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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the review in Zbl 0727.14022. finite Galois coverings; closed Riemann surfaces; complex manifolds , Finite branched coverings of complex manifolds, Sugaku Expositions 5 (1992), no. 2, 193-211. Riemann surfaces; Weierstrass points; gap sequences, Compact Riemann surfaces and uniformization, Coverings in algebraic geometry, Coverings of curves, fundamental group Finite branched coverings of complex manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The simplest analytic coverings of \({\mathbb{C}}^ 2\), ramified over \(u^ 3=v^ 2\), are the regular (Galois) coverings with polyhedral automorphism group. The natural correspondence between these coverings and these groups happens to be (essentially) one-to-one, a fact which does not hold in general.
It is thereby possible to find the analytic expressions of these coverings by means of the polynomial invariants of the polyhedral groups. The associated irregular coverings are the quotients of these expressions under the actions of the non-normal subgroups of the polyhedral groups. By computing actions of the fundamental group \(\pi _ 1({\mathbb{C}}^ 2\setminus \{u^ 3=v^ 2\})\) on the fibre, it is shown that all the non-cyclic coverings of degree less or equal to 5 are of this type.
As a corollary, the singularities of the (non-cyclic) covering spaces are cyclic quotients of the A-D-E surface singularities. elliptic genus; formal group laws associated to supersingular elliptic curves G. Teodosiu, A class of analytic coverings ramified over \(u^3=v^2\), J. London Math. Soc. (2) 38 (1988), 231-242. Zbl0619.14009 MR966295 Coverings in algebraic geometry, Ramification problems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Automorphisms of surfaces and higher-dimensional varieties, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Formal groups, \(p\)-divisible groups, Spherical harmonics, Special algebraic curves and curves of low genus, Elliptic curves A class of analytic coverings ramified over \(u^ 3=v^ 2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a number field. For any algebraic curve \(X\) of genus \(g\), \(n\)-punctured, the profinite or the pro-\(l\) completion of the topological fundamental group of \(X\) admits two actions: the action of the profinite completion of the mapping class group of the orientable surface of topological type \((g,n)\) and secondly the action of the absolute Galois group of \(K\). The aim of the paper under review is to compare these two actions, in particular to consider the intersection of the images of these actions. For any affine curve with non-abelian fundamental group the considered intersection is trivial, whereas in the pro-\(l\) case there exist many curves such that the image of the Galois group action contains the other image. Galois action; fundamental group; mapping class group Makoto Matsumoto & Akio Tamagawa, ''Mapping class group action versus Galois action on profinite fundamental groups'', Am. J. Math.122 (2000) no. 5, p. 1017-1026 Separable extensions, Galois theory, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Coverings of curves, fundamental group, Coverings in algebraic geometry Mapping-class-group action versus Galois action on profinite fundamental groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this work, we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the topology of cyclic coverings of such surfaces ramified along \(\mathbb Q\)-normal crossing divisors. Rational and ruled surfaces, Coverings in algebraic geometry Coverings of rational ruled normal surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Coverings in algebraic geometry, Families, moduli, classification: algebraic theory, Schemes and morphisms Propriétés de finitude du groupe fondamental | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies a particular class of algebraic varieties. They are two-sheeted covers of a projective space \(\mathbb P^m \) (\(m>2\)) over an algebraically closed field, with some restrictions on the set over which these covers are branched. The main theorem is that these varieties are birationally super-rigid. As corollaries the author obtains many information on the birational properties of this varieties. For instance, he finds the group of their birational automorphisms (in generic case) and that they are not rational. birational automorphism; rationality; covers; birationally super-rigid varieties Cheltsov, I.A., Shramov, K.A.: Log canonical thresholds of smooth Fano threefolds (with an appendix by Jean-Pierre Demailly), Uspekhi Mat. Nauk. \textbf{63}(5), 73-180 (2008); translation in Russian Math. Surveys \textbf{63}(5), 859-958 (2008) Birational automorphisms, Cremona group and generalizations, Rationality questions in algebraic geometry, Coverings in algebraic geometry Birational automorphisms of double spaces with singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This talk is concerned with the proof of Abhyankar's conjecture after \textit{M. Raynaud} [Invent. Math. 116, No. 1-3, 425-462 (1994; Zbl 0798.14013)]. Abhyankar's conjecture; covering of affine line; Sylow \(p\)-group; Galois group of covering Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings in algebraic geometry Raynaud's analysis of Abhyankar's conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a Galois cover \(\phi\colon X\to\mathbb{P}^2\) and a curve \(C\subset\mathbb{P}^2\) without common components with the branch locus \(B\) of \(\phi\), the author introduces the \textit{splitting graph} \(S_{\phi,C}\) of \(C\) for \(\phi\): its vertices are the singular points (outside \(\phi^*B\)) and irreducible components of the pull-back \(\phi^*C\), its edges are the local branches at the singular points, and the graph is equipped with the natural action of the deck translation group of \(\phi\). Obviously, \(S_{\phi,C}\) is a topological invariant of the pair \((\phi,C)\). The author applies this concept to the topological classification of the so-called \textit{Artal arrangements}, viz. curves of the form \(B+C\), where \(B\) is a smooth curve of a given degree \(d\ge3\) and \(C\) is the union of three non-concurrent lines with prescribed intersection multiplicities with \(B\) (and \(\phi\) is the standard cyclic \(d\)-fold covering ramified at \(B\)). It is asserted that two such arrangements are homeomorphic if and only if their splitting graphs are isomorphic. The (deformation) equivalence is established by a certain Menelaus type argument, taking for \(C\) the union of the coordinate lines and analyzing the equation for \(B\). embedded topology; Zariski pair; Galois cover of graphs; splitting graph; Artal arrangement Coverings in algebraic geometry, Plane and space curves Galois covers of graphs and embedded topology of plane curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0742.00065.]
