\mathscr E)\to Y\) and -\(\zeta_{\mathscr E}=c_1(\cO_{\mathbb{P}(\mathscr E)}(1))\). -Suppose that a real class \(D\) on \(Y\) satisfies -\[ - c_1(\mathscr H)\le kD - \quad\text{whenever}\quad - 0\ne\bigl(\mathscr H\longrightarrow\mathscr E^{\otimes k}\bigr) -\] -for a line bundle \(\mathscr H\) and integer \(k\ge0\). -For a real class \(A\) on \(Y\) and a real number \(b\ge0\), -\begin{equation}\label{zero:slope-descent-equation} - \pi_{\mathscr E}^*A+b\zeta_{\mathscr E}\ge0 - \quad\Longrightarrow\quad A+bD\ge0 . -\end{equation} -\end{corollary} - -\begin{proof} -If \(Y\) is a point, both \(A\) and \(D\) vanish and the assertion is -immediate. Suppose \(\dim Y\ge1\). Choose a real class \(H_0\) on \(Y\) so that -\(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0\) is Kähler; relative -ampleness of \(\cO(1)\) permits such a choice. Take numbers -\(\delta\downarrow0\) with \(b+\delta>0\) rational. Adding -\(\delta(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0)\) to the -pseudo-effective class in Equation~\eqref{zero:slope-descent-equation} makes it -big. A Fujita decomposition on a smooth projective modification -\(h:U\to\mathbb{P}_Y(\mathscr E)\), with the divisor coefficients rounded -upwards as in Equation~\eqref{zero:rational-fujita}, has the form -\[ - \beta=f^*(A+\delta H_0)+c_1(\Lambda),\qquad - \Lambda=(b+\delta)h^*\cO(1)-\cO_U(D') - \ \text{in }\Pic(U)\otimes\Q, -\] -where \(f=\pi_{\mathscr E}h\), \(\beta\) is Kähler, and \(D'\ge0\) -is rational. For divisible \(j\), -\[ - f_*\cO_U(j\Lambda)\ \subseteq\ - \operatorname{Sym}^{j(b+\delta)}\mathscr E . -\] -Its determinant of rank \(R_j\) consequently maps into -\(\mathscr E^{\otimes j(b+\delta)R_j}\). The assumed line inequality -gives -\[ - \frac{c_1(\mathcal F_j)}{jR_j}\le(b+\delta)D. -\] -Lemma~\ref{zero:direct-image} makes -\(A+\delta H_0+\lim c_1(\mathcal F_j)/(jR_j)\) pseudo-effective. -Adding the preceding pseudo-effective difference shows that -\(A+\delta H_0+(b+\delta)D\) is pseudo-effective. Let -\(\delta\downarrow0\). This proves the corollary. In this argument -only \(b+\delta\) and the coefficients of \(D'\) are rational; \(A\), -\(H_0\), and \(D\) remain real classes. -\end{proof} - -For the forthcoming application over the \(d\)-fold \(Z\), with fibers -of dimension \(d-1\), Lemma~\ref{zero:direct-image} has a useful concrete -consequence. Once we prove \(B_0\le C_nM\), the bound -\(\operatorname{vol}(M)\le C_nq\) gives -\[ - \int_U\beta^{2d-1}\le C_nwq. -\] -Thus an upper bound for the normalized determinant class will convert -a lower bound for total mass into a lower bound for the fiber volume -\(w\). The next calculation supplies the required determinant control. - -\subsubsection{Cotangent lines on the projective bundle} - -We return to \(Z=\mathbb P_{X^2}(\mathscr L_1\oplus\mathscr L_2)\). -The following calculation is where the chosen line bundle enters the -first determinant estimate. A line mapping into a cotangent tensor -has nonpositive degree on a general projective-line fiber; the -calculation also controls its horizontal class. - -\begin{lemma}\label{zero:cotangent-Z} -Let \(\mathscr H\) be a holomorphic line bundle on \(Z\), and suppose -there is a nonzero map \(\mathscr H\to\Omega_Z^{\otimes k}\), with -\(k\ge0\). Then -\[ - \mathscr H=\pi^*(\mathscr Q_1\boxtimes\mathscr Q_2) - \otimes\cO_Z(\ell) -\] -for lines \(\mathscr Q_i\) on \(X\) and an integer \(\ell\), and -\[ - \ell\le0,\qquad - c_1(\mathscr Q_1)_1+c_1(\mathscr Q_2)_2 - \le(k-\ell)(L_1+L_2). -\] -\end{lemma} - -\begin{proof} -Irregularity zero and Künneth give -\(\Pic(X^2)=\operatorname{pr}_1^*\Pic(X)\oplus -\operatorname{pr}_2^*\Pic(X)\), and the projective-bundle formula for -\(\Pic(Z)\) gives the asserted expression. Filter the target using -\[ - 0\longrightarrow\pi^*\Omega_{X^2}\longrightarrow\Omega_Z - \longrightarrow\cO_Z(-2)\otimes - \pi^*(\mathscr L_1\otimes\mathscr L_2)\longrightarrow0. -\] -Choose a nonzero associated graded component. Suppose it has \(i\) -relative factors and \(k_1,k_2\) cotangent factors from the two copies -of \(X\), so \(k_1+k_2=k-i\). Its relative degree is \(-2i\). -Pushing to \(X^2\) forces \(m=-2i-\ell\ge0\); hence -\(\ell\le-2i\le0\). A nonzero monomial in -\(\operatorname{Sym}^m(\mathscr L_1\oplus\mathscr L_2)\), with first -exponent \(m_1\), gives on the two general slices -\[ - c_1(\mathscr Q_1)\le(i+m_1+k_1)L,\qquad - c_1(\mathscr Q_2)\le(i+m-m_1+k_2)L. -\] -These are exactly the assumed cotangent-line bounds on \(X\), applied -after moving the indicated powers of \(\mathscr L\) to the source. -Each coefficient on the right is at most \(k-\ell\). Since \(L\ge0\), -pullback and addition prove the claimed bound. -\end{proof} - -\subsubsection{Extracting a positive rank fraction} - -\begin{lemma}\label{zero:diagonal-mass} -There are constants \(c_n,C_n>0\) such that, for every sufficiently -large \(q\), one can choose a rational number -\[ - c_nA_q\le s\le C_nA_q -\] -and a Kähler current \(T\) in \(C_s\) with analytic singularities, -not generically singular on \(E\), with the following property. On a -smooth projective modification \(h:U\to E\), its restricted mass has -a rational-divisor approximation -\begin{equation}\label{zero:selected-decomposition} - h^*(2M+s\zeta)=\beta+\{D'\}, - \qquad \beta\ \text{Kähler},\quad D'\ge0\ \text{rational}, -\end{equation} -which retains all log-ideal orders of \(T|_E\). If -\(f:U\to Z\) is the natural map and -\[ - w=\int_{U_z}\beta^{d-1} -\] -on a general fiber, then -\begin{equation}\label{zero:selected-w} - w\ge c_ns^{d-1}. -\end{equation} -\end{lemma} - -\begin{proof} -First let \(s\) be any rational number in \((0,s_{\max})\) with -\(v_E(s)>0\), and use the approximation in -Equation~\eqref{zero:rational-fujita} for any current used -to compute this restricted volume. In -Lemma~\ref{zero:direct-image}, the data for Equation~\eqref{zero:restriction-class} -are -\[ - b_0=d,\quad e=d-1,\quad G=2M,\quad - \Lambda=s h^*\cO_E(1)-\cO_U(D') - \ \text{in }\Pic(U)\otimes\Q. -\] -For large divisible \(j\), the associated sheaves satisfy -\begin{equation}\label{zero:symmetric-inclusion} - \mathcal F_j=f_*\cO_U(j\Lambda) - \subseteq\operatorname{Sym}^{js}\Omega_Z,\qquad - R_j=\operatorname{rk}\mathcal F_j,\qquad 00\). The contribution of -\(0(1-\delta)m\). Summing instead over their internal -degree gives -\[ - H_m(\delta) - \le \binom{m+n}{n}\binom{\lceil\delta m\rceil+n}{n},\qquad - \limsup_{m\to\infty}\frac{H_m(\delta)}{N_m} - \le\binom{2n}{n}\delta^n. -\] -On the other hand, \(w^+\ge w/2\) and -Equation~\eqref{zero:plus-rank} show that the approximation on \(E^+\) -retains at least half the rank fraction in -Equation~\eqref{zero:selected-rank-fraction}: -\[ - \lim_{j\to\infty}\frac{R_j^+}{N_{js}} - =\frac{w^+}{s^{d-1}} - \ge\frac{w}{2s^{d-1}}\ge c_n>0. -\] -Fix \(\delta>0\), depending only on \(n\), so small that, for each fixed -\(q\) and its selected data, fewer than \(R_j^+/2\) monomials have transverse -degree greater than \((1-\delta)js\) for all sufficiently large divisible -\(j\). For each -remaining monomial, Equation~\eqref{zero:coefficient-order} and -Lemma~\ref{zero:incidence-pole} give -\[ - \operatorname{ord}_u(c_{\alpha,\gamma}) - \ge j(s-C_n)-(1-\delta)js - =j(\delta s-C_n)\ge\frac{\delta js}{2}. -\] -The last inequality holds once \(q\) is large, since -\(s\ge c_nA_q\to\infty\). - -Over the same discrete valuation ring \(R\), the torsion-free sheaf -\(\mathcal F_j^+\) becomes a free module of rank \(R_j^+\). Write its -inclusion into the symmetric power as a matrix with the monomials -as rows. Every maximal minor uses at least \(R_j^+/2\) of the rows -whose coefficients have order at least \(\delta js/2\). All maximal -minors therefore have order at least -\(\delta jR_j^+s/4\). By the definition of \(h_j\), this proves -Equation~\eqref{zero:determinant-order-equation}. Dividing the determinant -map by \(\sigma_{J_+}^{h_j}\) gives a holomorphic map at every point of -codimension one: the coefficients have the required order along \(J_+\), -and its local equation is a unit away from \(J_+\). Hartogs' theorem -extends the divided map across the remaining set of codimension at least -two, giving the stated factorization through -\(\det\mathcal F_j^+(h_jJ_+)\). -\end{proof} - -\subsubsection{Cancellation on the point blowup} - -\begin{proof}[Completion of the proof of Proposition~\ref{zero:line-nonvanishing}] -Fix \(q\) sufficiently large for the preceding lemmas, with its chosen -\(P,s,T\) and modifications. Choose a very general point \(x\) in the first -factor of \(X^2\). The slice of \(Z^+\) above \(x\) is -\[ - S=\mathbb{P}_Y(\cO_Y\oplus a^*\mathscr L),\qquad - a:Y=\operatorname{Bl}_xX\longrightarrow X, -\] -and \(J_+|_S\) is the pullback of \(F\). We choose \(x\) so that -Equation~\eqref{zero:slope-blowup}, Equation~\eqref{zero:point-bound}, the restrictions of -the currents below, and all determinant maps for divisible \(j\) -can be tested on this slice. These exclude at most countably many -proper analytic sets and the measure-zero exceptional sets for -current restrictions. - -Write the restricted determinant line as -\[ - (\det\mathcal F_j^+)|_S - =\pi_S^*\mathscr Q_j^+\otimes\cO_S(\ell_j^+),\qquad - Q_j^+=c_1(\mathscr Q_j^+). -\] -Factoring the common zero of order \(h_j\) from the determinant -map gives a nonzero map whose source on \(S\) is -\[ - \pi_S^*(\mathscr Q_j^+\otimes\cO_Y(h_jF)) - \otimes\cO_S(\ell_j^+). -\] -The plus sign of \(h_jF\) follows because division of the map by -the local equation of \(J_+^{h_j}\) enlarges its source from -\(\det\mathcal F_j^+\) to -\(\det\mathcal F_j^+\otimes\cO(h_jJ_+)\). -Its target embeds into the \(jsR_j^+\)-fold tensor of the -restriction of \(p_+^*\Omega_Z\). In particular we use the -cotangent bundle of the original \(Z\), with its restricted -filtration -\[ - 0\longrightarrow - \pi_S^*(\cO_Y^{\oplus n}\oplus a^*\Omega_X) - \longrightarrow (p_+^*\Omega_Z)|_S - \longrightarrow - \cO_S(-2)\otimes\pi_S^*a^*\mathscr L - \longrightarrow0. -\] -The same homogeneous-monomial calculation as in -Equation~\eqref{zero:determinant-bound-Z}, now applying -Equation~\eqref{zero:slope-blowup} to the second factor of \(X^2\), gives -\begin{equation}\label{zero:determinant-bound-slice} - \ell_j^+\le0,\qquad - Q_j^++h_j\{F\} - \le(jsR_j^+-\ell_j^+)a^*L. -\end{equation} -More explicitly, if the chosen graded term has \(i\) relative -factors, the pushed polynomial has degree -\(-2i-\ell_j^+\ge0\). Its exponent in \(a^*\mathscr L\), together -with the \(i\) relative factors and the cotangent order from the second -factor of \(X^2\), is at most \(jsR_j^+-\ell_j^+\). The cotangent -factors from the first factor of \(X^2\) are constant. This proves the -displayed inequality using -the tensors of \(a^*\Omega_X\), without introducing \(\Omega_Y\) -or \(\Omega_{Z^+}\). - -Apply Lemma~\ref{zero:direct-image} to the decomposition in -Equation~\eqref{zero:plus-decomposition}. Equation~\eqref{zero:GRR-base} -shows that the limits of the restricted determinant components divided by -\(jR_j^+\) exist; denote them by \(Q^+\) and \(\ell^+\). Divide -Equation~\eqref{zero:determinant-bound-slice} by \(jR_j^+\), use -Equation~\eqref{zero:determinant-order-equation}, and pass to the closed -pseudo-effective cone. We obtain -\begin{equation}\label{zero:slice-upper} - \ell^+\le0,\qquad - Q^++c_ns\{F\}\le(s-\ell^+)a^*L. -\end{equation} -The pseudo-effective class in Equation~\eqref{zero:GRR-base}, restricted to the -very general slice \(S\), is -\begin{equation}\label{zero:slice-psef} - \pi_S^*(Q^++2a^*P)+(\ell^++2q)\xi\ge0. -\end{equation} -Testing on a general vertical line gives \(\ell^++2q\ge0\). -Let \(A\subset S\) be the axis of -\(\mathbb{P}_Y(\cO_Y\oplus a^*\mathscr L)\) with divisor class \(\{A\}=\xi\). -On \(A\simeq Y\), its normal class is \(\{A\}|_A=\xi|_A=a^*L\ge0\). -Add an arbitrarily small Kähler class to the class in -Equation~\eqref{zero:slice-psef}, and choose a Kähler current with analytic -singularities in the resulting big class. Remove its generic divisorial -pole along \(A\) and restrict -the residual current to \(A\). Adding back the removed nonnegative multiple -of \(a^*L\) preserves pseudo-effectivity. Passing to the limit gives -\begin{equation}\label{zero:slice-lower} - Q^++2a^*P+(\ell^++2q)a^*L\ge0. -\end{equation} - -The slope bound in Equation~\eqref{zero:slice-upper} supplies the -pseudo-effective class \((s-\ell^+)a^*L-Q^+-c_ns\{F\}\). Adding it to -Equation~\eqref{zero:slice-lower} cancels the entire normalized determinant -class restricted to the axis, \(Q^++\ell^+a^*L\), leaving -\[ - 2a^*P+(s+2q)a^*L-c_ns\{F\}\ge0. -\] -Since \(L\le P/r\), the point bound in Equation~\eqref{zero:point-bound} implies -\begin{equation}\label{zero:contradiction} - \frac{c_ns}{\,2+(s+2q)/r\,}\le C_n. -\end{equation} -The denominator is bounded independently of \(q\), because -\(s\le C_nq^{1/d}\) and \(r\ge q^2\). The numerator tends to -infinity, because \(s\ge c_nq^{1/d}\). Equation~\eqref{zero:contradiction} -is impossible for arbitrarily large \(q\). - -For each \(q\), the choice of \(t(q)\), the rational \(s\), the current -\(T\), and the modifications precedes the limit in \(j\). The monomial -cutoff \(\delta\) depends only on \(n\), while the required lower bound for -divisible \(j\) may depend on the fixed data. We choose the very general -points after those data are fixed. -Thus every simultaneous general-point test above involves at most -countably many conditions, and no bound depends on the point. -The contradiction disproves the contrary hypothesis and proves -Proposition~\ref{zero:line-nonvanishing}. -\end{proof} - -\subsection{Vanishing on the simple model} - -We now apply Proposition~\ref{zero:line-nonvanishing} to the sum of -the canonical bundle and the nef line bundle whose class we wish to -eliminate. The remaining use of the ordinary adjoint -decomposition is particularly short: it turns meromorphic -nonvanishing into both signs of pseudo-effectivity. - -\begin{proposition}\label{zero:simple-vanishing} -Let \(X\) be a smooth simple compact Kähler manifold of algebraic -dimension zero carrying a generically nondegenerate holomorphic -two-form. Every nef rational holomorphic line bundle on \(X\) has zero -first Chern class. -\end{proposition} - -\begin{proof} -The assertion is immediate for a point. A positive-dimensional space -of algebraic dimension zero cannot be a curve, so assume \(\dim X\ge2\). -If \(q(X)>0\), its Albanese map is generically finite onto its image, -by simplicity. Ueno's structure theorem for subvarieties of complex -tori makes this image a translate of a torus: its quotient by the -connected stabilizer is of general type, and positive dimension of that -quotient would contradict algebraic dimension zero \cite{Ueno}. -Lemma~\ref{zero:mass-test} proves the conclusion. - -The same argument applies after any smooth generically finite