Let \(X\), \(Y\) be algebraic varieties such that \(Y\) is normal and \(X\) has rational Gorenstein singularities. Denote by \(\omega_ X\) the canonical sheaf of \(X\). A projective morphism \(f:X\to Y\) is called a Fano map if \(\omega_ X^{-1}\) is relatively ample with respect to \(f\). Consider the induced map \(f_ *:\pi_ 1(\hat X)\to\pi_ 1(\hat Y)\) between the algebraic fundamental groups of \(X\) and \(Y\). The author proves that if \(f\) is a Fano map which is flat over an open set of \(Y\) containing all the singular points of \(Y\), then \(f_ *\) is an isomorphism. Moreover he shows that if the natural map \(\pi_ 1(X)\to\pi_ 1(\hat X)\) is injective, then \(\pi_ 1(X)\) and \(\pi_ 1(Y)\) are also isomorphic. rational Gorenstein singularities; Fano map; fundamental groups Birational geometry, Fano varieties, Coverings in algebraic geometry Fano maps and fundamental groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The group G of automorphisms of the configuration of 27 lines on a cubic surface has order 51,840. For this group the author finds the Loewy structures of the projective indecomposable modules over a splitting field of characteristic 3. The proof involves a variety of techniques from the representation theory of both algebraic and finite groups (exploring the fact that \(G\cong SO_ 5(3))\). configuration; cubic surface; Loewy structures; projective indecomposable modules Benson D.J.: Projective modules for the group of 27 lines on a cubic surface. Comm. Algebra 17(5), 1017--1068 (1989) Modular representations and characters, Representation theory for linear algebraic groups, Enumerative problems (combinatorial problems) in algebraic geometry, Other finite incidence structures (geometric aspects), Finite automorphism groups of algebraic, geometric, or combinatorial structures, Special surfaces Projective modules for the group of twenty-seven lines on a cubic surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we study the geometry of two-torsion points of elliptic curves in order to distinguish the embedded topology of reducible plane curves consisting of a smooth cubic and its tangent lines. As a result, we obtain a new family of Zariski tuples consisting of such curves. elliptic curves; torsion points; Zariski pairs; splitting numbers Elliptic curves, Coverings in algebraic geometry Zariski tuples for a smooth cubic and its tangent lines | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give an algorithm which computes \(r\), defined by \textit{K. Kato} in the paper [Am. J. Math. 116, No. 4, 757--784 (1994; Zbl 0864.11057)], which is an important invariant for Artin-Schreier extensions of surfaces \(X\) over fields of positive characteristic. The Swan conductor gives the invariant of ramifications concerning codimension 1 subvarieties of \(X\). This \(r\) gives the invariant of ramifications concerning codimension 2 subvarieties of \(X\). The invariant \(r\) is important to calculate the Euler Poincaré characteristic of some smooth \(l\)-adic sheaf of rank 1 on an open dense subscheme \(U\) of \(X\). Swan conductor; ramification; surface; Artin-Schreier extension Positive characteristic ground fields in algebraic geometry, Ramification and extension theory, Coverings in algebraic geometry, Ramification problems in algebraic geometry On ramifications of Artin-Schreier extensions of surfaces over algebraically closed fields of positive characteristic. I. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems After delving into its strong relationship with qcqs schemes, some applications like singularities and Prüfer spaces, coupled with a finding being an intuitive approach to Grothendieck's Algebraic Geometry, and grouping the fact that strictly topological finite models play the role of the Čech nerve of a cover, the authors established an algebraic construction of schematic spaces via Galois Category analogous to the world-renowned Grothendieck's étale fundamental group. The main achievement is the following theorem:
Theorem.
\begin{itemize}
\item[1.] If \(X\) is a schematic space and \(\mathscr{O}_X(X)\) has connected spectrum, for any geometric point \(\overline{x}\) of \(X\), the pair \((\mathbf{Qcoh}^{\mathrm{fet}}(X)^{\mathrm{op}}, \mathrm{Fib}_{\overline{x}})\) is a Galois Category.
\item[2.] If \(\pi: S\rightarrow X\) is a finite model of a scheme and \(\overline{s}\in S^{\bullet}(\Omega)\) is the corresponding geometric point, then there is an isomorphism of profinite groups
\[
\pi^{\mathrm{et}}_1(S,\overline{s})\simeq \pi^{\mathrm{et}}_1(X,\overline{x}),
\]
where
\(\pi^{\mathrm{et}}_1(X,\overline{x}):=\mathrm{Aut}_{[\mathbf{Qcoh}^{\mathrm{fet}}(X)^{\mathrm{op}}, \mathbf{Set}_f]}(\mathrm{Fib}_{\overline{x}})\).
\end{itemize}
The keys are the constructions of a series of connectedness for desired demands and a description of geometric points of schematic spaces. Plus, in the definition of geometric points, we employ an approach similar to scheme theory.
Specifically, in Section 3, the authors proved a proposition pertaining to the subcategory of pw-connected schematic spaces \(\mathbf{SchFin}^{\mathrm{pw}}\), which is crucial in Section 7:
Theorem. The functor \(\mathbf{pw}: \mathbf{SchFin}\rightarrow \mathbf{SchFin}^{\mathrm{pw}}\) is right adjoint to \(i:\mathbf{SchFin}^{\mathrm{pw}}\hookrightarrow\mathbf{SchFin}\). The map \(\mathbf{pw}(X)\rightarrow X\) is a qc-isomorphism for all \(X\) and verifies \(\mathbf{pw} \circ i=\mathrm{Id}\). In particular, \(\mathbf{SchFin}^{\mathrm{pw}}_{\mathrm{qc}}\simeq\mathbf{SchFin}_{\mathrm{qc}}\).
By a logical equivalence between finite étale covers and sheaves of algebras, the authors study the finite locally free sheaves on schematic finite spaces in Section 5. Afterward, they define the key fiber functor in Section 7 of our space \(X\) via \(\mathbf{SchFin}^{\mathrm{pw}}\):
\begin{align*}
\mathrm{Fib}_{\overline{x}}: \mathbf{Qcoh}^{\mathrm{fet}}(X)^{\mathrm{op}} & \rightarrow \mathbf{Set}_f\\
\mathcal{A} & \rightarrow |\mathbf{pw}((\star,\Omega)\times _X (X,\mathcal{A}))|.
\end{align*}
The rest is essentially the routine process of categorical theory. schematic finite space; ringed space; étale fundamental group; étale covers; Galois category; finite poset Schemes and morphisms, Categorical Galois theory, Coverings in algebraic geometry, Algebraic aspects of posets Étale covers and fundamental groups of schematic finite spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kähler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture. Kähler manifolds; Kähler groups; Shafarevich morphisms; holomorphic convexity Kähler manifolds, Variation of Hodge structures (algebro-geometric aspects), Compact Kähler manifolds: generalizations, classification, Homotopy theory and fundamental groups in algebraic geometry, Uniformization of complex manifolds, Coverings in algebraic geometry Linear representations of Kähler groups: factorizations and linear Shafarevich conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(Y\to X\) is a \(G\)-Galois branched cover of curves over an algebraically closed field \(k\), and if \(G\) is a quotient of a finite group \(\Gamma\), then \(Y\to X\) is dominated by a \(\Gamma\)-Galois branched cover \(Z\to X\). This is classical in characteristic 0, and was proven in characteristic \(p\) by the author [\textit{D. Harbater}, Contemp. Math. 186, 353-369 (1995; Zbl 0858.14013)] and \textit{F. Pop} [Invent. Math. 120, 555-578 (1995; Zbl 0842.14017)] in conjunction with the proof of the geometric case of the Shafarevich conjecture on free absolute Galois groups. The resulting cover \(Z\to X\), though, may acquire additional branch points. The present paper shows how many new branch points are needed, and shows that there is some control on the positions of these branch points and on the inertia groups of \(Z\to X\). Galois branched cover of curves; branch points D. Harbater, Embedding problems and adding branch points, in Aspects of Galois Theory, London Mathematical Society Lecture Note Series, Vol. 256, Cambridge University Press, Cambridge, 1999, pp. 119--143. Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Inverse Galois theory, Coverings in algebraic geometry Embedding problems and adding branch points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a discrete, complete valuation ring of characteristic \(p\geq 0\), with residue field \(k\) algebraically closed. The author studies tame coverings of a semi-stable curve \(X\) over \(R\).