Kähler -cover of \(X\) having positive irregularity. Such a cover is still -simple and has algebraic dimension zero; vanishing upstairs descends -by push--pull. We may therefore suppose that every such cover has -irregularity zero. Lemma~\ref{zero:no-correspondences} then describes -the subvarieties through a very general point of \(X^2\). - -Clear denominators and let \(N_0\) be the resulting nef holomorphic line -bundle. A top wedge of the generically nondegenerate two-form is a -nonzero holomorphic section of \(K_X\). In particular \(K_X\) is -effective. Set -\[ - \mathscr L=K_X+N_0,\qquad L=c_1(\mathscr L). -\] -The class \(L\) is pseudo-effective and dominates \(c_1(K_X)\). -Lemma~\ref{zero:cotangent-slope} supplies both slope inequalities. -Every hypothesis of Proposition~\ref{zero:line-nonvanishing} is now -satisfied. Some positive power of \(K_X+N_0\) therefore has a nonzero -meromorphic section. Dividing by the corresponding power of a -canonical section shows that \(N_0\), as a rational line bundle, is -represented by a signed divisor. - -Resolve its positive and negative parts. On the resulting smooth -Kähler model write -\[ - N_0\qsim E-G,\qquad E,G\ge0, -\] -with rational coefficients and simple normal crossing union of their -supports; we suppress the pullback notation for \(N_0\). The canonical -bundle \(K'\) of this model remains effective. Choose a small rational -\(u>0\) so that both \(uE\) and \(uG\) are klt boundaries. -The ordinary decomposition, Theorem~\ref{pre:ordinary}, applies to -\(K'+uE\) and \(K'+uG\). On a common higher model, denote their pulled -back classes by \(\alpha_E\) and \(\alpha_G\). Algebraic dimension zero -makes each semiample part rationally trivial, so that the pulled back -rational lines equal their negative divisors. Since -\(\alpha_E-\alpha_G=u\{N_0\}\) is nef, -Lemma~\ref{pre:negative}\,(2) gives -\[ - N(\alpha_E)\le N(\alpha_G),\qquad - u\{N_0\}=\{N(\alpha_E)-N(\alpha_G)\}. -\] -Thus \(-\{N_0\}\) is pseudo-effective, whereas \(\{N_0\}\) is nef. -Pairing both classes with a Kähler power gives zero mass, and hence -\(\{N_0\}=0\). Pullback injectivity yields the same conclusion on -\(X\). -\end{proof} - -\begin{proof}[Proof of Proposition~\ref{zero:vanishing}] -Lemma~\ref{zero:geometric-reduction} reduces the assertion to -Proposition~\ref{zero:simple-vanishing}, which proves it. -In particular, once the induction has reached dimension \(n\), the -ordinary decomposition of \(K_T+B\) also proves the decomposition -assertion of Theorem~\ref{thm:decomposition} in algebraic dimension -zero: adding the nef line changes no Chern class. Its semiample positive -part is rationally trivial, and its negative part is unchanged. -\end{proof} +\section{Nef classes in algebraic dimension zero}\label{sec:zero} + +A semiample line bundle on a space of algebraic dimension zero is +rationally trivial. The induction therefore requires a stronger +conclusion in this case: the additional nef class must itself vanish. +The ordinary adjoint decomposition reduces this assertion to a question +about nef line bundles on simple manifolds. We first carry out that +geometric reduction. The remaining argument adapts the meromorphic +nonvanishing construction of \cite[Section~6]{K} to a line bundle that +need not be canonical. + +\begin{proposition}\label{zero:vanishing} +Let \(n\ge1\). Assume that the algebraic-dimension-zero assertion of +Theorem~\ref{thm:decomposition} holds in every dimension less than \(n\). +Let \(T\) be a smooth connected compact Kähler \(n\)-fold with +\(a(T)=0\), and let \(B\) be an effective rational simple normal crossing +boundary with coefficients less than one. If \(K_T+B\) is +pseudo-effective, then every nef rational holomorphic line bundle \(M\) +on \(T\) satisfies \(c_1(M)=0\). +\end{proposition} + +For intersections in this Section, a line bundle and its first Chern +class are distinguished when necessary by braces: \(\{H\}=c_1(H)\). +An inequality between real \((1,1)\)-classes is in pseudo-effective +order. A compact manifold is \emph{simple} if no positive-dimensional +proper compact analytic subvariety passes through a very general point. +Here, as usual, ``very general'' permits deletion of a countable union +of proper analytic subsets. + +\subsection{Reduction to a simple symplectic manifold} + +The following elementary intersection test lets us use finite covers +without requiring the nef line bundle to descend through them. + +\begin{lemma}\label{zero:mass-test} +Let \(V\) be a smooth compact Kähler \(n\)-fold, where \(n\ge1\), +and let \(\gamma\) be a nef +class, and let \(C\) be a nef and big class. If +\(\gamma\cdot C^{n-1}=0\), then \(\gamma=0\). +In particular, if \(V\) admits a generically finite surjective morphism +to a complex torus of algebraic dimension zero, every nef rational line +class on \(V\) is zero. +\end{lemma} + +\begin{proof} +Choose a Kähler class \(\omega\) and \(\varepsilon>0\) such that +\(C-\varepsilon\omega\) is pseudo-effective. When \(n\ge2\), the +identity +\[ + C^{n-1}-(\varepsilon\omega)^{n-1} + =(C-\varepsilon\omega) + \sum_{j=0}^{n-2}C^{n-2-j}(\varepsilon\omega)^j +\] +and nonnegativity of intersections of a pseudo-effective class with +nef classes give +\[ + 0=\gamma\cdot C^{n-1} + \ge \varepsilon^{n-1}\gamma\cdot\omega^{n-1}\ge0. +\] +The same conclusion is immediate when \(n=1\). A positive current in +\(\gamma\) thus has zero mass, so it vanishes. + +For the last assertion, write \(f:V\to A\) for the morphism. The +pushforward of the nef rational line class is a rational +pseudo-effective \((1,1)\)-class on \(A\). Averaging a positive current +under translations represents it by a constant semipositive form. The +kernel of a rational such form is the tangent space of a subtorus; the +induced positive integral form, after clearing denominators, polarizes +the quotient. Since \(a(A)=0\), this quotient is a point, and hence +\(f_*\gamma=0\). For a Kähler class \(\eta\) on \(A\), the class +\(C=f^*\eta\) is nef and big and +\[ + \gamma\cdot C^{n-1}=(f_*\gamma)\cdot\eta^{n-1}=0. +\] +Apply the first assertion. +\end{proof} + +\begin{lemma}\label{zero:geometric-reduction} +Under the induction hypothesis of Proposition~\ref{zero:vanishing}, +it suffices to prove the following assertion: every nef rational line +class on a smooth simple compact Kähler manifold of algebraic dimension +zero carrying a generically nondegenerate holomorphic two-form is zero. +\end{lemma} + +\begin{proof} +Apply the ordinary good divisorial decomposition to \(K_T+B\). On a +smooth modification its semiample part is rationally trivial, because +\(a(T)=0\). After the klt modification convention of +Section~\ref{sec:preliminaries}, the resulting ordinary adjoint still +equals its rational negative divisor as an actual rational line bundle. +The contraction of a known negative part +\cite[Theorem~2.13 and Proposition~3.8]{K}, with nef part zero, produces +an ordinary compact Kähler klt pair \((T_0,B_0)\) such that +\(K_{T_0}+B_0\qsim0\). + +The decomposition theorem of Matsumura--Wang--Wu--Zhang +\cite[Corollary~1.4]{BBYv2} applies to this pair: its nef b-part is zero. +A finite quasi-étale cover of \(T_0\) is a product of a rationally +connected factor, strict Calabi--Yau factors, irreducible holomorphic +symplectic factors, and a torus. Algebraic dimension is unchanged by +proper generically finite maps and by modifications. No +positive-dimensional rationally connected factor can occur, since a +Kähler resolution of such a factor is projective by algebraic +connectedness \cite{Campana1981}. A strict Calabi--Yau factor of +dimension at least three has no holomorphic two-forms. Extension of +forms for klt spaces \cite{KebekusSchnell2021} gives the same vanishing on a resolution, which is +projective by Kodaira's criterion. Such factors are also excluded. +Dimension-two factors of Calabi--Yau type are counted among the +symplectic factors. A resolution of an irreducible symplectic factor +has \(h^{2,0}=1\), generated by a generically nondegenerate form. + +Resolve the cover and the maps to \(T\) and to smooth models of all its +factors. We obtain a smooth compact Kähler manifold \(V\), with the +pulled-back nef line class \(\gamma\), mapping generically finitely to +\(T\) and to a product \(X_1\times\cdots\times X_r\). If \(r\ge2\), +let \(C\) be the pullback of a sum of Kähler classes from the factors. +For the projection omitting \(X_i\), every component of a smooth general +fiber maps generically finitely to \(X_i\). This fiber has algebraic +dimension zero and a nonzero top form, obtained by pulling back either +a torus volume form or a power of a symplectic form. Its canonical class +is therefore pseudo-effective. Its dimension is less than \(n\), so +the induction hypothesis, with zero boundary, makes the restriction of +\(\gamma\) zero. + +Expand \(C^{n-1}\). A nonzero term has top powers from all factors +except one, where precisely one power is missing. Integration over the +fibers just considered shows that its pairing with \(\gamma\) is zero. +Thus \(\gamma\cdot C^{n-1}=0\), and Lemma~\ref{zero:mass-test} gives +\(\gamma=0\). Vanishing descends to \(T\) by push--pull for a +generically finite map. A single torus factor is covered by the last +assertion of that lemma. + +It remains to consider one symplectic factor. On a smooth model \(X\) +we have \(a(X)=0\) and \(H^0(X,\Omega_X^2)=\C\sigma\), with +\(\sigma\) generically nondegenerate. There is no dominant meromorphic +fibration from \(X\) to an intermediate-dimensional compact base. +Indeed, on smooth Kähler models such a base has algebraic dimension +zero, hence is nonprojective and has a nonzero holomorphic two-form by +Kodaira's criterion. Its pullback would be a nonzero degenerate +holomorphic two-form on \(X\), which is impossible. + +The minimal-fibration theorem of Campana, in the form +\cite[Theorem~2.3(3) and Corollary~2.5(2)]{COP}, now says that \(X\) +is isotypically semi-simple. This means that \(X\) and a power +\(S^r\) of a simple manifold have a common generically finite cover; +we may take smooth Kähler models. If \(r\ge2\), repeat the preceding +product intersection argument. The smaller-dimensional fibers are +simple: simplicity is preserved by generically finite maps between +spaces of algebraic dimension zero. Indeed, a covering family of +proper subvarieties upstairs or downstairs gives such a family on the +other space by images or inverse images at points where the map is +finite; the countability of compact cycle components gives the very +general formulation. They are not uniruled, and hence +their canonical bundles are pseudo-effective by \cite[Theorem~1.1]{Ou}. +The induction hypothesis again applies. If \(r=1\), the common smooth +cover is simple and carries the pullback of \(\sigma\), a generically +nondegenerate holomorphic two-form. This is precisely the assertion +stated in the lemma. +\end{proof} + +\subsection{Finite correspondences and cotangent slopes} + +We next isolate the two geometric properties needed for the +nonvanishing argument. The first controls subvarieties of a product; +the second controls line subsheaves of cotangent tensors. + +\begin{lemma}\label{zero:no-correspondences} +Let \(X\) be a smooth simple compact Kähler manifold of algebraic +dimension zero, carrying a generically nondegenerate holomorphic +two-form. Suppose that every smooth connected compact Kähler manifold +mapping generically finitely onto \(X\) has irregularity zero. +Through a very general point of \(X^2\), the only positive-dimensional +proper irreducible compact analytic subvarieties are the two factor +slices. +\end{lemma} + +\begin{proof} +Fix a generically nondegenerate holomorphic two-form \(\sigma\) on +\(X\). A moving correspondence would produce a holomorphic one-form +on a fixed finite cover of \(X\). To see this, we identify its first projection +with a fixed finite cover over a parameter ball, and then contract the +pulled-back two-form in a direction varying the second projection. + +For a subvariety through a pair whose coordinates are very general, +simplicity makes each projection image either a point or all of \(X\). +If both projections dominate, their fibers through general points are +either zero-dimensional or the full other factor. A proper such +subvariety must therefore be a generically finite correspondence. + +Suppose these correspondences sweep \(X^2\). The compact cycle spaces +of a compact Kähler manifold have countably many compact irreducible +components \cite{LiebermanCycles}. One component must have incidence +image equal to \(X^2\). Resolve that component and the relevant +incidence component, obtaining a compact Kähler parameter space \(S\), +a smooth incidence space \(Y\), and maps +\[ + h:Y\longrightarrow S,\qquad F_1,F_2:Y\longrightarrow X. +\] +For a general parameter, the fiber is smooth and irreducible, and each +\(F_i\) restricts to a generically finite map. The map +\((F_1,h):Y\to X\times S\) is generically finite. Its normal finite +Stein factor is a finite cover of the smooth product. + +By purity, the branch locus of this cover is divisorial. Every branch +component dominating \(S\) has a proper image in \(X\): otherwise its +divisorial fibers over \(S\) would sweep \(X\), contradicting +simplicity. Compactness of \(S\) makes these images closed analytic +subsets. Remove the images in \(S\) of the remaining branch components +and take a small ball \(B\) about a general parameter. There is a fixed +proper analytic subset \(A\subset X\) such that the cover is étale +over \((X\setminus A)\times B\). Since \(B\) is simply connected, +this étale cover is pulled back from a fixed cover of \(X\setminus A\). +Normalize \(X\) in that cover. Uniqueness of normal finite extension +then identifies the cover over \(X\times B\) with the product of this +fixed normal cover and \(B\); see +\cite[Exposé~XII, Proposition~5.3 and Theorem~5.4]{SGA1}. + +Let \(\widetilde X\) be a smooth Kähler resolution of the fixed cover. +The second projection gives a meromorphic map +\(G:\widetilde X\times B\dashrightarrow X\). The pullback \(G^*\sigma\) +is a holomorphic two-form: resolve the meromorphic map and descend +holomorphic forms across a modification of a smooth space. Dominance +of \((F_1,F_2)\) implies that, at suitable general points, the +derivatives of \(G\) in the parameter directions span \(T_X\), while +the derivative along \(\widetilde X\) is an isomorphism. Choose a +constant tangent vector on \(B\) for which this parameter derivative +is nonzero at such a point. Contracting \(G^*\sigma\) with that vector +and restricting to \(\widetilde X\times\{t\}\) yields a nonzero global +holomorphic one-form on \(\widetilde X\). This contradicts the +hypothesis on irregularity. + +Each component parametrizing generically finite correspondences thus +has proper incidence image. Their