First, he considers semi-stable Kummerian coverings of the special fiber and proves that the fundamental Kummerian group \(\pi_1^{\text{kum}}(X_s)\), which classifies these coverings, is isomorphic to the fundamental group of a graph of groups (in the sense of Bass-Serre), this graph being constructed with the help of the classical graph associated to \(X_s\).
Second, the author compares finite coverings \(Y\to X\) of semi-stable curves over \(R\) and the induced coverings between the special fiber \(Y_s\to X_s\). He proves that coverings \(Y\to X\), which are étale outside the double points of \(X\), give Kummerian coverings \(Y_s\to X_s\), and that each of these last coverings can be lifted to a tame covering \(Y\to X\). As an application, the author proves the analog in characteristic \(p>2\) of Belyi's theorem. characteristic \(p\); tame coverings; fundamental group; graph of groups; semi-stable curves; Belyi's theorem; semi-stable Kummerian coverings Saïdi, M., Rev\hat etements modérés et groupe fondamental de graphe de groupes, Compositio Math., 107 (1997), 319-338. Coverings of curves, fundamental group, Coverings in algebraic geometry, Finite ground fields in algebraic geometry Tame coverings and the fundamental group of graphs of groups. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth connected projective curve defined over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(G\) be a finite group whose order is divisible by \(p\). Suppose that \(G\) has a normal \(p\)-Sylow subgroup. We give a necessary and sufficient condition for \(G\) to be a quotient of the algebraic fundamental group \(\pi_1(X)\) of \(X\). projective curve; quotient of the algebraic fundamental group Pacheco, A.; Stevenson, K. F., \textit{finite quotients of the algebraic fundamental group of projective curves in positive characteristic}, Pacific J. Math., 192, 143-158, (2000) Coverings of curves, fundamental group, Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Fundamental group, presentations, free differential calculus, Homogeneous spaces and generalizations Finite quotients of the algebraic fundamental group of projective curves in positive characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a rational surface with \(\dim|-K_s| \geq 1\) and let \(\pi: X \rightarrow S\) be a ramified cyclic covering from a nonruled smooth surface \(X\). The author show that, for any integer \(k \geq 3\) and ample divisor \(A\) on \(S\), the adjoint divisor \(K_X + k \pi^*A\) is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings. cylic covering; nonruled surface; ample; normally generated Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Coverings in algebraic geometry, Projective techniques in algebraic geometry On the projective normality of cyclic coverings over a rational surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a finite group scheme over an algebraically closed field \(k\). An \(S\)-module \(M\) is said to be endotrivial if the \(S\)-module \(\Hom_k(M,M)\) is isomorphic to the direct sum of the trivial module \(k\) and a projective \(S\)-module. In other words, \(\Hom_k(M,M)\) is isomorphic to the trivial module \(k\) in the stable module category. The set \(T(S)\) of equivalence classes of endotrivial \(S\)-modules in the stable module category of \(S\) admits the structure of a group. In the case that \(S\) is a finite group, \textit{L. Puig} [J. Algebra 131, No. 2, 513-526 (1990; Zbl 0699.20004)] showed that \(T(S)\) is finitely generated, and the paper under review is directed in part towards showing that this holds in the general setting of a finite group scheme. As a first step towards this, in previous work [part I, J. Reine Angew. Math. 653, 149-178 (2011; Zbl 1226.20039)], the authors showed that if \(S\) is unipotent, then for any natural number \(n\), there are finitely many isomorphism classes of endotrivial \(S\)-modules of dimension \(n\). Here, the authors extend this finiteness condition to arbitrary finite group schemes.
Much of the paper is then devoted to applications of this result to the question of lifting module structures. Suppose that \(S\) is a normal subgroup scheme of a group scheme \(H\), and that \(M\) is an \(S\)-module. There are many situations when one would like to know whether \(M\) admits the structure of an \(H\)-module which extends the action of \(S\). The authors consider a weaker notion: \(M\) is said to \textit{stably lift} to \(H\) if \(M\) plus a projective \(S\)-module lifts to \(H\). If \(H\) is a connected affine algebraic group scheme and \(S\) is a finite normal subgroup scheme, then it is first shown that any endotrivial \(S\)-module \(M\) is at least \(H\)-stable (\(M\) is preserved under twisting by an element of \(H\)). The authors then focus on syzygy modules \(\Omega_S^n(M)\), noting that if \(M\) is an endotrivial \(S\)-module then so is \(\Omega_S^n(M)\) for any integer \(n\). For a finite-dimensional \(H\)-module \(M\) (considered as an \(S\)-module), the authors give a set of conditions to guarantee that \(\Omega_S^n(M)\) stably lifts to \(H\) as well as a second set of conditions that would guarantee it in fact lifts to \(H\).
Let \(G\) be a semisimple, simply connected algebraic group over a field of prime characteristic. Associated to \(G\), let \(B\) denote a Borel subgroup containing a maximal torus \(T\) with unipotent radical \(U\), and let \(P\) denote a standard parabolic subgroup. A classical context for the above considerations would be setting \(H\) to be \(G\), \(B\), \(T\), \(U\), or \(P\), and setting \(S\) to be the \(r\)-th Frobenius kernel \(H_r\) of \(H\). In this context, it follows that \(\Omega_{H_r}^n(k)\) stably lifts from \(H_r\) to \(H\) for all \(n\). Moreover, any endotrivial \(B_1\)-module (or \(U_1\)-module) stably lifts to \(B\) (or \(U\)). However, the authors show (for \(G=\mathrm{SL}_3\) and the prime being 2) that while \(\Omega_{B_1}^2(k)\) stably lifts to \(B\) it does not lift to \(B\). On the other hand, it is shown, for any finite-dimensional \(G_rT\)-module \(M\), that \(\Omega_{G_r}^n(M)\) lifts to a \(G_rT\)-module structure; a statement that also holds for \(P_r\) to \(P_rT\). It is also shown, for \(G=\mathrm{SL}_2\), that every endotrivial \(G_1\)-module lifts to \(G\).