countable union misses a very +general pair, proving the assertion. +\end{proof} + +\begin{lemma}\label{zero:cotangent-slope} +Let \(X\) be a smooth simple compact Kähler manifold of dimension +\(n\ge2\), algebraic dimension zero and irregularity zero, and suppose \(K_X\) is +pseudo-effective. Let \(\mathscr L\) be a holomorphic line bundle such +that \(L=c_1(\mathscr L)\ge c_1(K_X)\) in pseudo-effective order. +For every nonzero map \(\mathscr H\to\Omega_X^{\otimes k}\), where +\(\mathscr H\) is a line bundle and \(k\ge0\), +\begin{equation}\label{zero:slope-X} + c_1(\mathscr H)\le kL. +\end{equation} +For every very general \(x\in X\), write +\(a:Y=\operatorname{Bl}_xX\to X\). Every nonzero line map +\(\mathscr H_Y\to(a^*\Omega_X)^{\otimes k}\) satisfies +\begin{equation}\label{zero:slope-blowup} + c_1(\mathscr H_Y)\le ka^*L. +\end{equation} +The exceptional set of points can be chosen simultaneously for all +\(\mathscr H_Y\) and \(k\). +\end{lemma} + +\begin{proof} +We give the slope argument from \cite[Lemma~6.2]{K}, indicating why it +does not require any hypothesis about canonical sections. Let +\(\gamma\) belong to the full dual of the pseudo-effective cone. +Ou's convention calls precisely these classes movable, and +Harder--Narasimhan filtrations exist for them +\cite[Definition~4.2 and Lemma~4.3]{Ou}. If the minimum +\(\gamma\)-slope of \(\Omega_X\) were negative, the first +Harder--Narasimhan subsheaf \(\mathscr T\subset T_X\) would have +positive slope and be semistable. Tensor slope inequalities make its +bracket into \(T_X/\mathscr T\) zero, so it is a foliation. Its dual +has negative maximum slope and is non-pseudo-effective. By +\cite[Theorem~1.4, Lemmas~4.3--4.4 and Proposition~4.5]{Ou}, its leaves have +compact closures given by a meromorphic fibration. The rank of +\(\mathscr T\) is strictly less than \(\dim X\), since +\(c_1(T_X)\cdot\gamma\le0\). A general leaf closure contradicts +simplicity. + +All Harder--Narasimhan quotient slopes of \(\Omega_X\) are consequently +nonnegative, and its maximum slope is at most +\(c_1(K_X)\cdot\gamma\). Tensor slopes give +\[ + c_1(\mathscr H)\cdot\gamma + \le k c_1(K_X)\cdot\gamma\le kL\cdot\gamma. +\] +Separation by the dual cone proves \eqref{zero:slope-X}. + +Irregularity zero embeds \(\Pic(X)\) in the countable group +\(H^2(X,\Z)\). For any vector bundle on \(X\), linearly independent +global sections are generically pointwise independent: the coefficients +expressing one section in a maximal pointwise independent subfamily +are meromorphic functions, hence constants because \(a(X)=0\). +Evaluation is therefore injective outside a proper analytic subset. +Apply this simultaneously to +\(\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) for all lines +\(\mathscr H\) and all \(k\). + +For a point outside the resulting countable exceptional union, write +\(\mathscr H_Y=a^*\mathscr H\otimes\cO_Y(mF)\), where \(F\) is the +exceptional divisor. A nonzero map as in the statement extends away +from \(F\) to a section on \(X\) by Hartogs. If \(m>0\), that section +vanishes at \(x\), because \(a_*\cO_Y(-mF)=\mathcal I_x^m\), +contrary to injectivity of evaluation. Thus \(m\le0\), and +\eqref{zero:slope-X} proves \eqref{zero:slope-blowup}. +\end{proof} + +\subsection{Meromorphic nonvanishing from cotangent slope bounds} + +The next proposition extracts the line-bundle content of the +construction in \cite[Section~6]{K}. Its hypotheses deliberately separate +cotangent slopes from the identity of the chosen line bundle. This is +what permits the application to the sum of the canonical bundle and a +nef line bundle. + +\begin{proposition}\label{zero:line-nonvanishing} +Let \(X\) be a smooth simple compact Kähler manifold of dimension +\(n\ge2\), with \(a(X)=0\) and irregularity zero. Assume that the only +positive-dimensional proper compact irreducible analytic subvarieties +through a very general point of \(X^2\) are its factor slices. +Let \(\mathscr L\) be a holomorphic line bundle with pseudo-effective +class \(L\). Assume the following line-subsheaf bounds: +\begin{enumerate} +\item For every line bundle \(\mathscr H\), every integer \(k\ge0\), +and every nonzero map \(\mathscr H\to\Omega_X^{\otimes k}\), one has +\(c_1(\mathscr H)\le kL\). +\item Outside a countable union of proper analytic subsets of \(X\), +every point \(x\) has this property: if +\(a:Y=\operatorname{Bl}_xX\to X\), then for every line bundle +\(\mathscr H_Y\), every \(k\ge0\), and every nonzero map +\(\mathscr H_Y\to(a^*\Omega_X)^{\otimes k}\), one has +\(c_1(\mathscr H_Y)\le ka^*L\). +\end{enumerate} +Then some positive power of \(\mathscr L\) has a nonzero meromorphic +section. +\end{proposition} + +We prove the proposition by comparing point poles with vanishing along +a diagonal. Assume, throughout the proof, that no positive power of +\(\mathscr L\) has a nonzero meromorphic section. A normalized big class +on \(X\) has bounded pole order at a very general point. On the rank-two projective bundle over \(X^2\) formed from the two +pullbacks of \(\mathscr L\), a class of growing volume produces a +large family of symmetric cotangent tensors. +The diagonal in \(X^2\) forces a large common vanishing order in +their determinant. The slope inequalities then turn that vanishing into an +impossible point pole on \(X\). + +The proof below follows the volume and two-diagonal construction of +\cite[results~6.3--6.10]{K}. We retain its analytic estimates and give +their proofs along with the determinant calculations, so that the +chosen line bundle enters only through the hypotheses stated above. + +\subsubsection{Point pole thresholds} + +We will use several standard facts about volumes of real \((1,1)\)-classes. +Our normalization is \(\operatorname{vol}(\alpha)=\int\alpha^e\) for a nef class on an +\(e\)-fold. Volume is continuous, homogeneous, monotone in +pseudo-effective order, invariant under modification, and its \(e\)-th +root is concave on the big cone. Analytic Fujita approximation computes +it by Kähler parts on smooth projective modifications. Here a +\emph{smooth projective modification} means a projective modification +whose source is smooth; the sources used here are compact Kähler +manifolds obtained by resolving coherent analytic ideals. For a smooth +irreducible divisor \(D\) and a big class \(\alpha\), write +\(\operatorname{vol}_{\,|D}(\alpha)\) for the numerical restricted volume. +It is zero when \(D\) is contained in the non-Kähler locus +\cite[Theorem~1.1]{Vu2023}; otherwise it is +the supremum of the masses on \(D\) of restrictions of Kähler currents +with analytic singularities that are not generically singular on \(D\) +\cite[Lemma~2.7]{CollinsTosatti2018}. The divisorial derivative and its +continuity on the big cone are +\begin{equation}\label{zero:volume-derivative} + \frac{\mathrm{d}}{\mathrm{d} u}\operatorname{vol}(\alpha-u\{D\}) + =-e\,\operatorname{vol}_{\,|D}(\alpha-u\{D\}); +\end{equation} +see \cite[Theorem~1.1]{Vu2023}. We apply this formula only on smooth +manifolds while the varying class is big. + +Here is a useful precise form of the approximation in the restricted +volume formula. Resolve the log-ideal singularities of a current +restricted to \(D\). Its pullback is an effective real divisor plus a +positive residual current with locally bounded potentials. The latter +dominates a positive multiple of the pulled-back Kähler form. A current +with locally bounded potentials has zero Lelong numbers, so Demailly +regularization makes its class, after subtracting that multiple, nef +\cite[Theorem~1.1]{DemaillyRegularization1992}. +On a projective modification there is an effective exceptional divisor +whose negative is relatively ample. Subtracting a sufficiently small +multiple of this divisor from the residual class therefore makes that +class Kähler; add the same multiple to the divisor part. Round all +divisor coefficients slightly upwards to rational numbers. Openness of +the Kähler cone preserves the Kähler property. These changes can be +arbitrarily small in top intersections, which compute the original mass +by the bounded-potential product formula. Thus we may approximate a +restricted mass by +\begin{equation}\label{zero:rational-fujita} + h^*(\alpha|_D)=\beta+\{D'\},\qquad + \beta\ \text{Kähler},\quad D'\ge0\ \text{a rational divisor}. +\end{equation} +Moreover, \(D'\) has at least the log-ideal orders of the original +restriction on every further resolution. We will use this last +property to turn poles into vanishing conditions. Only \(D'\) is made +rational; the class \(\beta\) and the horizontal part of \(\alpha\) remain +real. + +\begin{lemma}[A point pole bound]\label{zero:pole-threshold} +Let \(W\) be a smooth compact Kähler manifold of dimension \(e\ge2\), +\(\alpha\) a big real \((1,1)\)-class, and \(z\in W\). Suppose a smooth +projective modification \(\mu:W'\to W\), which is an isomorphism near +\(z\), admits a decomposition +\[ + \mu^*\alpha=K+\{D\},\qquad K\ \text{Kähler},\quad D\ge0, +\] +where \(D\) misses the point \(z'\) over \(z\). If the ordinary analytic +Seshadri constant \(\epsilon(K,z')\) is at least \(\eta>0\), then, on the +blowup \(b:\widehat W=\operatorname{Bl}_zW\to W\) with exceptional divisor \(G\), +\begin{equation}\label{zero:pole-formula} + \sup\{u\ge0:b^*\alpha-u\{G\}\ge0\} + \le \frac{\eta}{2} + +2^{e-1}\operatorname{vol}(\alpha)\eta^{-(e-1)} . +\end{equation} +\end{lemma} + +\begin{proof} +Set \(u_0=\eta/2\). Blowing up \(z'\), the class +\(K-u_0\{G'\}\) is Kähler. The Fujita decomposition is unchanged near +\(G'\), so it supplies a Kähler current for +\(\alpha_{u_0}=b^*\alpha-u_0\{G\}\) that is smooth near \(G\). +Its restriction there has class \(u_0c_1(\cO_{\mathbb{P}^{e-1}}(1))\). +The restricted volume is consequently \(u_0^{e-1}\): the current gives +this lower bound, and the volume of the restricted class gives the +opposite bound. + +Let \(\tau\) denote the left side of Equation~\eqref{zero:pole-formula}. The classes +\(\alpha_u=b^*\alpha-u\{G\}\) are big for \(0\le u<\tau\), since +\(\alpha_0\) is big and the pseudo-effective cone is convex. The +concave function \(f(u)=\operatorname{vol}(\alpha_u)^{1/e}\) satisfies, by +Equation~\eqref{zero:volume-derivative}, +\[ + f'(u_0)=-\frac{u_0^{e-1}}{f(u_0)^{e-1}}. +\] +Its tangent line at \(u_0\) must remain positive up to \(\tau\). +Therefore +\[ + \tau\le u_0+\frac{f(u_0)^e}{u_0^{e-1}} + \le u_0+\frac{\operatorname{vol}(\alpha)}{u_0^{e-1}}, +\] +which is Equation~\eqref{zero:pole-formula}. +\end{proof} + +Fix a Kähler class \(\omega\) on \(X\). The class \(L\) is not big: +a big holomorphic line bundle would make \(X\) Moishezon, contrary to +\(a(X)=0\). For \(t>0\) define +\begin{equation}\label{zero:normalization} + P=r(L+t\omega),\qquad + r=\operatorname{vol}(L+t\omega)^{-1/n}. +\end{equation} +Then \(P\) is a big real class, \(\operatorname{vol}(P)=1\), \(L\le P/r\), and +\(r\to\infty\) as \(t\downarrow0\). Choose a smooth Fujita model +\(\mu:Y_P\to X\) with Kähler part \(P'\) satisfying +\begin{equation}\label{zero:P-prime} + \mu^*P=P'+\{D_P\},\qquad v:=\int_{Y_P}(P')^n>\frac12. +\end{equation} +At a very general point \(y\) of this model there is no +positive-dimensional proper subvariety: its image would be one through a +very general point of \(X\), and \(y\) avoids the exceptional locus. +The ordinary Seshadri formula for a Kähler class, +\[ + \epsilon(K,y) + =\inf_{\substack{W\ni y\\ \dim W>0}} + \left(\frac{\int_W K^{\dim W}} + {\operatorname{mult}_y W}\right)^{1/\dim W}, +\] +therefore gives \(\epsilon(P',y)=v^{1/n}\ge2^{-1/n}\). +This is the ordinary nef/Kähler formula +\cite[Theorem~2.8]{Tosatti2016}; see also +\cite[Equation~(1.5)]{CollinsTosatti2018} and the +Kähler cone criterion of \cite{DemaillyPaun2004}. We use this ordinary +formula in both applications below. Lemma~\ref{zero:pole-threshold}, with +\(\operatorname{vol}(P)=1\), now yields a constant \(C_n\) depending only on \(n\) +such that, for a very general \(x\), +\begin{equation}\label{zero:point-bound} + \tau(P,x):=\sup\{u\ge0:a^*P-u\{F\}\ge0\}\le C_n, + \qquad a:\operatorname{Bl}_xX\to X. +\end{equation} +Here and below positive constants denoted \(c_n,C_n\) may be decreased or +increased from one occurrence to the next. They are independent of all +parameters and choices of currents and modifications. + +\subsubsection{A projective bundle with controlled volume} + +We now produce a class whose volume grows, while its point pole threshold +grows much more slowly. On \(X^2\), let \(\mathscr L_i\) and \(L_i\) +denote the pullbacks of \(\mathscr L\) and \(L\) from the \(i\)-th factor. +Use the quotient convention and put +\[ + \pi:Z=\mathbb{P}_{X^2}(\mathscr L_1\oplus\mathscr L_2)\longrightarrow X^2, + \qquad \xi=c_1(\cO_Z(1)),\qquad d=\dim Z=2n+1 . +\] +Thus \(\pi_*\cO_Z(m)=\operatorname{Sym}^m(\mathscr L_1\oplus\mathscr L_2)\) for +\(m\ge0\). The zero divisor \(A_i\) of +\(\pi^*\mathscr L_i\to\cO_Z(1)\) has class \(\xi-L_i\); it is the +section on which \(\xi\) restricts to \(L_{3-i}\). In particular +\(\xi=\{A_i\}+L_i\ge0\). + +\begin{lemma}\label{zero:subvarieties} +For the projective bundle \(Z\) just defined, the only +positive-dimensional compact irreducible analytic subvarieties through +a very general point are a fiber of \(\pi\), the full inverse images of +the two factor slices, and \(Z\) itself. +\end{lemma} + +\begin{proof} +The image of such a subvariety in \(X^2\) is a point, a factor slice, +or all of \(X^2\), by the hypothesis of +Proposition~\ref{zero:line-nonvanishing}. Generic relative dimension +one gives the full inverse image. Otherwise, except for a point, the +subvariety is a multisection of the restricted projective line bundle. +On the slice or product \(S\), its divisor line is +\(\cO(k)\otimes\pi^*\mathscr H\), with \(k>0\). Its homogeneous +equation has coefficients in +\[ + H^0\bigl(S,\mathscr H\otimes + \mathscr L_1^{\otimes i}\otimes + \mathscr L_2^{\otimes(k-i)}\bigr),\qquad 0\le i\le k. +\] +A single nonzero monomial cuts out only axes and vertical divisors. +A non-axis multisection therefore has two nonzero coefficients. Their +quotient is a meromorphic section of a nonzero power of +\(\mathscr L_1\otimes\mathscr L_2^{-1}\). Restricting to a general +factor slice gives a meromorphic section of a nonzero power of +\(\mathscr L\); inversion makes the power positive if necessary. +This contradicts the standing contrary hypothesis. The axes +avoid a very general point of \(Z\), proving the lemma. +\end{proof} + +Let \(q\) tend to infinity through positive integers. For each sufficiently +large \(q\), choose \(t=t(q)\) in Equation~\eqref{zero:normalization} so that +\(r=r(q)\ge q^2\), and use the resulting normalized class \(P=P(q)\). +Define the real class and the scale +\begin{equation}\label{zero:M-definition} + M=P_1+P_2+q\xi,\qquad A_q=q^{1/d}. +\end{equation} + +The quantitative