The authors also relate their ideas on lifting of module structures to recent work of \textit{B. J. Parshall} and \textit{L. L. Scott} [Algebr. Represent. Theory 16, No. 3, 793-817 (2013; Zbl 1286.20059)] who introduced two equivalent notions of stability that are stronger than being \(H\)-stable: numerically \(H\)-stable and tensor \(H\)-stable. It is noted that stable lifting seems to be a stronger condition than tensor or numerical stability. endotrivial modules; finite group schemes; Frobenius kernels; cohomology; syzygies; lifting module structures; stable liftings; stable module categories; numerical stability; tensor stability; projective modules; simply connected algebraic groups Carlson, Jon F.; Nakano, Daniel K.: Endotrivial modules for finite groups schemes II, Bull. inst. Math. acad. Sin. 7, No. 2, 271-289 (2012) Modular representations and characters, Group schemes, Representation theory for linear algebraic groups Endotrivial modules for finite group schemes. II. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems After the classical work of Hurwitz, the Hurwitz scheme in characteristic 0, parametrizing simply branched covers of the projective line, was first rigorously introduced by \textit{W. Fulton} [Ann. Math. (2) 90, 542--575 (1969; Zbl 0194.21901)]. Successively several variants of this scheme have been introduced and studied, e.g. the unparametrized Hurwitz scheme, which is the quotient of the Hurwitz scheme by the PGL(2) action on \(\mathbb P^1\). A natural compactification of this scheme, the space of admissible covers, was given by \textit{J. Harris} and \textit{D. Mumford} [Invent. Math. 67, 23--86 (1982; Zbl 0506.14016)]. Recently \textit{R. Pandharipande} [Math. Ann. 313, 715--729 (1999; Zbl 0933.14035)] identified it as a closed subscheme in the space of stable maps into the stack \(\overline{\mathcal M}_{0,n+1}\).
In the paper under review, following the idea of Pandharipande, the authors introduce compactified Hurwitz stacks in mixed characteristic. They study then in detail the example of the compactified stack of double covers of \(\mathbb P^1\) branched in 4 points, in characteristic 2, and compare it with the characteristic 0 scheme. branched cover; admissible cover; Hurwitz scheme; Hurwitz stacks D. Abramovich and F. Oort, Stable maps and Hurwitz schemes in mixed characteristics, Advances in algebraic geometry motivated by physics (Lowell 2000), Contemp. Math. 276, American Mathematical Society, Providence (2001), 89-100. Families, moduli of curves (algebraic), Generalizations (algebraic spaces, stacks), Coverings in algebraic geometry, Coverings of curves, fundamental group Stable maps and Hurwitz schemes in mixed characteristics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E\) be an elliptic curve over an algebraically closed field \(k\) of characteristic \(p \geq 0\). A locally free sheaf \(\mathcal F\) on \(E\) is called \textit{unipotent} if it admits a filtration with all the successive quotients isomorphic to \({\mathcal O}_E\). According to the classification of \textit{M. F. Atiyah} [Proc. Lond. Math. Soc., III. Ser. 7, 414--452 (1957; Zbl 0084.17305)], there exists, for each \(n \geq 1\), precisely one irreducible unipotent sheaf \({\mathcal F}_n\) of rank \(n\). Moreover, \({\mathcal F}_m \otimes {\mathcal F}_n\) is a direct sum of irreducible unipotent sheaves hence the tensor product of sheaves endows the free abelian group \(R\) generated by the isomorphism classes of \({\mathcal F}_n\), \(n \geq 1\), with a ring structure. In characteristic 0, Atiyah showed that \(R\) is isomorphic to the polynomial ring \({\mathbb Z}[{\mathcal F}_2]\).
In the paper under review, the author investigates the structure of \(R\) when \(p > 0\). Let \(\mathcal C\) be the category of finite length \(k[t]\)-modules and \(K({\mathcal C})\) the free abelian group generated by the isomorphism classes \([M]\) of objects of \(\mathcal C\), modulo the relations \([M \oplus M^{\prime}] = [M] + [M^{\prime}]\). If \(M\), \(M^{\prime} \in {\mathcal C}\), the \textit{convolution product} \(M \star M^{\prime}\) is the \(k[t]\)-module obtained from the \(k[t] \otimes_k k[t]\)-module \(M \otimes_k M^{\prime}\) by the restriction of scalars \(k[t] \rightarrow k[t] \otimes_k k[t]\), \(t \mapsto t\otimes 1 + 1\otimes t\). The convolution product induces a ring structure on \(K({\mathcal C})\). Using \textit{Fourier-Mukai transforms} (actually, the Proposition 2.8 of the paper of \textit{T. Oda} [Nagoya Math. J. 43, 41--72 (1971; Zbl 0201.53603)] would suffice) and an ingenious elementary argument, the author shows that \(R \simeq K({\mathcal C})\) as rings. In particular, the ring structure of \(R\) depends only on \(p\), and not on the elliptic curve.