goal can now be stated. On the blowup +\(a:\operatorname{Bl}_xX\to X\) of a very general point, with exceptional +divisor \(F\), we will produce a positive rational scale \(s\) satisfying +\(c_nA_q\le s\le C_nA_q\) and +\[ + 2a^*P+(s+2q)a^*L-c_ns\{F\}\ge0. +\] +Since \(L\le P/r\), this would imply +\[ + \frac{c_ns}{2+(s+2q)/r}\le\tau(P,x)\le C_n. +\] +The denominator stays bounded because \(r\ge q^2\), while the numerator +tends to infinity. To obtain the class inequality, the first diagonal +will produce a subsheaf occupying a fixed positive fraction of a +symmetric cotangent power. An incidence construction will then force +its determinant to vanish to order proportional to \(s\). We begin +with the volume that supplies this rank fraction. + +\begin{lemma}\label{zero:two-slot-volume} +For these choices, \(M\) is big and +\begin{equation}\label{zero:M-bounds} + c_nq\le\operatorname{vol}(M)\le C_nq,\qquad + \tau(M,z):=\sup\{u\ge0:b_z^*M-u\{F_z\}\ge0\}\le C_nA_q +\end{equation} +at a very general point \(z\in Z\), where +\(b_z:\operatorname{Bl}_zZ\to Z\) has exceptional divisor \(F_z\). +\end{lemma} + +\begin{proof} +Pull \(Z\) to the Fujita model \(Y_P^2\) from +Equation~\eqref{zero:P-prime}, and put \(H=P'_1+P'_2\) on this pulled-back bundle. +There is a further smooth projective modification +\(\nu:\widehat Z\to\mathbb{P}_{Y_P^2}(\mu^*\mathscr L_1\oplus +\mu^*\mathscr L_2)\) and a Fujita decomposition of the pullback of \(M\) +whose Kähler part is exactly +\begin{equation}\label{zero:exact-K} + K=\frac12\nu^*H+\Theta , +\end{equation} +where \(\Theta\) is Kähler and has degree \(q\) on a general vertical +line. The modification and the divisor part miss that line. + +Here is the construction. It suffices to decompose \(H/2+q\xi\) as +\(\Theta\) plus an effective divisor, clean on a whole general vertical +line. Since \(\cO(1)\) is relatively ample, a small positive class of +the form \(\delta(\xi+cH)\) is Kähler for some \(c>0\). Choose +\(\delta>0\) so small that \(\delta0\) and \(C_s\) is big for +\(0\le s0 +\] +on a general fiber. For sufficiently large divisible integers \(j\), +let \(\mathcal F_j=f_*\cO_U(j\Lambda)\) and \(R_j=\operatorname{rk}\mathcal F_j\). +Here \(j\Lambda\) is an actual line bundle and +\(c_1(\mathcal F_j)\) means \(c_1(\det\mathcal F_j)\). Then +\begin{align} + R_j&=\frac{w}{e!}j^e+O(j^{e-1}),\label{zero:GRR-rank}\\ + B_0:=G+\lim_j\frac{c_1(\mathcal F_j)}{jR_j} + &=\frac{f_*\beta^{e+1}}{(e+1)w}\ge0,\label{zero:GRR-base}\\ + \int_U\beta^{b_0+e} + &\le (e+1)^{b_0}w\,\operatorname{vol}(B_0).\label{zero:mixed-bound} +\end{align} +The limit in Equation~\eqref{zero:GRR-base} is a limit of real Bott--Chern classes. +\end{lemma} + +\begin{proof} +The restriction of \(c_1(\Lambda)\) to each fiber is represented by the +restriction of the Kähler form \(\beta\). The fiberwise criterion for +relative ampleness makes \(\Lambda\) relatively ample after clearing +denominators. Relative Serre vanishing then kills the higher direct +images for all sufficiently large divisible \(j\). Analytic +Grothendieck--Riemann--Roch \cite{LevyGRR}, in degrees zero and two, gives +\[ + R_j=\frac{f_*c_1(\Lambda)^e}{e!}j^e+O(j^{e-1}),\qquad + c_1(\mathcal F_j) + =\frac{f_*c_1(\Lambda)^{e+1}}{(e+1)!}j^{e+1}+O(j^e). +\] +The degree-zero pushforward is \(w\). Expanding +\(\beta=f^*G+c_1(\Lambda)\) shows that +\[ + f_*\beta^{e+1} + =f_*c_1(\Lambda)^{e+1}+(e+1)wG, +\] +because terms with at least two horizontal factors have negative +fiber degree after pushforward. This proves the equality in +Equation~\eqref{zero:GRR-base}; the cohomological GRR equality is an equality in +Bott--Chern cohomology by the \(\partial\bar\partial\)-lemma on compact +Kähler manifolds. Pushforward of the positive form +\(\beta^{e+1}\) is a positive closed \((1,1)\)-current, so \(B_0\) is +pseudo-effective. + +Apply Lemma~\ref{zero:nef-pushforward} to choose \(p:Z'_0\to Z_0\), +\(q:U'\to U\), and \(f':U'\to Z'_0\), and put \(\beta'=q^*\beta\). +The resulting class \(B'_0\) is nef and satisfies +Equation~\eqref{zero:flattened-volume}. +Choose a Kähler class \(\omega'\) on \(Z'_0\) and put +\(C=B'_0+\varepsilon\omega'\). For \(0\le k\le b_0\), set +\[ + I_k=\int_{U'}(\beta')^{e+k}(f'^*C)^{b_0-k}. +\] +These are mixed intersections of nef classes. The first two satisfy +\[ + I_0=w\int_{Z'_0}C^{b_0},\qquad + I_1=(e+1)w\int_{Z'_0}B'_0C^{b_0-1} + \le(e+1)w\int_{Z'_0}C^{b_0}. +\] +The mixed nef inequalities make the sequence \(I_k\) log-concave. +Every \(I_k\) is positive: on the dense open where \(q\) is a local +biholomorphism and \(f'\) is a submersion, \(\beta'\) is positive +definite and \(f'^*C\) has rank \(b_0\), so the defining top form +is strictly positive. The successive ratios are therefore at most +\(I_1/I_0\le e+1\). Thus +\[ + \int_U\beta^{b_0+e}=I_{b_0} + \le(e+1)^{b_0}w\int_{Z'_0}C^{b_0}. +\] +Letting \(\varepsilon\downarrow0\) and using +Equation~\eqref{zero:flattened-volume} proves Equation~\eqref{zero:mixed-bound}. +\end{proof} + +We will also use the determinant formula without its volume bound. + +\begin{corollary}\label{zero:slope-descent} +Let \(Y\) be a smooth compact Kähler manifold and \(\mathscr E\) a +holomorphic vector bundle of rank at least two. Write +\(\pi_{\mathscr E}:\mathbb{P}_Y(\mathscr E)\to Y\) and +\(\zeta_{\mathscr E}=c_1(\cO_{\mathbb{P}(\mathscr E)}(1))\). +Suppose that a real class \(D\) on \(Y\) satisfies +\[ + c_1(\mathscr H)\le kD + \quad\text{whenever}\quad + 0\ne\bigl(\mathscr H\longrightarrow\mathscr E^{\otimes k}\bigr) +\] +for a line bundle \(\mathscr H\) and integer \(k\ge0\). +For a real class \(A\) on \(Y\) and a real number \(b\ge0\), +\begin{equation}\label{zero:slope-descent-equation} + \pi_{\mathscr E}^*A+b\zeta_{\mathscr E}\ge0 + \quad\Longrightarrow\quad A+bD\ge0 . +\end{equation} +\end{corollary} + +\begin{proof} +If \(Y\) is a point, both \(A\) and \(D\) vanish and the assertion is +immediate. Suppose \(\dim Y\ge1\). Choose a real class \(H_0\) on \(Y\) so that +\(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0\) is Kähler; relative +ampleness of \(\cO(1)\) permits such a choice. Take numbers +\(\delta\downarrow0\) with \(b+\delta>0\) rational. Adding +\(\delta(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0)\) to the +pseudo-effective class in Equation~\eqref{zero:slope-descent-equation} makes it +big. A Fujita decomposition on a smooth projective modification +\(h:U\to\mathbb{P}_Y(\mathscr E)\), with the divisor coefficients rounded +upwards as in Equation~\eqref{zero:rational-fujita}, has the form +\[ + \beta=f^*(A+\delta H_0)+c_1(\Lambda),\qquad + \Lambda=(b+\delta)h^*\cO(1)-\cO_U(D') + \ \text{in }\Pic(U)\otimes\Q, +\] +where \(f=\pi_{\mathscr E}h\), \(\beta\) is Kähler, and \(D'\ge0\) +is rational. For divisible \(j\), +\[ + f_*\cO_U(j\Lambda)\ \subseteq\ + \operatorname{Sym}^{j(b+\delta)}\mathscr E . +\] +Its determinant of rank \(R_j\) consequently maps into +\(\mathscr E^{\otimes j(b+\delta)R_j}\). The assumed line inequality +gives +\[ + \frac{c_1(\mathcal F_j)}{jR_j}\le(b+\delta)D. +\] +Lemma~\ref{zero:direct-image} makes +\(A+\delta H_0+\lim c_1(\mathcal F_j)/(jR_j)\) pseudo-effective. +Adding the preceding pseudo-effective difference shows that +\(A+\delta H_0+(b+\delta)D\) is pseudo-effective. Let +\(\delta\downarrow0\). This proves the corollary. In this argument +only \(b+\delta\) and the coefficients of \(D'\) are rational; \(A\), +\(H_0\), and \(D\) remain real classes. +\end{proof} + +For the forthcoming application over the \(d\)-fold \(Z\), with fibers +of dimension \(d-1\), Lemma~\ref{zero:direct-image} has a useful concrete +consequence. Once we prove \(B_0\le C_nM\), the bound +\(\operatorname{vol}(M)\le C_nq\) gives +\[ + \int_U\beta^{2d-1}\le C_nwq. +\] +Thus an upper bound for the normalized determinant class will convert +a lower bound for total mass into a lower bound for the fiber volume +\(w\). The next calculation supplies the required determinant control. + +\subsubsection{Cotangent lines on the projective bundle} + +We return to \(Z=\mathbb P_{X^2}(\mathscr L_1\oplus\mathscr L_2)\). +The following calculation is where the chosen line bundle enters the +first determinant estimate. A line mapping into a cotangent tensor +has nonpositive degree on a general projective-line fiber; the +calculation also controls its horizontal class. + +\begin{lemma}\label{zero:cotangent-Z} +Let \(\mathscr H\) be a holomorphic line bundle on \(Z\), and suppose +there is a nonzero map \(\mathscr H\to\Omega_Z^{\otimes k}\), with +\(k\ge0\). Then +\[ + \mathscr H=\pi^*(\mathscr Q_1\boxtimes\mathscr Q_2) + \otimes\cO_Z(\ell) +\] +for lines \(\mathscr Q_i\) on \(X\) and an integer \(\ell\), and +\[ + \ell\le0,\qquad + c_1(\mathscr Q_1)_1+c_1(\mathscr Q_2)_2 + \le(k-\ell)(L_1+L_2). +\] +\end{lemma} + +\begin{proof} +Irregularity zero and Künneth give +\(\Pic(X^2)=\operatorname{pr}_1^*\Pic(X)\oplus +\operatorname{pr}_2^*\Pic(X)\), and the projective-bundle formula for +\(\Pic(Z)\) gives the asserted expression. Filter the target using +\[ + 0\longrightarrow\pi^*\Omega_{X^2}\longrightarrow\Omega_Z + \longrightarrow\cO_Z(-2)\otimes + \pi^*(\mathscr L_1\otimes\mathscr L_2)\longrightarrow0. +\] +Choose a nonzero associated graded component. Suppose it has \(i\) +relative factors and \(k_1,k_2\) cotangent factors from the two copies +of \(X\), so \(k_1+k_2=k-i\). Its relative degree is \(-2i\). +Pushing to \(X^2\) forces \(m=-2i-\ell\ge0\); hence +\(\ell\le-2i\le0\). A nonzero monomial in +\(\operatorname{Sym}^m(\mathscr L_1\oplus\mathscr L_2)\), with first +exponent \(m_1\), gives on the two general slices +\[ + c_1(\mathscr Q_1)\le(i+m_1+k_1)L,\qquad + c_1(\mathscr Q_2)\le(i+m-m_1+k_2)L. +\] +These are exactly the assumed cotangent-line bounds on \(X\), applied +after moving the indicated powers of \(\mathscr L\) to the source. +Each coefficient on the right is at most \(k-\ell\). Since \(L\ge0\), +pullback and addition prove the claimed bound. +\end{proof} + +\subsubsection{Extracting a positive rank fraction} + +\begin{lemma}\label{zero:diagonal-mass} +There are constants \(c_n,C_n>0\) such that, for every sufficiently +large \(q\), one can choose a rational number +\[ + c_nA_q\le s\le C_nA_q +\] +and a Kähler current \(T\) in \(C_s\) with analytic singularities, +not generically singular on \(E\), with the following property. On a +smooth projective modification \(h:U\to E\), its restricted mass has +a rational-divisor approximation +\begin{equation}\label{zero:selected-decomposition} + h^*(2M+s\zeta)=\beta+\{D'\}, + \qquad \beta\ \text{Kähler},\quad D'\ge0\ \text{rational}, +\end{equation} +which retains all log-ideal orders of \(T|_E\). If +\(f:U\to Z\) is the natural map and +\[ + w=\int_{U_z}\beta^{d-1} +\] +on a general fiber, then +\begin{equation}\label{zero:selected-w} + w\ge c_ns^{d-1}. +\end{equation} +\end{lemma} + +\begin{proof} +First let \(s\) be any rational number in \((0,s_{\max})\) with +\(v_E(s)>0\), and use the approximation in +Equation~\eqref{zero:rational-fujita} for any current used +to compute this restricted volume. In +Lemma~\ref{zero:direct-image}, the data for Equation~\eqref{zero:restriction-class} +are +\[ + b_0=d,\quad e=d-1,\quad G=2M,\quad + \Lambda=s h^*\cO_E(1)-\cO_U(D') + \ \text{in }\Pic(U)\otimes\Q. +\] +For large divisible \(j\), the associated sheaves satisfy +\begin{equation}\label{zero:symmetric-inclusion} + \mathcal F_j=f_*\cO_U(j\Lambda) + \subseteq\operatorname{Sym}^{js}\Omega_Z,\qquad + R_j=\operatorname{rk}\mathcal F_j,\qquad 00\). The contribution of +\(0(1-\delta)m\). Summing instead over their internal +degree gives +\[ + H_m(\delta) + \le \binom{m+n}{n}\binom{\lceil\delta m\rceil+n}{n},\qquad + \limsup_{m\to\infty}\frac{H_m(\delta)}{N_m} + \le\binom{2n}{n}\delta^n. +\] +On the other hand, \(w^+\ge w/2\) and +Equation~\eqref{zero:plus-rank} show that the approximation on \(E^+\) +retains at least half the rank fraction in +Equation~\eqref{zero:selected-rank-fraction}: +\[ + \lim_{j\to\infty}\frac{R_j^+}{N_{js}} + =\frac{w^+}{s^{d-1}} + \ge\frac{w}{2s^{d-1}}\ge c_n>0. +\] +Fix \(\delta>0\), depending only on \(n\), so small that, for each fixed +\(q\) and its selected data, fewer than \(R_j^+/2\) monomials have transverse +degree greater than \((1-\delta)js\) for all sufficiently large divisible +\(j\). For each +remaining monomial, Equation~\eqref{zero:coefficient-order} and +Lemma~\ref{zero:incidence-pole} give +\[ + \operatorname{ord}_u(c_{\alpha,\gamma}) + \ge j(s-C_n)-(1-\delta)js + =j(\delta s-C_n)\ge\frac{\delta js}{2}. +\] +The last inequality holds once \(q\) is large, since +\(s\ge c_nA_q\to\infty\). + +Over the same discrete valuation ring \(R\), the torsion-free sheaf +\(\mathcal F_j^+\) becomes a free module of rank \(R_j^+\). Write its +inclusion into the symmetric power as a matrix with the monomials +as rows. Every maximal minor uses at least \(R_j^+/2\) of the rows +whose coefficients have order at least \(\delta js/2\). All maximal +minors therefore have order at least +\(\delta jR_j^+s/4\). By the definition of \(h_j\), this proves +Equation~\eqref{zero:determinant-order-equation}. Dividing the determinant +map by \(\sigma_{J_+}^{h_j}\) gives a holomorphic map at every point of +codimension one: the coefficients have the required order along \(J_+\), +and its local equation is a unit away from \(J_+\). Hartogs' theorem +extends the divided map across the remaining set of codimension at least +two, giving the stated factorization through +\(\det\mathcal F_j^+(h_jJ_+)\). +\end{proof} + +\subsubsection{Cancellation on the point blowup} + +\begin{proof}[Completion of the proof of Proposition~\ref{zero:line-nonvanishing}] +Fix \(q\) sufficiently large for the preceding lemmas, with its chosen +\(P,s,T\) and modifications. Choose a very general point \(x\) in the first +factor of \(X^2\). The slice of \(Z^+\) above \(x\) is +\[ + S=\mathbb{P}_Y(\cO_Y\oplus a^*\mathscr L),\qquad + a:Y=\operatorname{Bl}_xX\longrightarrow X, +\] +and \(J_+|_S\) is the pullback of \(F\). We choose \(x\) so that +Equation~\eqref{zero:slope-blowup}, Equation~\eqref{zero:point-bound}, the restrictions of +the currents below, and all determinant maps for divisible \(j\) +can be tested on this slice. These exclude at most countably many +proper analytic sets and the measure-zero exceptional sets for +current restrictions. + +Write the restricted determinant line as +\[ + (\det\mathcal F_j^+)|_S + =\pi_S^*\mathscr Q_j^+\otimes\cO_S(\ell_j^+),\qquad + Q_j^+=c_1(\mathscr Q_j^+). +\] +Factoring the common zero of order \(h_j\) from the determinant +map gives a nonzero map whose source on \(S\) is +\[ + \pi_S^*(\mathscr Q_j^+\otimes\cO_Y(h_jF)) + \otimes\cO_S(\ell_j^+). +\] +The plus sign of \(h_jF\) follows because division of the map by +the local equation of \(J_+^{h_j}\) enlarges its source from +\(\det\mathcal F_j^+\) to +\(\det\mathcal F_j^+\otimes\cO(h_jJ_+)\). +Its