The author deduces from this description several properties of \(R\) when \(p > 0\). \(R\) is not an integral domain, the extension \({\mathbb Z} \subset R\) is integral and \(\text{Spec}\, R\) has infinitely many irreducible components, such that \(R\) is non-Noetherian. Moreover, the abelian subgroup \(R_\infty\) of \(R\) generated by \([{\mathcal F}_{p^i}]\), \(i \geq 0\), is a subring of \(R\) and the author describes explicitly its spectrum. elliptic curve; positive characteristic; ring of unipotent bundles; tensor product of matrices; modular representation ring Vector bundles on curves and their moduli, Elliptic curves, Canonical forms, reductions, classification, Modular representations and characters On the ring of unipotent vector bundles on elliptic curves in positive characteristics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper the critical value mapping is considered, which associates with any polynomial from underdiagonal miniversal deformation of the family of parabolic singularities the set of its critical values. It is shown that the restriction of this mapping to the subset of polynomials with \(k\) different critical values, \(k\geq 2\), is a covering of the space of unordered \(k\)-tuples of different complex numbers. In particular, it is proved that the connected components of such subsets are \(K[\pi,1]\) spaces. critical value; miniversal deformation; parabolic singularities Jaworski, P.: Distribution of critical values of miniversal deformations of parabolic singularities. Invent. Math. \textbf{86}, 19-33 (1986) Deformations of complex singularities; vanishing cycles, Local complex singularities, Complex singularities, Coverings in algebraic geometry, Singularities in algebraic geometry Distribution of critical values of miniversal deformations of parabolic singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In der vorliegenden Arbeit werden ``\(n\)-rationale'' Riemannsche Flächen untersucht, die definiert werden als kompakte Riemannsche Flächen vom Geschlecht \(g\geq 1\), die dargestellt werden können als \(n\)-blättrige Galoisüberlagerung der Riemannschen Zahlenkugel, deren Decktransformationsgruppe zyklisch ist. Coverings in algebraic geometry, Compact Riemann surfaces and uniformization On cyclic Galois coverings of the number sphere | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author derives intersection formulas for curves on an abelian branched covering of a smooth complex projective surface. These formulas are applied to certain naturally arising sequences of covering surfaces to show that the first Betti numbers of their desingularizations are polynomial periodic. That is, they can be described by a polynomial with periodically varying coefficients. The sequences of branched coverings arise as follows. One fixes a collection of irreducible curves \({\mathcal B}\) lying on a smooth surface \(Y\). For each integer \(n\) there is a canonically defined finite surjective morphism \(\rho_ n : X_ n \to Y\) with Galois group equal to the abelianization of \(\pi_ 1(Y \backslash {\mathcal B})\) tensored with \(\mathbb{Z}/n \mathbb{Z}\). The first Betti number of any desingularization \(\widetilde X_ n \to X_ n\) is the difference \(b_ 1(X_ n \backslash \rho_ n^{-1} ({\mathcal B})) - \text{Null} (\widetilde \rho_ n^{-1} ({\mathcal B}))\) where \(\widetilde \rho_ n : \widetilde X_ n \to Y\) is the composition map and Null\((\widetilde \rho_ n^{-1} ({\mathcal B}))\) is the nullity of the intersection matrix. (Here we assume for simplicity that \({\mathcal B}\) supports an ample divisor.) \textit{P. Sarnak} has shown that the first Betti number of the unbranched coverings \(X_ n \backslash \rho_ n^{-1} ({\mathcal B})\) is polynomial periodic [cf. \textit{P. Sarnak} and \textit{S. Adams}, Isr. J. Mth. 88, 31-72 (1994)]. Thus, to prove the main theorem, it suffices to show the same for Null\((\widetilde \rho_ n^{-1} ({\mathcal B}))\).
In previous work, the author studied the intersection matrix on the desingularizations \(\widetilde X_ n\) and proved the main theorem given certain restrictions on the branch locus. In this paper Mumford's rational intersection theory on normal surfaces is used to find cleaner formulas for intersections on the canonically defined \(X_ n\). The corresponding matrices have the same nullity as those for \(\widetilde X_ n\) and are easier to analyze. This leads to a proof of the theorem without any restrictions on the branch locus. intersection; curves on abelian branched covering of smooth complex projective surface; Betti numbers; desingularizations Hironaka, E.: Intersection theory on branched covering surfaces and polynomial periodicity. Internat. math. Res. notices 6, 185-196 (1993) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Coverings in algebraic geometry Intersection theory on branched covering surfaces and polynomial periodicity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a curve \(X\) of the form \(y^p=h(x)\) over a number field, one can use descents to obtain explicit bounds on the Mordell-Weil rank of the Jacobian or to prove that the curve has no rational points. We show how, having performed such a descent, one can easily obtain additional information which may rule out the existence of rational divisors on \(X\) of degree prime to \(p\). This can yield sharper bounds on the Mordell-Weil rank by demonstrating the existence of nontrivial elements in the Shafarevich-Tate group. As an example we compute the Mordell-Weil rank of the Jacobian of a genus 4 curve over \(\mathbb Q\) by determining that the 3-primary part of the Shafarevich-Tate group is isomorphic to \(\mathbb Z/3\times\mathbb Z/3\). abelian variety; Mordell-Weil group; explicit descent Creutz, B.: Explicit descent in the Picard group of a cyclic cover of the projective line. In: Howe, E.W., Kedlaya K.S. (eds.) ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, San Diego 2012. Open Book Series, vol. 1, pp. 295-315. Mathematical Science, Berkeley, California (2013) Abelian varieties of dimension \(> 1\), Picard groups, Coverings in algebraic geometry, Computer solution of Diophantine equations Explicit descent in the Picard group of a cyclic cover of the projective line | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(n \geq 2\) is an integer, then a closed Riemann surface \(S\) is called \textit{cyclic \(n\)-gonal} if if it admits a conformal automorphism \(\tau\) of order \(n\) so that the quotient orbifold \(S/\langle \tau \rangle\) has genus zero. These surfaces can be described by algebraic curves of the form
\[
y^{n}=\prod_{j=1}^{s}(x-a_{j})^{t_{j}},
\]
where (i) \(a_{1},\ldots,a_{s} \in {\mathbb C}\) are distinct, (ii) \(t_{j} \in \{1,\ldots,n-1\}\), for \(j=1,\ldots,s\), (iii) \(\gcd(n,t_{1},\ldots,t_{s})=1\) and (iv) \(t_{1}+\cdots+t_{s} \equiv 0 \mod(n)\). In this case, \(\tau(x,y)=(x,\omega_{n}y)\), where \(\omega_{n}=\exp(2\pi i/n)\), and \(\pi(x,y)=x\) is a regular branched cover, with \(\langle \tau \rangle\) as its deck group, whose branch value set is \(\{a_{1},\ldots,a_{s}\}\) and \(a_{j}\) has branch order \(n/\gcd(n,t_{j})\); the tuple \(T:=(t_{1},\ldots,t_{s})\) is called the multi-degree of the curve. The cyclic \(n\)-gonal Riemann surface is called \textit{generalized superelliptic} if, moreover, \(\gcd(n,t_{j})=1\), for every \(j=1,\ldots,s\), that is, the quotient orbifold \(S/\langle \tau\rangle\) has all of its cone points of order \(n\).
This paper provides a short survey on groups of automorphisms of cyclic \(n\)-gonal Riemann surfaces in the case that \(\langle \tau \rangle\) is a normal subgroup of \(A=\text{Aut}(S)\). This last property is known to hold, for example, (i) if \(n=p\) is a prime so that \(g>(p-1)^{2}\), where \(g\) is the genus of \(S\) [\textit{R. D. M. Accola}, Proc. Am. Math. Soc. 26, 315--322 (1970; Zbl 0212.42501)] and (ii) in the generalized superelliptic case and \(g>(n-1)^{2}\) [\textit{A. Kontogeorgis}, J. Algebra 216, No. 2, 665--706, Art. No. jabr.1998.7804 (1999; Zbl 0938.11056)].