target embeds into the \(jsR_j^+\)-fold tensor of the +restriction of \(p_+^*\Omega_Z\). In particular we use the +cotangent bundle of the original \(Z\), with its restricted +filtration +\[ + 0\longrightarrow + \pi_S^*(\cO_Y^{\oplus n}\oplus a^*\Omega_X) + \longrightarrow (p_+^*\Omega_Z)|_S + \longrightarrow + \cO_S(-2)\otimes\pi_S^*a^*\mathscr L + \longrightarrow0. +\] +The same homogeneous-monomial calculation as in +Equation~\eqref{zero:determinant-bound-Z}, now applying +Equation~\eqref{zero:slope-blowup} to the second factor of \(X^2\), gives +\begin{equation}\label{zero:determinant-bound-slice} + \ell_j^+\le0,\qquad + Q_j^++h_j\{F\} + \le(jsR_j^+-\ell_j^+)a^*L. +\end{equation} +More explicitly, if the chosen graded term has \(i\) relative +factors, the pushed polynomial has degree +\(-2i-\ell_j^+\ge0\). Its exponent in \(a^*\mathscr L\), together +with the \(i\) relative factors and the cotangent order from the second +factor of \(X^2\), is at most \(jsR_j^+-\ell_j^+\). The cotangent +factors from the first factor of \(X^2\) are constant. This proves the +displayed inequality using +the tensors of \(a^*\Omega_X\), without introducing \(\Omega_Y\) +or \(\Omega_{Z^+}\). + +Apply Lemma~\ref{zero:direct-image} to the decomposition in +Equation~\eqref{zero:plus-decomposition}. Equation~\eqref{zero:GRR-base} +shows that the limits of the restricted determinant components divided by +\(jR_j^+\) exist; denote them by \(Q^+\) and \(\ell^+\). Divide +Equation~\eqref{zero:determinant-bound-slice} by \(jR_j^+\), use +Equation~\eqref{zero:determinant-order-equation}, and pass to the closed +pseudo-effective cone. We obtain +\begin{equation}\label{zero:slice-upper} + \ell^+\le0,\qquad + Q^++c_ns\{F\}\le(s-\ell^+)a^*L. +\end{equation} +The pseudo-effective class in Equation~\eqref{zero:GRR-base}, restricted to the +very general slice \(S\), is +\begin{equation}\label{zero:slice-psef} + \pi_S^*(Q^++2a^*P)+(\ell^++2q)\xi\ge0. +\end{equation} +Testing on a general vertical line gives \(\ell^++2q\ge0\). +Let \(A\subset S\) be the axis of +\(\mathbb{P}_Y(\cO_Y\oplus a^*\mathscr L)\) with divisor class \(\{A\}=\xi\). +On \(A\simeq Y\), its normal class is \(\{A\}|_A=\xi|_A=a^*L\ge0\). +Add an arbitrarily small Kähler class to the class in +Equation~\eqref{zero:slice-psef}, and choose a Kähler current with analytic +singularities in the resulting big class. Remove its generic divisorial +pole along \(A\) and restrict +the residual current to \(A\). Adding back the removed nonnegative multiple +of \(a^*L\) preserves pseudo-effectivity. Passing to the limit gives +\begin{equation}\label{zero:slice-lower} + Q^++2a^*P+(\ell^++2q)a^*L\ge0. +\end{equation} + +The slope bound in Equation~\eqref{zero:slice-upper} supplies the +pseudo-effective class \((s-\ell^+)a^*L-Q^+-c_ns\{F\}\). Adding it to +Equation~\eqref{zero:slice-lower} cancels the entire normalized determinant +class restricted to the axis, \(Q^++\ell^+a^*L\), leaving +\[ + 2a^*P+(s+2q)a^*L-c_ns\{F\}\ge0. +\] +Since \(L\le P/r\), the point bound in Equation~\eqref{zero:point-bound} implies +\begin{equation}\label{zero:contradiction} + \frac{c_ns}{\,2+(s+2q)/r\,}\le C_n. +\end{equation} +The denominator is bounded independently of \(q\), because +\(s\le C_nq^{1/d}\) and \(r\ge q^2\). The numerator tends to +infinity, because \(s\ge c_nq^{1/d}\). Equation~\eqref{zero:contradiction} +is impossible for arbitrarily large \(q\). + +For each \(q\), the choice of \(t(q)\), the rational \(s\), the current +\(T\), and the modifications precedes the limit in \(j\). The monomial +cutoff \(\delta\) depends only on \(n\), while the required lower bound for +divisible \(j\) may depend on the fixed data. We choose the very general +points after those data are fixed. +Thus every simultaneous general-point test above involves at most +countably many conditions, and no bound depends on the point. +The contradiction disproves the contrary hypothesis and proves +Proposition~\ref{zero:line-nonvanishing}. +\end{proof} + +\subsection{Vanishing on the simple model} + +We now apply Proposition~\ref{zero:line-nonvanishing} to the sum of +the canonical bundle and the nef line bundle whose class we wish to +eliminate. The remaining use of the ordinary adjoint +decomposition is particularly short: it turns meromorphic +nonvanishing into both signs of pseudo-effectivity. + +\begin{proposition}\label{zero:simple-vanishing} +Let \(X\) be a smooth simple compact Kähler manifold of algebraic +dimension zero carrying a generically nondegenerate holomorphic +two-form. Every nef rational holomorphic line bundle on \(X\) has zero +first Chern class. +\end{proposition} + +\begin{proof} +The assertion is immediate for a point. A positive-dimensional space +of algebraic dimension zero cannot be a curve, so assume \(\dim X\ge2\). +If \(q(X)>0\), its Albanese map is generically finite onto its image, +by simplicity. Ueno's structure theorem for subvarieties of complex +tori makes this image a translate of a torus: its quotient by the +connected stabilizer is of general type, and positive dimension of that +quotient would contradict algebraic dimension zero \cite{Ueno}. +Lemma~\ref{zero:mass-test} proves the conclusion. + +The same argument applies after any smooth generically finite Kähler +cover of \(X\) having positive irregularity. Such a cover is still +simple and has algebraic dimension zero; vanishing upstairs descends +by push--pull. We may therefore suppose that every such cover has +irregularity zero. Lemma~\ref{zero:no-correspondences} then describes +the subvarieties through a very general point of \(X^2\). + +Clear denominators and let \(N_0\) be the resulting nef holomorphic line +bundle. A top wedge of the generically nondegenerate two-form is a +nonzero holomorphic section of \(K_X\). In particular \(K_X\) is +effective. Set +\[ + \mathscr L=K_X+N_0,\qquad L=c_1(\mathscr L). +\] +The class \(L\) is pseudo-effective and dominates \(c_1(K_X)\). +Lemma~\ref{zero:cotangent-slope} supplies both slope inequalities. +Every hypothesis of Proposition~\ref{zero:line-nonvanishing} is now +satisfied. Some positive power of \(K_X+N_0\) therefore has a nonzero +meromorphic section. Dividing by the corresponding power of a +canonical section shows that \(N_0\), as a rational line bundle, is +represented by a signed divisor. + +Resolve its positive and negative parts. On the resulting smooth +Kähler model write +\[ + N_0\qsim E-G,\qquad E,G\ge0, +\] +with rational coefficients and simple normal crossing union of their +supports; we suppress the pullback notation for \(N_0\). The canonical +bundle \(K'\) of this model remains effective. Choose a small rational +\(u>0\) so that both \(uE\) and \(uG\) are klt boundaries. +The ordinary decomposition, Theorem~\ref{pre:ordinary}, applies to +\(K'+uE\) and \(K'+uG\). On a common higher model, denote their pulled +back classes by \(\alpha_E\) and \(\alpha_G\). Algebraic dimension zero +makes each semiample part rationally trivial, so that the pulled back +rational lines equal their negative divisors. Since +\(\alpha_E-\alpha_G=u\{N_0\}\) is nef, +Lemma~\ref{pre:negative}\,(2) gives +\[ + N(\alpha_E)\le N(\alpha_G),\qquad + u\{N_0\}=\{N(\alpha_E)-N(\alpha_G)\}. +\] +Thus \(-\{N_0\}\) is pseudo-effective, whereas \(\{N_0\}\) is nef. +Pairing both classes with a Kähler power gives zero mass, and hence +\(\{N_0\}=0\). Pullback injectivity yields the same conclusion on +\(X\). +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{zero:vanishing}] +Lemma~\ref{zero:geometric-reduction} reduces the assertion to +Proposition~\ref{zero:simple-vanishing}, which proves it. +In particular, once the induction has reached dimension \(n\), the +ordinary decomposition of \(K_T+B\) also proves the decomposition +assertion of Theorem~\ref{thm:decomposition} in algebraic dimension +zero: adding the nef line changes no Chern class. Its semiample positive +part is rationally trivial, and its negative part is unchanged. +\end{proof} diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/06-cycles.tex b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/06-cycles.tex index fe16f444e38eb064602c3b23c6d1f27a26a5373f..5d3889c3b98ee42a309f1819aa0e54421103f62f 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/06-cycles.tex +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/06-cycles.tex @@ -1,488 +1,488 @@ -\section{The nef summand on the algebraic-reduction fibers} -\label{sec:cycles} - -Let $T$ be a smooth compact K\"ahler manifold of dimension $n$, and suppose -that $00$, some divisible $m$ has two sections whose -ratio is nonconstant on the connected fiber $X_w$. Lifting these -sections as above gives a meromorphic function on $X$ whose restriction -to $X_w$ is that ratio. Every meromorphic function on $X$ factors -through its algebraic reduction. Such a function, when defined on a -nonempty open subset of the smooth fiber $X_w$, is constant there: -a local holomorphic section of $h$ through a point where the quotient -is holomorphic shows that the base function extends holomorphically -at $w$. This contradicts the chosen nonconstant ratio. - -The countable simultaneous choice is legitimate by Baire's theorem. -If a generic assertion is initially established on a dense open set, -a proper analytic exceptional subset of that open is still nowhere -dense in $W$. Thus such generic assertions can also be included in the -choice. -\end{proof} - -\subsection{Divisor incidence and a countable collection of fibrations} - -We use the cycle-space results of Barlet, Lieberman, and Fujiki -\cite{Barlet,LiebermanCycles,Fujiki1978,Fujiki}. In the form needed here, -the spaces of -effective cycles of a compact K\"ahler manifold have countably many -irreducible components; each component is compact and belongs to -Fujiki's class $\mathcal C$; see \cite[p.~189]{Fujiki}. The locus of -cycles contained in a fiber of a proper map is an analytic relative -cycle space \cite[Proposition~3.2]{Fujiki1978}. We also use the -meromorphic fiber map obtained by analytic flattening: after modifying -the base, the flat strict transform has an analytic family of fiber -cycles and hence a holomorphic map to the cycle space -\cite[Chapter~V]{Barlet}. All parameter spaces and images in class $\mathcal C$ -will be replaced by smooth compact K\"ahler models when needed. - -The following observation explains why the numerical class of a divisor -can detect a semiample fibration without fixing its flat twist. - -\begin{lemma}\label{cycle:divisors} -Let $F$ be a smooth connected compact K\"ahler manifold, let $A$ be a -rational line bundle on $F$, and suppose that a modification -$\rho:\widehat F\to F$ from a smooth compact K\"ahler manifold satisfies -\begin{equation}\label{cycle:fiber-decomposition} - \rho^*A\num P+R, - \qquad P\text{ semiample}, - \qquad R=N(c_1(\rho^*A))\text{ a rational divisor}. -\end{equation} -Let $g:\widehat F\to Q$ be the fibration defined by $P$, so that $Q$ is -normal projective and $P\qsim g^*H_Q$ for an ample rational line bundle -$H_Q$. There is a dense open subset $U\subset F$, independent of the -divisors below, with the following properties for every sufficiently -divisible positive integer $m$. -\begin{enumerate} -\item There exists an analytic family of effective -divisors on $F$ with class $m c_1(A)$ that distinguishes the general fibers -of the meromorphic fibration $g\circ\rho^{-1}$. -\item For every effective divisor $D$ with class -$m c_1(A)$, membership -in $\operatorname{Supp}D$ is constant along the intersection of $U$ -with a general smooth fiber of that meromorphic fibration. -\end{enumerate} -\end{lemma} - -\begin{proof} -The difference between the two sides of -\eqref{cycle:fiber-decomposition} is flat as a rational line bundle. -By Lemma~\ref{pre:picard}, there is a flat rational line -bundle $\eta$ on $F$ such that -\[ - \rho^*(A+\eta)\qsim P+R. -\] -Choose $m$ clearing the denominators and such that $mH_Q$ is very ample. -The divisors -\[ - g^*D_Q+mR, - \qquad D_Q\in|mH_Q|, -\] -descend to an analytic family of divisors on $F$ with class $m c_1(A)$. -Indeed $\rho_*\cO_{\widehat F}=\cO_F$, so the projection formula -identifies sections of $m(A+\eta)$ with sections of its pullback. -On the isomorphism locus of $\rho$, away from $\operatorname{Supp}R$, -they distinguish general $g$-fibers, since hyperplanes distinguish -points of $Q$. - -For an arbitrary effective divisor $D$ of this class, -Lemma~\ref{pre:negative}(1) gives -\[ - \rho^*D\geq mR. -\] -The residual divisor $D'=\rho^*D-mR$ is effective and has class -$m c_1(P)$. Let $G$ be a smooth compact connected fiber of $g$. If -$D'$ does not contain $G$, its restriction is an effective divisor of -class zero on $G$, hence is empty: in positive dimension this follows -by integrating against a K\"ahler form to the appropriate power. -Consequently $\operatorname{Supp}D'$ either contains $G$ or misses it. -For a point fiber the same alternative is immediate. Taking the image -of the isomorphism locus of $\rho$, and deleting -$\operatorname{Supp}R$ and the nonsmooth locus of $g$, gives the stated -open set $U$. This choice uses only $\rho$, $R$, and $g$, and works for -all the divisors simultaneously. -\end{proof} - -\begin{lemma}\label{cycle:detection} -Let $h:X\to W$ be a fibration of smooth connected compact K\"ahler -manifolds with $\dim W<\dim X$, and let $A$ be a rational line bundle -on $X$. There is a countable collection of diagrams -\begin{equation}\label{cycle:diagrams} - \begin{tikzpicture}[baseline=(current bounding box.center),>=Stealth] - \node (Xi) at (0,1.15) {$X_i$}; - \node (Zi) at (2.7,1.15) {$Z_i$}; - \node (X) at (0,0) {$X$}; - \node (W) at (2.7,0) {$W$}; - \draw[->] (Xi) -- node[above] {$f_i$} (Zi); - \draw[->] (Xi) -- node[left] {$p_i$} (X); - \draw[->] (Zi) -- node[right] {$k_i$} (W); - \draw[->] (X) -- node[below] {$h$} (W); - \end{tikzpicture} -\end{equation} -where $p_i$ is a modification, $f_i$ is a fibration, and $X_i,Z_i$ are -smooth compact K\"ahler manifolds, with the following property. - -For a very general smooth fiber $F=X_w$, suppose that a smooth compact -K\"ahler modification $\rho:\widehat F\to F$ has a decomposition -\eqref{cycle:fiber-decomposition} for $A|_F$, and that the fibration -$g:\widehat F\to Q$ defined by $P$ has positive-dimensional base. -For some $i$, the restriction of $f_i$ over $w$ is bimeromorphically -equivalent to $g$. In particular, on a common resolution, their -general fibers coincide. The collection and the exceptional subsets -of $W$ can be fixed before $F$, $\rho$, $P$, and $R$ are chosen. -\end{lemma} - -\begin{proof} -We associate a map to each component of the relative divisor space by -recording all its divisors through a point. On a fiber with the stated -decomposition, Lemma~\ref{cycle:divisors} makes these incidence supports -constant along the semiample fibration. Integral multiplicities will -then give constancy of the cycle maps themselves, while the -distinguishing linear system will give the converse. - -Put $d=\dim X-\dim W$, and let $W^\circ$ be the smooth-fibration locus -of $h$. Consider the relative cycle spaces whose points are pairs -$(w,D)$, where $D$ is an effective $(d-1)$-cycle in $X_w$. They are -closed subspaces of the product of $W$ with the corresponding compact -cycle components of $X$. Thus they have countably many compact -irreducible components in class $\mathcal C$. - -For each positive integer $m$ clearing the denominator of $A$, retain -the components meeting $W^\circ$ on which the divisor class is -$m c_1(A|_{X_w})$. This is a condition on an entire component over -$W^\circ$: in a smooth family the cohomology class of an analytic -family of cycles is locally constant, and $c_1(A)$ supplies a global -section of the same cohomology local system. After resolving a -component, its open part over $W^\circ$ is connected, so equality at -one point implies equality throughout that part. Components not -dominating $W$ have proper analytic images; exclude all these images -from the eventual choice of $w$. - -Fix a retained dominating component and a smooth compact K\"ahler -resolution $S$ of it. Denote its divisor at $s\in S$ by $D_s$ and form -the incidence -\[ - \mathcal I_S - =\{(x,s)\in X\times S:x\in\operatorname{Supp}D_s\}. -\] -We take the incidence of the analytic cycle family, including its -limits from the open part over $W^\circ$. Over that open part it is a -family of pure $(d-1)$-dimensional cycles. Hence every irreducible -component there dominates $S$ and has dimension $\dim S+d-1$. -Let $I_1,\ldots,I_t$ be the incidence components that dominate $X$. -The other components have proper analytic images in $X$; outside -those images they contribute nothing to incidence. - -For each $I_j\to X$, analytic flattening gives a meromorphic map -recording its generic fiber as an effective cycle in $S$. All these -cycles have dimension -\[ - e=\dim S+d-1-\dim X. -\] -Taking the tuple of these maps, together with $h$, gives a meromorphic -map $\Phi_S$ from $X$ into $W$ times a finite product of compact cycle -components of $S$. On a dense open subset of $X$, the union of the -supports of the recorded cycles is precisely -\begin{equation}\label{cycle:incidence-support} - \{s\in S:x\in\operatorname{Supp}D_s\}. -\end{equation} -In obtaining this open set we discard the images of the nondominating -incidence components and the exceptional sets of the fiber maps. -The target components and the image of $\Phi_S$ belong to class -$\mathcal C$. Resolving the graph, taking Stein factorization, and -replacing the base by a smooth compact K\"ahler model therefore gives -a diagram \eqref{cycle:diagrams}. The coordinate $h$ ensures that its -base maps to $W$. If there is no dominating incidence component, we -retain only the coordinate $h$; such a component will not be needed -to detect a positive-dimensional $Q$. - -This construction gives countably many diagrams. Fix their models, -their incidence-recording open sets, and their Stein factorizations. -We may simultaneously require that their fibers over the eventual -$w$ be smooth, that $p_i$ restrict to a modification over $w$, and that -each of the dense open sets used meet the corresponding fiber densely. -For any one diagram these are generic conditions, by properness, -generic smoothness, and the fiber-dimension theorem. In particular, -let $U_i\subset Z_i$ be the locus over which $f_i$ is smooth. We -require that $k_i^{-1}(w)$ meet $U_i$ densely. To justify this -condition, a component of $Z_i\setminus U_i$ that does not dominate -$W$ has proper image, whereas a component dominating $W$ has general -fiber of dimension strictly less than $\dim Z_i-\dim W$. -Since the general $k_i$-fiber is smooth of that pure dimension, it -cannot have a component contained in $Z_i\setminus U_i$. The same -conditions may be imposed after any preselected countable collection -of further modifications of these diagrams. - -We now choose such a very general $w$ and a decomposition on $F=X_w$ -as in the statement. By Lemma~\ref{cycle:divisors}, a projective -linear system of divisors of class $m c_1(A|_F)$ distinguishes the -general $g$-fibers for some $m$. Its image in the relative cycle -space is irreducible and lies in one irreducible component. That -component dominates $W$, since all images of nondominating components -were excluded. It is therefore one of the components used above. -Passing to its parameter resolution $S$ preserves the distinguishing -property: every divisor still has a preimage in $S$. - -The map $\Phi_S|_F$ distinguishes general $g$-fibers. Indeed two -different general points of $Q$ are separated by a divisor in the -chosen linear system, so their incidence supports -\eqref{cycle:incidence-support} differ. Conversely, fix a general -smooth connected $g$-fiber $G$. On the common open set of -Lemma~\ref{cycle:divisors} and the incidence recording, membership in -every divisor $D_s$ is constant along $G$. Thus the support in -\eqref{cycle:incidence-support} is fixed as the point moves in that -open part of $G$. - -This support assertion also gives constancy of the recorded -\emph{cycles}. Their fixed union of supports is pure $e$-dimensional -and has finitely many irreducible components. Since all recorded -cycles have dimension $e$, there are only countably many possible tuples of -effective cycles supported there: their multiplicities are -nonnegative integers. A holomorphic map from a connected complex -manifold with countable image is constant. Removing a proper -analytic subset from the smooth connected $G$ leaves a connected -open set, so $\Phi_S$ is constant on a general $g$-fiber. Point -fibers require no separate argument. - -Constancy on the general connected fibers gives a meromorphic -factorization through $Q$. One can see this by taking the image of -the graph in $Q$ times the target: its projection to $Q$ has one-point -general fibers and is bimeromorphic. The distinguishing property -makes the induced map from $Q$ generically one-to-one onto its image. -Hence $\Phi_S|_F$ defines the same meromorphic fibration as $g$. - -Finally, the restriction of $f_i$ over $w$ still has connected fibers: -its fibers are exactly the corresponding fibers of $f_i$. Its base -over $w$ is connected, being the image of the connected source fiber. -The finite map in Stein factorization cannot split a connected -general $g$-fiber: a map from that fiber into a finite set is constant. -The preselected resolutions restrict to modifications over $w$. -Thus the same comparison holds for $f_i$, proving the assertion. -\end{proof} - -\subsection{Vanishing of the nef class on the fibers} - -We can now choose the flat twists on the total space. The order of -choice is essential: the following proof fixes a countable collection -of global representatives before it chooses the smooth fiber. - -\begin{proposition}\label{cycle:nef-summand} -Assume Theorem~\ref{thm:decomposition} in dimensions less than $n$. -Let $T$ be a smooth connected compact K\"ahler $n$-fold with -$00$. - -On a general smooth $g$-fiber $G$, the class of $P_F$ is zero. -Restricting the last equality in -\eqref{cycle:residual-effective} and using the same positivity -argument shows that -\begin{equation}\label{cycle:restriction-zero} - c_1(\rho^*(M|_F)|_G)=0, - \qquad - G\cap\operatorname{Supp}(\rho^*E_F-R_F)=\varnothing. -\end{equation} -If $G$ is a point, the first assertion is automatic and the second -holds for a general point. In positive dimension, choose a general -fiber not contained in the effective divisor, restrict it, and -integrate the effective and nef classes against a K\"ahler power. -Both nonnegative masses must vanish. - -Lemma~\ref{cycle:detection} gives an index $i$ such that the restriction -of $f_i$ over $w$ and $g$ have the same general fibers up to modifications. By -\eqref{cycle:restriction-zero} and injectivity of pullback on -cohomology, $p_i^*M$ has class zero on a smooth $f_i$-fiber above -$w$. Such a fiber can be chosen in the smooth-fibration locus of -$f_i$, by the generic conditions already imposed. Over the -connected smooth locus in $Z_i$, the restrictions of this Chern -class form a locally constant section of $R^2(f_i)_*\R$. Vanishing -at one fiber therefore implies vanishing at every fiber in this -locus. Thus $i\in I_0$; its representative $M_i^*$ was already -chosen before $w$. - -We claim that -\begin{equation}\label{cycle:positive-kappa} - \kappa(F,J_F+M_i^*|_F)>0. -\end{equation} -On a common resolution of the maps on $F$, the line $M_i^*|_F$ -is rationally trivial on a general $g$-fiber because it is a -pullback from the intermediate base of $f_i$. This triviality -descends between smooth models of the fiber: for a modification, -pullback on line bundles is injective by normality and the -projection formula. The difference -\[ - \Lambda=\rho^*(J_F+M_i^*|_F)-P_F-R_F -\] -is a flat rational line bundle by -\eqref{cycle:inductive-decomposition}. Its restriction to a -general $G$ is rationally trivial. Indeed $P_F|_G$ is trivial, -$M_i^*|_G$ is trivial, and -\eqref{cycle:restriction-zero} makes -$\cO_{\widehat F}(\rho^*E_F-R_F)|_G$ trivial, while -$J_F\qsim E_F$. - -Resolve $Q$ and the induced map, and continue to denote the -resulting fibration between smooth models by $g$. By Lemma~\ref{desc:flat}, -$\Lambda\qsim g^*\lambda$ for a flat rational line bundle -$\lambda$ on the smooth projective base. The line $P_F$ is the -pullback of the ample rational line on the original $Q$; on its -resolution the corresponding base line $H_Q'$ is nef and big. -Twisting by $\lambda$ preserves bigness. On these models we have -an actual rational identity -\[ - \rho^*(J_F+M_i^*|_F) - \qsim g^*(H_Q'+\lambda)+R_F. -\] -Multiplication by the canonical section of a divisible multiple -of $R_F$ embeds the section spaces of the big base line into the -section spaces on $\widehat F$. Since $\dim Q>0$, this proves -\eqref{cycle:positive-kappa}. - -This contradicts the instance of Lemma~\ref{cycle:sections} imposed -on $J+M_i^*$ when $w$ was chosen. Hence $M|_F\num0$ for the -chosen fiber. Finally, restrictions of $c_1(M)$ form a -locally constant section of $R^2h_*\R$ on the connected smooth -locus of $h$. Their vanishing therefore holds on every fiber in -that locus, and in particular on a general smooth fiber. -\end{proof} - -\begin{corollary}\label{cycle:base-descent} -Under the assumptions of Proposition~\ref{cycle:nef-summand}, -there are modifications $p:T'\to T$ and $b:W'\to W$, with -$T'$ smooth compact K\"ahler and $W'$ smooth projective, a -fibration $h':T'\to W'$ lifting $h$, and a nef rational line -bundle $N$ on $W'$ such that -\[ - p^*M\num(h')^*N. -\] -\end{corollary} - -\begin{proof} -Proposition~\ref{cycle:nef-summand} supplies the fiberwise -vanishing required in Proposition~\ref{desc:nef}. Apply that -proposition to $h$ and $M$. The modified base remains projective, -so the descended rational class is nef. -\end{proof} +\section{The nef summand on the algebraic-reduction fibers} +\label{sec:cycles} + +Let $T$ be a smooth compact K\"ahler manifold of dimension $n$, and suppose +that $00$, some divisible $m$ has two sections whose +ratio is nonconstant on the connected fiber $X_w$. Lifting these +sections as above gives a meromorphic function on $X$ whose restriction +to $X_w$ is that ratio. Every meromorphic function on $X$ factors +through its algebraic reduction. Such a function, when defined on a +nonempty open subset of the smooth fiber $X_w$, is constant there: +a local holomorphic section of $h$ through a point where the quotient +is holomorphic shows that the base function extends holomorphically +at $w$. This contradicts the chosen nonconstant ratio. + +The countable simultaneous choice is legitimate by Baire's theorem. +If a generic assertion is initially established on a dense open set, +a proper analytic exceptional subset of that open is still nowhere +dense in $W$. Thus such generic assertions can also be included in the +choice. +\end{proof} + +\subsection{Divisor incidence and a countable collection of fibrations} + +We use the cycle-space results of Barlet, Lieberman, and Fujiki +\cite{Barlet,LiebermanCycles,Fujiki1978,Fujiki}. In the form needed here, +the spaces of +effective cycles of a compact K\"ahler manifold have countably many +irreducible components; each component is compact and belongs to +Fujiki's class $\mathcal C$; see \cite[p.~189]{Fujiki}. The locus of +cycles contained in a fiber of a proper map is an analytic relative +cycle space \cite[Proposition~3.2]{Fujiki1978}. We also use the +meromorphic fiber map obtained by analytic flattening: after modifying +the base, the flat strict transform has an analytic family of fiber +cycles and hence a holomorphic map to the cycle space +\cite[Chapter~V]{Barlet}. All parameter spaces and images in class $\mathcal C$ +will be replaced by smooth compact K\"ahler models when needed. + +The following observation explains why the numerical class of a divisor +can detect a semiample fibration without fixing its flat twist. + +\begin{lemma}\label{cycle:divisors} +Let $F$ be a smooth connected compact K\"ahler manifold, let $A$ be a +rational line bundle on $F$, and suppose that a modification +$\rho:\widehat F\to F$ from a smooth compact K\"ahler manifold satisfies +\begin{equation}\label{cycle:fiber-decomposition} + \rho^*A\num P+R, + \qquad P\text{ semiample}, + \qquad R=N(c_1(\rho^*A))\text{ a rational divisor}. +\end{equation} +Let $g:\widehat F\to Q$ be the fibration defined by $P$, so that $Q$ is +normal projective and $P\qsim g^*H_Q$ for an ample rational line bundle +$H_Q$. There is a dense open subset $U\subset F$, independent of the +divisors below, with the following properties for every sufficiently +divisible positive integer $m$. +\begin{enumerate} +\item There exists an analytic family of effective +divisors on $F$ with class $m c_1(A)$ that distinguishes the general fibers +of the meromorphic fibration $g\circ\rho^{-1}$. +\item For every effective divisor $D$ with class +$m c_1(A)$, membership +in $\operatorname{Supp}D$ is constant along the intersection of $U$ +with a general smooth fiber of that meromorphic fibration. +\end{enumerate} +\end{lemma} + +\begin{proof} +The difference between the two sides of +\eqref{cycle:fiber-decomposition} is flat as a rational line bundle. +By Lemma~\ref{pre:picard}, there is a flat rational line +bundle $\eta$ on $F$ such that +\[ + \rho^*(A+\eta)\qsim P+R. +\] +Choose $m$ clearing the denominators and such that $mH_Q$ is very ample. +The divisors +\[ + g^*D_Q+mR, + \qquad D_Q\in|mH_Q|, +\] +descend to an analytic family of divisors on $F$ with class $m c_1(A)$. +Indeed $\rho_*\cO_{\widehat F}=\cO_F$, so the projection formula +identifies sections of $m(A+\eta)$ with sections