In the particular case that \(n=p\) is a prime, \(t_{1}=\cdots=t_{s-1}=1\) and the polynomial
\[
f(x)=\prod_{j=1}^{s-1}\left(x-\frac{a_{j}}{a_{j}-a_{s}}\right)^{t_{j}}
\]
is square free, then the cyclic \(p\)-gonal Riemann surface is called \textit{superelliptic}. In this case, it is observed that \(\tau\) is central in the normalizer of \(\langle \tau \rangle\) in \(A\).
In the case of general cyclic \(p\)-gonal Riemann surfaces, where \(p\) is a prime, for a fixed multi-degree \(T\), a description of the moduli space of them is presented (together its complex dimension as orbifold). Riemann surface; automorphisms of Riemann surfaces; \(p\)-gonal curve, hyperelliptic surface; superelliptic curve Broughton, S. A.: Superelliptic surfaces as p-gonal surfaces. Contemp. math. 629, 15-28 (2014) Automorphisms of curves, Compact Riemann surfaces and uniformization, Classification theory of Riemann surfaces, Coverings in algebraic geometry Superelliptic surfaces as \(p\)-gonal surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is well known that each basic, finite dimensional algebra A over an algebraically closed field k can be presented as a factor algebra of the path algebra kQ of the quiver \(Q=(Q_ 0,Q_ 1)\) determined by A, modulo some ideal generated by a finite set R of relations, i.e. k-linear combinations of oriented paths of length at least two in Q with the same origins and endpoints. Analogously as in algebraic topology one constructs a universal covering quiver with relations \((\tilde Q,\tilde R)\) of \((Q,R)\), and attaches to it a quadratic form \(f: \oplus_{q\in Q_ 0}{\mathbb{Z}}\to {\mathbb{Z}}\) given by the formula
\[
f(x)=\sum_{q\in \tilde Q_ 0}x^ 2_ q-\sum_{d\in \tilde Q_ 1}x_{o(d)}x_{t(d)}+\sum_{r\in \tilde R}x_{o(r)}x_{t(r)}
\]
where o(. ) (resp. t(.)) denotes origin (resp. endpoint) of an arrow or oriented path in \(\tilde Q.\)
The author announces the following theorem: Let (Q,R) be a quiver with relations such that Q has only two vertices. Then \(A=kQ/<R>\) is tame, that is, in each dimension all indecomposable finitely generated A- modules up to isomorphism are contained in a finite number of 1-parameter families, iff the quadratic form f is weakly nonnegative. - The author gives also a full list of ''maximal'' quivers with relations with the above properties. The representation type of many algebras from the list follows from \textit{P. Dowbor} and \textit{A. Skowroński} [Comment. Math. Helvet. 62, 311-337 (1987)]. basic, finite dimensional algebra; path algebra; universal covering quiver with relations; quadratic form; tame; representation type V. I. Bekkert, Tame two-point quivers with relations. Izv. Vyssh. Uchebn. Zaved. Mat.12, 62-64 (1986). Representation theory of associative rings and algebras, Finite rings and finite-dimensional associative algebras, Coverings in algebraic geometry Tame two-point quivers with relations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let G denote a Chevalley group over finite field \(F_ q\) of characteristic p with a fixed Borel subgroup B. Consider an \(F_ qG\)- module V satiyfing \(V=<C_ V(O_ p(B))^ g:\) \(g\in G>\). Following the 0-th homology construction in \textit{M. A. Ronan} and \textit{S. D. Smith} [ibid. 96, 319-346 (1985; Zbl 0604.20043)] one can define a sheaf \({\mathcal F}_ V\) on V in terms of subspaces \(V_ 0\) of \(C_ V(O_ p(P_ 0))\) when \(P_ 0\) runs over the parabolics of G containing B.
The problem of splitting of V with only two composition factors S and \(T\cong V/S\) reduces to the splitting of \({\mathcal F}_ V\) in the direct sum of \({\mathcal F}_ S\) and \({\mathcal F}_ T\) and some homological questions for \(H_ 0({\mathcal F}_ T)\). Then the author gives an elegant sufficient condition for complete reducibility of V: all composition factors of V are modules \(T_ i\) with minimal highest weights \(\lambda_ i\) not pairwise adjacent in the Dynkin diagram (if \(p=2\), G cannot be of type \(BC_ 3)\), or all \(T_ i\) are pairwise isomorphic. Chevalley group; Borel subgroup; composition factors; splitting; complete reducibility; weights; Dynkin diagram Smith, S. D.: Sheaf homology and complete reducibility. J. algebra 95, 72-80 (1985) Cohomology theory for linear algebraic groups, Linear algebraic groups over finite fields, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Modular representations and characters Sheaf homology and complete reducibility | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a compact Kähler manifold and \(\widetilde{X}\) its universal covering: The Shafarevich conjecture states that \(\widetilde{X}\) is holomorphically convex, namely for every infinite discrete \(S\subset \widetilde{X}\) there exists a holomorphic function on \(\widetilde{X}\) that is unbounded on \(S\). In the paper under consideration, the authors prove the following result, that, by the above remarks, can be regarded as a partial verification of the Shafarevich conjecture in the case of surfaces:
Theorem: Let \(X\) be a compact Kähler surface and \(X'\to X\) a Galois covering with Galois group \(\Gamma\); if \(\Gamma\) is reductive and \(X'\) does not have two ends, then \(X'\) is holomorphically convex.
(A finitely generated group \(\Gamma\) is \textit{reductive} if it has an almost faithful Zariski dense representation in a reductive complex Lie group \(G\)). It follows that if \(X\) is a compact Kähler surface and the universal covering \(\widetilde{X}\to X\) admits a factorization \(\widetilde{X}\to X'\to X\), with \(X'\to X\) satisfying the assumptions of the theorem above, then \(\widetilde{X}\) admits nonconstant holomorphic functions. The authors also show that the assumptions of the theorem are satisfied by a large number of examples. universal covering; algebraic surface; Shafarevich conjecture; compact Kähler surface; Galois covering; holomorphically convex covering Katzarkov, L.; Ramachandran, M., On the universal coverings of algebraic surfaces, Ann. Sci. École Norm. Sup., 31, 525-535, (1998) Coverings in algebraic geometry, Kähler manifolds, Compact complex surfaces On the universal coverings of algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove that every projective, geometrically reduced scheme of dimension \(n\) over an infinite field \(k\) of positive characteristic admits a finite morphism over some finite radicial extension \(k'\) of \(k\) to projective \(n\)-space, étale away from the hyperplane \(H\) at infinity, which maps a chosen Weil divisor into \(H\) and a chosen smooth geometric point of \(X\) not on the divisor to some point not in \(H\). Noether normalization; finite radiacl extensions; Abhyankar map Kedlaya, KS, Étale covers of affine spaces in positive characteristic, C. R. Math. Acad. Sci. Paris, 335, 921-926, (2002) Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry, Separable extensions, Galois theory, Coverings in algebraic geometry Étale covers of affine spaces in positive characteristic. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C=\{f(x,y)=0\}\) be an algebraic curve in \({\mathbb{C}}{\mathbb{P}}^ 2\) such that the projection \(\pi_ x:\quad {\mathbb{C}}^ 2\to {\mathbb{C}}_ x\) onto the x-axis is generic for C. Set \(S(C,\pi)=\{p\in C| (\partial f/\partial y(p)=0\}\), \(D(C,\pi)=\pi (S(C,\pi))\), and choose \(M\in {\mathbb{C}}_ x\setminus D(C,\pi)\). The braid monodromy of C is the homomorphism \(\theta:\quad \pi_ 1({\mathbb{C}}_ x\setminus D(C,\pi),M)\to B[\pi^{-1}(M),C\cap \pi^{-1}(M)],\) this last being the group of homotopy classes of compact-supported homeomorphisms of \(\pi^{-1}(M)\) which preserve \(C\cap \pi^{-1}(M).\)
In the paper we give an a priori construction of the braid monodromy for certain classes of curves with ordinary singularities of branch points. In particular we obtain that the monodromy of an arrangement of real lines is determined by its ``dual graph''.
Some applications to the study of the fundamental group of the complement are given. In the last part, we exploit the above result to conclude that the complement to certain arrangements is not a K(\(\pi\),1). arrangements of hyperplanes; braid monodromy; curves with ordinary singularities; arrangement of real lines; fundamental group of the complement M. Salvetti,Arrangements of lines and monodromy of plane curves, Comp. Math.,68 (1988), pp. 103--122. Projective techniques in algebraic geometry, Singularities of curves, local rings, Coverings of curves, fundamental group, Coverings in algebraic geometry Arrangements of lines and monodromy of plane curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Schottky-relations on the thetanullwerte for Jacobi varieties have an extensive history beginning with Riemann's formulas for the ''Prym'' theta- functions, and culminating in the recent work of \textit{H. M. Farkas} and \textit{H. E. Rauch} [Ann. Math., II. Ser. 92, 434-461 (1970; Zbl 0204.096)], \textit{D. Mumford} [''Prym varieties. II'', Contribut. to Analysis, Collect. Papers dedic. L. Bers, 325-350 (1974; Zbl 0299.14018)] and \textit{J. Igusa} [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 531-545 (1981; Zbl 0501.14026)]. These relations are established from an analysis of the configuration of Jacobian-Prym theta-functions on smooth double coverings. In this paper, we obtain these results directly from a special even-order vanishing property of Jacobian theta-functions: this phenomenon enables one to construct certain linear systems on an (intrinsic) Abelian covering leading to irrational identities on theta- functions of type first observed in genus 3 by Frobenius and Schottky. The vanishing property can naturally also be explained in terms of the Prym theta-functions for general double coverings; this is indicated explicitly in {\S} 3 with a version of the Schottky-Jung-Farkas-Rauch proportionality theorem for a class of generalized theta-functions on singular curves. thetanullwerte; Jacobi varieties; vanishing property; double coverings J. D. Fay, \textit{Theta Functions on Riemann Surfaces} (Lect. Notes Math., Vol. 352), Springer, Berlin (1973); D. Mumford, Tata Lectures on Theta (Progr. Math., Vol. 28), Vol. 1, \textit{Introduction and Motivation: Theta Functions in One Variable. Basic Results on Theta Functions in Several Variables}, Birkhäuser, Boston, Mass. (1983); Vol. 2, Jaconian Theta Functions and Differential Equations (Progr. Math., Vol. 43), Birkhäuser, Boston, Mass. (1984). Theta functions and abelian varieties, Elliptic functions and integrals, Riemann surfaces, Coverings in algebraic geometry On the even-order vanishing of Jacobian theta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Every non-singular projective surface \(S\) over \(\mathbb C\) defines three structures \(vS\), \(dS\), \(tS\), where \(tS\) is the topological type of \(S\), \(dS\) is the related smooth 4-manifold and \(vS\) is the deformation type of \(S\).
The aim of this article is to give a short survey of some results related to the program on investigation of smooth structures on projective surfaces and their deformation types, initiated by Boris Moishezon. It has three sources: classical (Italian) algebraic geometry including Picard-Lefschetz theory, braid group theory, topology of smooth manifolds. Chisini's problem; projective surface; Picard-Lefschetz theory; braid group Coverings in algebraic geometry On generic coverings of the plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A general structure theorem for Gorenstein covers, i.e., finite flat maps such such that each fiber is Gorenstein, was given in [\textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, No. 3, 439--460 (1996; Zbl 0866.14009)] (the case of degree \(3\) covers was the only one that had been previously studied, see \textit{R. Miranda} [Am. J. Math. 107, 1123-1158 (1985; Zbl 0611.14011)] and also \textit{R. Pardini} [Ark. Mat. 27, No. 2, 319--341 (1989; Zbl 0707.14010)]. Using this general result, it is possible to give a complete characterization for \(d\leq 5\) (cf. [Zbl 0921.14006] for the case \(d=5\)). On the other hand, such a complete description seems hard to obtain: the paper under consideration belongs to a series of works by the same author, all describing special families of covers of degree \(>5\). Here the author studies a family of degree 6 covers, that he calls ``Anglo-American'' covers, proving in particular a Bertini type statement for these covers. Also, an incorrect statement contained in [\textit{G. Casnati}, J. Pure Appl. Algebra 182, No.1, 17--32 (2003; Zbl 1024.14007)] is rectified. ramified cover; degree 6 cover; Anglo-American cover G. Casnati, Covers of algebraic varieties VI. A Bertini theorem for Anglo-American covers, preprint Coverings in algebraic geometry, Ramification problems in algebraic geometry Covers of algebraic varieties VI. Anglo--American covers and \((1,3)\)--polarized abelian surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This short note studies the problem that whether a generical finite morphism between projective varieties has a ``good'' biratioanl model, in the sense that the birational model is a finite cover with smooth branching divisor.