of its pullback. +On the isomorphism locus of $\rho$, away from $\operatorname{Supp}R$, +they distinguish general $g$-fibers, since hyperplanes distinguish +points of $Q$. + +For an arbitrary effective divisor $D$ of this class, +Lemma~\ref{pre:negative}(1) gives +\[ + \rho^*D\geq mR. +\] +The residual divisor $D'=\rho^*D-mR$ is effective and has class +$m c_1(P)$. Let $G$ be a smooth compact connected fiber of $g$. If +$D'$ does not contain $G$, its restriction is an effective divisor of +class zero on $G$, hence is empty: in positive dimension this follows +by integrating against a K\"ahler form to the appropriate power. +Consequently $\operatorname{Supp}D'$ either contains $G$ or misses it. +For a point fiber the same alternative is immediate. Taking the image +of the isomorphism locus of $\rho$, and deleting +$\operatorname{Supp}R$ and the nonsmooth locus of $g$, gives the stated +open set $U$. This choice uses only $\rho$, $R$, and $g$, and works for +all the divisors simultaneously. +\end{proof} + +\begin{lemma}\label{cycle:detection} +Let $h:X\to W$ be a fibration of smooth connected compact K\"ahler +manifolds with $\dim W<\dim X$, and let $A$ be a rational line bundle +on $X$. There is a countable collection of diagrams +\begin{equation}\label{cycle:diagrams} + \begin{tikzpicture}[baseline=(current bounding box.center),>=Stealth] + \node (Xi) at (0,1.15) {$X_i$}; + \node (Zi) at (2.7,1.15) {$Z_i$}; + \node (X) at (0,0) {$X$}; + \node (W) at (2.7,0) {$W$}; + \draw[->] (Xi) -- node[above] {$f_i$} (Zi); + \draw[->] (Xi) -- node[left] {$p_i$} (X); + \draw[->] (Zi) -- node[right] {$k_i$} (W); + \draw[->] (X) -- node[below] {$h$} (W); + \end{tikzpicture} +\end{equation} +where $p_i$ is a modification, $f_i$ is a fibration, and $X_i,Z_i$ are +smooth compact K\"ahler manifolds, with the following property. + +For a very general smooth fiber $F=X_w$, suppose that a smooth compact +K\"ahler modification $\rho:\widehat F\to F$ has a decomposition +\eqref{cycle:fiber-decomposition} for $A|_F$, and that the fibration +$g:\widehat F\to Q$ defined by $P$ has positive-dimensional base. +For some $i$, the restriction of $f_i$ over $w$ is bimeromorphically +equivalent to $g$. In particular, on a common resolution, their +general fibers coincide. The collection and the exceptional subsets +of $W$ can be fixed before $F$, $\rho$, $P$, and $R$ are chosen. +\end{lemma} + +\begin{proof} +We associate a map to each component of the relative divisor space by +recording all its divisors through a point. On a fiber with the stated +decomposition, Lemma~\ref{cycle:divisors} makes these incidence supports +constant along the semiample fibration. Integral multiplicities will +then give constancy of the cycle maps themselves, while the +distinguishing linear system will give the converse. + +Put $d=\dim X-\dim W$, and let $W^\circ$ be the smooth-fibration locus +of $h$. Consider the relative cycle spaces whose points are pairs +$(w,D)$, where $D$ is an effective $(d-1)$-cycle in $X_w$. They are +closed subspaces of the product of $W$ with the corresponding compact +cycle components of $X$. Thus they have countably many compact +irreducible components in class $\mathcal C$. + +For each positive integer $m$ clearing the denominator of $A$, retain +the components meeting $W^\circ$ on which the divisor class is +$m c_1(A|_{X_w})$. This is a condition on an entire component over +$W^\circ$: in a smooth family the cohomology class of an analytic +family of cycles is locally constant, and $c_1(A)$ supplies a global +section of the same cohomology local system. After resolving a +component, its open part over $W^\circ$ is connected, so equality at +one point implies equality throughout that part. Components not +dominating $W$ have proper analytic images; exclude all these images +from the eventual choice of $w$. + +Fix a retained dominating component and a smooth compact K\"ahler +resolution $S$ of it. Denote its divisor at $s\in S$ by $D_s$ and form +the incidence +\[ + \mathcal I_S + =\{(x,s)\in X\times S:x\in\operatorname{Supp}D_s\}. +\] +We take the incidence of the analytic cycle family, including its +limits from the open part over $W^\circ$. Over that open part it is a +family of pure $(d-1)$-dimensional cycles. Hence every irreducible +component there dominates $S$ and has dimension $\dim S+d-1$. +Let $I_1,\ldots,I_t$ be the incidence components that dominate $X$. +The other components have proper analytic images in $X$; outside +those images they contribute nothing to incidence. + +For each $I_j\to X$, analytic flattening gives a meromorphic map +recording its generic fiber as an effective cycle in $S$. All these +cycles have dimension +\[ + e=\dim S+d-1-\dim X. +\] +Taking the tuple of these maps, together with $h$, gives a meromorphic +map $\Phi_S$ from $X$ into $W$ times a finite product of compact cycle +components of $S$. On a dense open subset of $X$, the union of the +supports of the recorded cycles is precisely +\begin{equation}\label{cycle:incidence-support} + \{s\in S:x\in\operatorname{Supp}D_s\}. +\end{equation} +In obtaining this open set we discard the images of the nondominating +incidence components and the exceptional sets of the fiber maps. +The target components and the image of $\Phi_S$ belong to class +$\mathcal C$. Resolving the graph, taking Stein factorization, and +replacing the base by a smooth compact K\"ahler model therefore gives +a diagram \eqref{cycle:diagrams}. The coordinate $h$ ensures that its +base maps to $W$. If there is no dominating incidence component, we +retain only the coordinate $h$; such a component will not be needed +to detect a positive-dimensional $Q$. + +This construction gives countably many diagrams. Fix their models, +their incidence-recording open sets, and their Stein factorizations. +We may simultaneously require that their fibers over the eventual +$w$ be smooth, that $p_i$ restrict to a modification over $w$, and that +each of the dense open sets used meet the corresponding fiber densely. +For any one diagram these are generic conditions, by properness, +generic smoothness, and the fiber-dimension theorem. In particular, +let $U_i\subset Z_i$ be the locus over which $f_i$ is smooth. We +require that $k_i^{-1}(w)$ meet $U_i$ densely. To justify this +condition, a component of $Z_i\setminus U_i$ that does not dominate +$W$ has proper image, whereas a component dominating $W$ has general +fiber of dimension strictly less than $\dim Z_i-\dim W$. +Since the general $k_i$-fiber is smooth of that pure dimension, it +cannot have a component contained in $Z_i\setminus U_i$. The same +conditions may be imposed after any preselected countable collection +of further modifications of these diagrams. + +We now choose such a very general $w$ and a decomposition on $F=X_w$ +as in the statement. By Lemma~\ref{cycle:divisors}, a projective +linear system of divisors of class $m c_1(A|_F)$ distinguishes the +general $g$-fibers for some $m$. Its image in the relative cycle +space is irreducible and lies in one irreducible component. That +component dominates $W$, since all images of nondominating components +were excluded. It is therefore one of the components used above. +Passing to its parameter resolution $S$ preserves the distinguishing +property: every divisor still has a preimage in $S$. + +The map $\Phi_S|_F$ distinguishes general $g$-fibers. Indeed two +different general points of $Q$ are separated by a divisor in the +chosen linear system, so their incidence supports +\eqref{cycle:incidence-support} differ. Conversely, fix a general +smooth connected $g$-fiber $G$. On the common open set of +Lemma~\ref{cycle:divisors} and the incidence recording, membership in +every divisor $D_s$ is constant along $G$. Thus the support in +\eqref{cycle:incidence-support} is fixed as the point moves in that +open part of $G$. + +This support assertion also gives constancy of the recorded +\emph{cycles}. Their fixed union of supports is pure $e$-dimensional +and has finitely many irreducible components. Since all recorded +cycles have dimension $e$, there are only countably many possible tuples of +effective cycles supported there: their multiplicities are +nonnegative integers. A holomorphic map from a connected complex +manifold with countable image is constant. Removing a proper +analytic subset from the smooth connected $G$ leaves a connected +open set, so $\Phi_S$ is constant on a general $g$-fiber. Point +fibers require no separate argument. + +Constancy on the general connected fibers gives a meromorphic +factorization through $Q$. One can see this by taking the image of +the graph in $Q$ times the target: its projection to $Q$ has one-point +general fibers and is bimeromorphic. The distinguishing property +makes the induced map from $Q$ generically one-to-one onto its image. +Hence $\Phi_S|_F$ defines the same meromorphic fibration as $g$. + +Finally, the restriction of $f_i$ over $w$ still has connected fibers: +its fibers are exactly the corresponding fibers of $f_i$. Its base +over $w$ is connected, being the image of the connected source fiber. +The finite map in Stein factorization cannot split a connected +general $g$-fiber: a map from that fiber into a finite set is constant. +The preselected resolutions restrict to modifications over $w$. +Thus the same comparison holds for $f_i$, proving the assertion. +\end{proof} + +\subsection{Vanishing of the nef class on the fibers} + +We can now choose the flat twists on the total space. The order of +choice is essential: the following proof fixes a countable collection +of global representatives before it chooses the smooth fiber. + +\begin{proposition}\label{cycle:nef-summand} +Assume Theorem~\ref{thm:decomposition} in dimensions less than $n$. +Let $T$ be a smooth connected compact K\"ahler $n$-fold with +$00$. + +On a general smooth $g$-fiber $G$, the class of $P_F$ is zero. +Restricting the last equality in +\eqref{cycle:residual-effective} and using the same positivity +argument shows that +\begin{equation}\label{cycle:restriction-zero} + c_1(\rho^*(M|_F)|_G)=0, + \qquad + G\cap\operatorname{Supp}(\rho^*E_F-R_F)=\varnothing. +\end{equation} +If $G$ is a point, the first assertion is automatic and the second +holds for a general point. In positive dimension, choose a general +fiber not contained in the effective divisor, restrict it, and +integrate the effective and nef classes against a K\"ahler power. +Both nonnegative masses must vanish. + +Lemma~\ref{cycle:detection} gives an index $i$ such that the restriction +of $f_i$ over $w$ and $g$ have the same general fibers up to modifications. By +\eqref{cycle:restriction-zero} and injectivity of pullback on +cohomology, $p_i^*M$ has class zero on a smooth $f_i$-fiber above +$w$. Such a fiber can be chosen in the smooth-fibration locus of +$f_i$, by the generic conditions already imposed. Over the +connected smooth locus in $Z_i$, the restrictions of this Chern +class form a locally constant section of $R^2(f_i)_*\R$. Vanishing +at one fiber therefore implies vanishing at every fiber in this +locus. Thus $i\in I_0$; its representative $M_i^*$ was already +chosen before $w$. + +We claim that +\begin{equation}\label{cycle:positive-kappa} + \kappa(F,J_F+M_i^*|_F)>0. +\end{equation} +On a common resolution of the maps on $F$, the line $M_i^*|_F$ +is rationally trivial on a general $g$-fiber because it is a +pullback from the intermediate base of $f_i$. This triviality +descends between smooth models of the fiber: for a modification, +pullback on line bundles is injective by normality and the +projection formula. The difference +\[ + \Lambda=\rho^*(J_F+M_i^*|_F)-P_F-R_F +\] +is a flat rational line bundle by +\eqref{cycle:inductive-decomposition}. Its restriction to a +general $G$ is rationally trivial. Indeed $P_F|_G$ is trivial, +$M_i^*|_G$ is trivial, and +\eqref{cycle:restriction-zero} makes +$\cO_{\widehat F}(\rho^*E_F-R_F)|_G$ trivial, while +$J_F\qsim E_F$. + +Resolve $Q$ and the induced map, and continue to denote the +resulting fibration between smooth models by $g$. By Lemma~\ref{desc:flat}, +$\Lambda\qsim g^*\lambda$ for a flat rational line bundle +$\lambda$ on the smooth projective base. The line $P_F$ is the +pullback of the ample rational line on the original $Q$; on its +resolution the corresponding base line $H_Q'$ is nef and big. +Twisting by $\lambda$ preserves bigness. On these models we have +an actual rational identity +\[ + \rho^*(J_F+M_i^*|_F) + \qsim g^*(H_Q'+\lambda)+R_F. +\] +Multiplication by the canonical section of a divisible multiple +of $R_F$ embeds the section spaces of the big base line into the +section spaces on $\widehat F$. Since $\dim Q>0$, this proves +\eqref{cycle:positive-kappa}. + +This contradicts the instance of Lemma~\ref{cycle:sections} imposed +on $J+M_i^*$ when $w$ was chosen. Hence $M|_F\num0$ for the +chosen fiber. Finally, restrictions of $c_1(M)$ form a +locally constant section of $R^2h_*\R$ on the connected smooth +locus of $h$. Their vanishing therefore holds on every fiber in +that locus, and in particular on a general smooth fiber. +\end{proof} + +\begin{corollary}\label{cycle:base-descent} +Under the assumptions of Proposition~\ref{cycle:nef-summand}, +there are modifications $p:T'\to T$ and $b:W'\to W$, with +$T'$ smooth compact K\"ahler and $W'$ smooth projective, a +fibration $h':T'\to W'$ lifting $h$, and a nef rational line +bundle $N$ on $W'$ such that +\[ + p^*M\num(h')^*N. +\] +\end{corollary} + +\begin{proof} +Proposition~\ref{cycle:nef-summand} supplies the fiberwise +vanishing required in Proposition~\ref{desc:nef}. Apply that +proposition to $h$ and $M$. The modified base remains projective, +so the descended rational class is nef. +\end{proof} diff --git a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/07-completion.tex b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/07-completion.tex index 8956d27da5d058b6fb3a4b4acb64ebcec60e0703..9a66881313ff7d2c4f80aae9a437f55f9ced3a0b 100644 --- a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/07-completion.tex +++ b/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/sections/07-completion.tex @@ -1,305 +1,305 @@ -\section{Completion of the induction}\label{sec:completion} - -We now combine the preceding constructions. The main geometric step is -to make the ordinary adjoint semiample after adding a sufficiently -positive line from the algebraic base, while preserving the map to that -base. The adjoint formula of Proposition~\ref{adj:base} then puts the -problem on a projective variety. We first isolate the negative-part -calculation needed to pull the projective answer back. - -\subsection{Divisors above subsets of codimension at least two} - -\begin{lemma}\label{finish:exceptional} -Let \(r:V\to S\) be a surjective morphism from a smooth compact K\"ahler -manifold to a normal projective variety. Let \(H\) be a nef rational -Cartier divisor on \(S\). If \(E\geq0\) is a real divisor on \(V\) whose -every component has image of codimension at least two in \(S\), then -\[ - N(r^*H+E)=E. -\] -\end{lemma} -\begin{proof} -Put \(n=\dim V\), \(\beta=c_1(r^*H)\), and -\(\alpha=\beta+\{E\}\). Nefness gives \(N(\alpha)\leq E\). -Suppose -\[ - U=E-N(\alpha)>0. -\] -Then \(\beta+\{U\}=Z(\alpha)\) is modified nef. Write -\(U=\sum_i u_iU_i\), with \(u_i>0\), and choose -\[ - s=\max_i\dim r(U_i)\leq\dim S-2. -\] -In particular \(0\leq s\leq n-2\). Let \(\eta\) be the pullback of an -ample class on \(S\), and let \(\omega\) be a K\"ahler class on \(V\). -Consider the symmetric bilinear form -\[ - Q(\gamma,\delta)=\int_V\gamma\,\delta\,\eta^s\omega^{n-s-2} - \quad\text{on }H^{1,1}(V,\R). -\] -Since the image of each \(U_i\) has dimension at most \(s\), any pairing -on \(U_i\) with \(s+1\) classes from \(S\) vanishes. Therefore -\begin{equation}\label{finish:orthogonal} - Q(\{U\},\eta)=0,\qquad Q(\{U\},\beta)=0. -\end{equation} -By Lemma~\ref{pre:negative}(3), \(Z(\alpha)\) restricts -pseudo-effectively to a resolution of \(U_i\). Pairing there with the -nef classes \(\eta^s\omega^{n-s-2}\) and summing over \(i\) gives -\[ - 0\leq Q(\{U\},Z(\alpha))=Q(\{U\},\{U\}). -\] -On the other hand, -\[ - Q(\eta,\eta)=\int_V\eta^{s+2}\omega^{n-s-2}>0, -\] -because \(s+2\leq\dim S\). The mixed Hodge--Riemann relations -\cite[Theorem~A]{DinhNguyen}, applied with -\(\eta+\varepsilon\omega\) and then passed to the limit, show that \(Q\) -has at most one positive eigenvalue. Thus its restriction to -\(\eta^\perp\) is negative semidefinite. Equations -\eqref{finish:orthogonal} force \(Q(\{U\},\{U\})=0\). -A zero-square vector for a negative semidefinite form belongs to its -radical; decomposing every class into a multiple of \(\eta\) and an -element of \(\eta^\perp\) shows that \(\{U\}\) belongs to the radical of -the whole form \(Q\). - -This contradicts -\[ - Q(\{U\},\omega)= - \sum_i u_i\int_{U_i}\eta^s\omega^{n-s-1}>0. -\] -Indeed every term is nonnegative, and a component with image dimension -\(s\) gives a strictly positive integral on its smooth generic locus. -Consequently \(U=0\). -\end{proof} - -This proof is the codimension-two part of the lifting argument in -\cite[Section~5]{K}. Its formulation above also applies to maps with -positive-dimensional fibers; birational exceptional translation alone -would not suffice at that point of our proof. - -\subsection{Keeping the ordinary program over the algebraic base} - -\begin{proposition}\label{finish:over-base} -Let \(T\) be a smooth compact K\"ahler manifold of dimension \(n\), let -\((T,B)\) be a rational simple normal crossing klt pair with -\(J=K_T+B\) pseudo-effective, and let -\(h:T\to W\) be a surjective morphism to a smooth projective variety. -There are a smooth compact K\"ahler modification \(T_1\to T\), an -effective klt adjoint \(J_1\) on \(T_1\) as in -\eqref{pre:resolution-error}, and a finite ordinary \(J_1\)-negative -program over \(W\), ending at a normal compact K\"ahler klt pair -\((Y,\Delta)\), such that for some positive integer \(b\) and an ample -Cartier divisor \(A_W\) on \(W\), -\[ - K_Y+\Delta+b h_Y^*A_W -\] -is semiample. Here \(h_Y:Y\to W\) is the descended morphism. -\end{proposition} -\begin{proof} -Choose \(A_W\) very ample and an integer \(b>2n\), and set -\(C=h^*A_W\). A general member of a sufficiently high multiple of -the free system \(|C|\), divided by that multiple and multiplied by -\(b\), gives an effective rational representative of \(bC\). -Bertini's theorem, with the multiple chosen large enough, makes its -sum with \(B\) an effective simple normal crossing klt boundary. -Theorem~\ref{pre:ordinary} therefore gives a decomposition of \(J+bC\). - -Pass to a higher smooth model \(T_1\) and apply the convention -\eqref{pre:resolution-error} to the original pair \((T,B)\). Write -\(J_1=K_{T_1}+B_1\) and \(C_1=h_1^*A_W\). The exceptional error and -Lemma~\ref{pre:negative} retain an actual identity -\begin{equation}\label{finish:augmented-decomposition} - J_1+bC_1\qsim P_1+R_1,\qquad - P_1\text{ semiample},\qquad R_1=N(J_1+bC_1)\geq0, -\end{equation} -with rational terms. On this fixed model choose a fresh general -fractional representative of \(bC_1\), so that its sum with \(B_1\) -is again simple normal crossing and klt. -Apply Proposition~\ref{pre:contract} to this augmented ordinary -adjoint and its known negative part in -\eqref{finish:augmented-decomposition}. Its finite ordinary scaling -contracts or flips detected analytic extremal rays negative for the -augmented adjoint. It preserves the actual semiample positive line -\(P_1\), with a fixed generated Cartier multiple, and contracts the -negative part \(R_1\). - -The positive line \(P_1\) and the fixed algebraic-base line -\(C_1=h_1^*A_W\) have distinct roles; they need not be equal or -proportional. Preservation of \(C_1\) and of the specific morphism -to \(W\) follows from a separate ray estimate. We check inductively -that each step is over \(W\). Suppose the current -model \(T_i\) still has a morphism \(h_i:T_i\to W\) and the unaugmented -pair is klt. Put -\[ - J_i=K_{T_i}+B_i,\qquad C_i=h_i^*A_W. -\] -An extremal ray negative for \(J_i+bC_i\) is \(J_i\)-negative because -\(C_i\) is nef. The K\"ahler klt cone theorem -\cite[Theorem~1.3]{HX}, with zero nef b-part, supplies a rational curve -\(\Gamma\) generating this ray and satisfying -\[ - 0<-J_i\cdot\Gamma\leq2n. -\] -If \(C_i\cdot\Gamma>0\), Cartier integrality gives -\[ - (J_i+bC_i)\cdot\Gamma\geq-2n+b>0, -\] -a contradiction. Hence \(C_i\) is numerically trivial on the contracted -ray. - -The contracted fibers are projective, and all their curves have class -in the contracted ray. If their image under \(h_i\) had positive -dimension, a curve on a fiber would have positive degree against -\(h_i^*A_W\); thus \(h_i\) is constant on each contracted fiber. -Connected-fiber descent to the normal contraction base factors \(h_i\) -through it. For a flip, composing the morphism from the flipped space -with this base map gives the next \(h_i\). The added line is consequently -the same Cartier pullback \(C_{i+1}=h_{i+1}^*A_W\) on the next model. -Each step is also \(J_i\)-negative, and the unaugmented ordinary klt -pair persists. This proves the induction through the finite program. -On a common resolution the two pullbacks of the line from \(W\) -agree. Subtracting \(b\) times this pullback from the augmented -comparison therefore gives an effective exceptional comparison for -the unaugmented adjoints as well. -Writing \(P_Y\) for the descended positive line, the final actual -rational-line identity is -\[ - K_Y+\Delta+b h_Y^*A_W\qsim P_Y. -\] -The line \(P_Y\) is semiample by Proposition~\ref{pre:contract}, which -proves the required semiampleness of the augmented adjoint. -\end{proof} - -The degree argument is the familiar way of preserving a nef Cartier -line in an adjoint program; compare \cite[Lemma~3.6]{K} and -\cite[Section~8.2]{P}. Here preservation of the line also preserves the -specific morphism to \(W\). - -\subsection{The strengthened decomposition} - -\begin{proof}[Proof of Theorem~\ref{thm:decomposition}] -We induct simultaneously on \(n=\dim T\) for both assertions of the -theorem. Dimension zero is immediate. Suppose both assertions hold in -every dimension less than \(n\). The projective case of the decomposition -is Proposition~\ref{pre:projective}. If \(a(T)=0\), -Proposition~\ref{zero:vanishing} gives \(M\num0\); ordinary decomposition -then proves the required statement for \(J+M\). It remains to treat -\[ - 00. +\] +Then \(\beta+\{U\}=Z(\alpha)\) is modified nef. Write +\(U=\sum_i u_iU_i\), with \(u_i>0\), and choose +\[ + s=\max_i\dim r(U_i)\leq\dim S-2. +\] +In particular \(0\leq s\leq n-2\). Let \(\eta\) be the pullback of an +ample class on \(S\), and let \(\omega\) be a K\"ahler class on \(V\). +Consider the symmetric bilinear form +\[ + Q(\gamma,\delta)=\int_V\gamma\,\delta\,\eta^s\omega^{n-s-2} + \quad\text{on }H^{1,1}(V,\R). +\] +Since the image of each \(U_i\) has dimension at most \(s\), any pairing +on \(U_i\) with \(s+1\) classes from \(S\) vanishes. Therefore +\begin{equation}\label{finish:orthogonal} + Q(\{U\},\eta)=0,\qquad Q(\{U\},\beta)=0. +\end{equation} +By Lemma~\ref{pre:negative}(3), \(Z(\alpha)\) restricts +pseudo-effectively to a resolution of \(U_i\). Pairing there with the +nef classes \(\eta^s\omega^{n-s-2}\) and summing over \(i\) gives +\[ + 0\leq Q(\{U\},Z(\alpha))=Q(\{U\},\{U\}). +\] +On the other hand, +\[ + Q(\eta,\eta)=\int_V\eta^{s+2}\omega^{n-s-2}>0, +\] +because \(s+2\leq\dim S\). The mixed Hodge--Riemann relations +\cite[Theorem~A]{DinhNguyen}, applied with +\(\eta+\varepsilon\omega\) and then passed to the limit, show that \(Q\) +has at most one positive eigenvalue. Thus its restriction to +\(\eta^\perp\) is negative semidefinite. Equations +\eqref{finish:orthogonal} force \(Q(\{U\},\{U\})=0\). +A zero-square vector for a negative semidefinite form belongs to its +radical; decomposing every class into a multiple of \(\eta\) and an +element of \(\eta^\perp\) shows that \(\{U\}\) belongs to the radical of +the whole form \(Q\). + +This contradicts +\[ + Q(\{U\},\omega)= + \sum_i u_i\int_{U_i}\eta^s\omega^{n-s-1}>0. +\] +Indeed every term is nonnegative, and a component with image dimension +\(s\) gives a strictly positive integral on its smooth generic locus. +Consequently \(U=0\). +\end{proof} + +This proof is the codimension-two part of the lifting argument in +\cite[Section~5]{K}. Its formulation above also applies to maps with +positive-dimensional fibers; birational exceptional translation alone +would not suffice at that point of our proof. + +\subsection{Keeping the ordinary program over the algebraic base} + +\begin{proposition}\label{finish:over-base} +Let \(T\) be a smooth compact K\"ahler manifold of dimension \(n\), let +\((T,B)\) be a rational simple normal crossing klt pair with +\(J=K_T+B\) pseudo-effective, and let +\(h:T\to W\) be a surjective morphism to a smooth projective variety. +There are a smooth compact K\"ahler modification \(T_1\to T\), an +effective klt adjoint \(J_1\) on \(T_1\) as in +\eqref{pre:resolution-error}, and a finite ordinary \(J_1\)-negative +program over \(W\), ending at a normal compact K\"ahler klt pair +\((Y,\Delta)\), such that for some positive integer \(b\) and an ample +Cartier divisor \(A_W\) on \(W\), +\[ + K_Y+\Delta+b h_Y^*A_W +\] +is semiample. Here \(h_Y:Y\to W\) is the descended morphism. +\end{proposition} +\begin{proof} +Choose \(A_W\) very ample and an integer \(b>2n\), and set +\(C=h^*A_W\). A general member of a sufficiently high multiple of +the free system \(|C|\), divided by that multiple and multiplied by +\(b\), gives an effective rational representative of \(bC\). +Bertini's theorem, with the multiple chosen large enough, makes its +sum with \(B\) an effective simple normal crossing klt boundary. +Theorem~\ref{pre:ordinary} therefore gives a decomposition of \(J+bC\). + +Pass to a higher smooth model \(T_1\) and apply the convention +\eqref{pre:resolution-error} to the original pair \((T,B)\). Write +\(J_1=K_{T_1}+B_1\) and \(C_1=h_1^*A_W\). The exceptional error and +Lemma~\ref{pre:negative} retain an actual identity +\begin{equation}\label{finish:augmented-decomposition} + J_1+bC_1\qsim P_1+R_1,\qquad + P_1\text{ semiample},\qquad R_1=N(J_1+bC_1)\geq0, +\end{equation} +with rational terms. On this fixed model choose a fresh general +fractional representative of \(bC_1\), so that its sum with \(B_1\) +is again simple normal crossing and klt. +Apply Proposition~\ref{pre:contract} to this augmented ordinary +adjoint and its known negative part in +\eqref{finish:augmented-decomposition}. Its finite ordinary scaling +contracts or flips detected analytic extremal rays negative for the +augmented adjoint. It preserves the actual semiample positive line +\(P_1\), with a fixed generated Cartier multiple, and contracts the +negative part \(R_1\). + +The positive line \(P_1\) and the fixed algebraic-base line +\(C_1=h_1^*A_W\) have distinct roles; they need not be equal or +proportional. Preservation of \(C_1\) and of the specific morphism +to \(W\) follows from a separate ray estimate. We check inductively +that each step is over \(W\). Suppose the current +model \(T_i\) still has a morphism \(h_i:T_i\to W\) and the unaugmented +pair is klt. Put +\[ + J_i=K_{T_i}+B_i,\qquad C_i=h_i^*A_W. +\] +An extremal ray negative for \(J_i+bC_i\) is \(J_i\)-negative because +\(C_i\) is nef. The K\"ahler klt cone theorem +\cite[Theorem~1.3]{HX}, with zero nef b-part, supplies a rational curve +\(\Gamma\) generating this ray and satisfying +\[ + 0<-J_i\cdot\Gamma\leq2n. +\] +If \(C_i\cdot\Gamma>0\), Cartier integrality gives +\[ + (J_i+bC_i)\cdot\Gamma\geq-2n+b>0, +\] +a contradiction. Hence \(C_i\) is numerically trivial on the contracted +ray. + +The contracted fibers are projective, and all their curves have class +in the contracted ray. If their image under \(h_i\) had positive +dimension, a curve on a fiber would have positive degree against +\(h_i^*A_W\); thus \(h_i\) is constant on each contracted fiber. +Connected-fiber descent to the normal contraction base factors \(h_i\) +through it. For a flip, composing the morphism from the flipped space +with this base map gives the next \(h_i\). The added line is consequently +the same Cartier pullback \(C_{i+1}=h_{i+1}^*A_W\) on the next model. +Each step is also \(J_i\)-negative, and the unaugmented ordinary klt +pair persists. This proves the induction through the finite program. +On a common resolution the two pullbacks of the line from \(W\) +agree. Subtracting \(b\) times this pullback from the augmented +comparison therefore gives an effective exceptional comparison for +the unaugmented adjoints as well. +Writing \(P_Y\) for the descended positive line, the final actual +rational-line identity is +\[ + K_Y+\Delta+b h_Y^*A_W\qsim P_Y. +\] +The line \(P_Y\) is semiample by Proposition~\ref{pre:contract}, which +proves the required semiampleness of the augmented adjoint. +\end{proof} + +The degree argument is the familiar way of preserving a nef Cartier +line in an adjoint program; compare \cite[Lemma~3.6]{K} and +\cite[Section~8.2]{P}. Here preservation of the line also preserves the +specific morphism to \(W\). + +\subsection{The strengthened decomposition} + +\begin{proof}[Proof of Theorem~\ref{thm:decomposition}] +We induct simultaneously on \(n=\dim T\) for both assertions of the +theorem. Dimension zero is immediate. Suppose both assertions hold in +every dimension less than \(n\). The projective case of the decomposition +is Proposition~\ref{pre:projective}. If \(a(T)=0\), +Proposition~\ref{zero:vanishing} gives \(M\num0\); ordinary decomposition +then proves the required statement for \(J+M\). It remains to treat +\[ + 0