It is well-known that such a good model exists for generically double covers over fields of characteristic 0. As the main result of this note, such a good model exists for generically triple covers over perfect fields of characteristic not \(2\) or \(3\) under a natural assumption. To be more precise, if \(T\to S\) is a generically finite morphism of degree 3 between projective varieties, over a perfect field \(k\) with \(\text{char}\,k\neq 2, 3\), suppose that we have an embedded resolution of singularities of divisors on \(S\) by iterated blowing ups with smooth centers, then a good model exists. algebraic varieties; triple coverings Coverings in algebraic geometry Models of triple covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite group and let \(K\) be an Hilbertian field. The group \(G\) is said to satisfy the arithmetic lifting property over \(K\) if every Galois extension of \(K\) with group \(G\) arises as a specialization of a Galois branched covering of the projective line defined over \(K\) and with the same Galois group \(G\). This property is known to hold in some cases, among which is the case when \(G\) is abelian and \(K\) is a number field [see \textit{S. Beckmann}, J. Algebra 164, 430-451 (1994; Zbl 0802.12003).]
The main achievement of the paper is the following. Assume that \(A\) is a finite cyclic group and that \(H\) is a finite group satisfying the arithmetic lifting property over \(K\) and acting on \(A\); assume also that the orders of \(A\) and \(H\) are relatively prime and that the characteristic of \(K\) does not divide the order of \(A\). Then \(G=A\rtimes H\) satisfies the arithmetic lifting property over \(K\). In this case, an arithmetic lifting for any Galois extension of \(K\) with Galois group \(A\wr H\) is explicitly constructed and the result is deduced. It is also shown that if \(A\) is any finite abelian group and \(H\) is a group with the arithmetic lifting property over \(K\), then the group \(A\wr H\) has this property as well, provided that \(K\) satisfies some additional assumptions if \(A\) has nontrivial \(2\)-torsion. Hilbertian fields; inverse Galois theory E. Black, On semidirect products and the arithmetic lifting property, J. London Math. Soc. (2) 60 (1999), 677-688. CMP 2000:11 Hilbertian fields; Hilbert's irreducibility theorem, Inverse Galois theory, Coverings in algebraic geometry On semidirect products and the arithmetic lifting property | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An \(S_3\)-cover is a finite flat map of schemes \(\pi: Y\to X\) that is the quotient map for a faithful action of the symmetric group \(S_3\) on \(X\). In this paper the author aims at giving a bottom-up description, in the spirit of what is done in [\textit{R. Miranda}, Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)] for triple covers and in [\textit{R. Pardini}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26, No. 4, 719--735 (1998; Zbl 1040.14005)] for abelian covers, of \(S_3\)-covers of integral noetherian schemes over a domain \(R\) such that \(6\in R\) is invertible.
However the description is necessarily less explicit in this case, since the situation is much more complicated. The building data for \(\pi\) are a line bundle \(L\) and a rank 2 vector bundle \(E\) on \(X\), an \(S_3\) action on \(\bar{E}:=E\oplus E\) that fiberwise consists of two copies of the irreducible \(2\)-dimensional representation of \(S_3\), and a section \(\sigma\in\text{Hom}(S^2L,{\mathcal O}_X)\oplus \text{Hom}(\bar E\otimes L, \bar E)\oplus \text{Hom}(S^2\bar E, {\mathcal O}_X\oplus L\oplus \bar E)\). The section \(\sigma\) corresponds to an \(S_3\)-equivariant structure of commmutative and associative \({\mathcal O}_X\)-algebra on \({\mathcal O}_X\oplus L\oplus \bar E\), and therefore must satisfy some compatibility conditions. Indeed, the bulk of the paper is devoted to investigating these compatibility conditions. cover; Galois cover; dihedral cover; group action; symmetric group; \(S_3\)-action Easton, R.W., \(S_3\)-covers of schemes., Canad. J. math., 63, 5, 1058-1082, (2011) Coverings in algebraic geometry, Group actions on varieties or schemes (quotients) \(S_3\)-covers of schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper deals with the relationship between the normal subgroups of the fundamental group of a complex analytic manifold and their associated covering spaces. No new result is given. fundamental group; covering Coverings in algebraic geometry, Homotopy groups of special spaces Algebraic structure of the restricted analytic sheaves and restricted ideal sheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author constructs new examples of surfaces of general type as double planes, that is as double covers of \(\mathbb{P}^2\). More precisely he constructs: a surface with \(p_g=q=0\), \(P_2=4\) as double plane with branch locus of degree \(12\) having five irreducible components; a surface with \(p_g=q=1\), \(P_2=4\) as double plane with branch locus of degree \(12\) having four irreducible components; a surface with \(p_g=q=0\), \(P_2=5\) as double plane with branch locus of degree \(14\) having six irreducible components; a surface with \(p_g=q=0\), \(P_2=4\) as double plane with irreducible branch locus of degree \(22\).
The first three examples have a bicanonical map that is not birational, and a torsion group different from zero (more precisely there are non trivial 2-torsion elements in the Picard group arising from irreducible components of the branch locus). All constructions are explicit: the author gives the equations of the branch curves. The results are presented without complete proofs, which will be published elsewhere. surfaces of general type; double planes; branch locus; bicanonical transformation E. Stagnaro, Numerical Burniat and irregular surfaces, Acta Appl. Math. 75 (2003), 167--181, Monodromy and differential equations (Moscow, 2001). Surfaces of general type, Rational and birational maps, Coverings in algebraic geometry, Special algebraic curves and curves of low genus Numerical Burniat and irregular surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article describes a general construction of double covers over quadratic degeneracy loci (Theorem \(3.1\)) and Lagrangian intersection loci (Theorem \(4.2\)) using reflexive sheaves.
\begin{itemize}
\item[(1)] Under some regularity assumptions on the family of quadratic forms, the authors obtain (in Proposition \(3.7\)) a criterion for the double cover over the corank-\(k\) degeneracy locus of the family to be nonsingular. The double covers obtained are shown (in Proposition \(3.10\)) to be the normalization of the double covers obtained by the Stein factorization of the projections of the Hilbert schemes of linear isotropic spaces for the quadric fibration corresponding to the family of quadratic forms. Applications of the results give constructions of natural double covers of symmetroid hypersurfaces of odd degree (Theorem \(3.11\)).
\item[(2)] The double covers over Lagrangian intersection loci are shown to remain unchanged under (appropriately defined) isotropic reduction (Proposition \(4.5\)). Moreover étale locally these double covers coincide with the double covers over quadratic degeneracy loci for appropriately defined families of quadratic forms (Proposition \(4.7\)) and hence the nonsingularity criterion of double covers of quadratic degeneracy loci are applicable.
\item[(3)] Constructions of double covers of Lagrangian intersection loci are applied to obtain double covers of EPW varieties, which generalize the constructions of double EPW sextics [\textit{K. G. O'Grady}, Mich. Math. J. 62, No. 1, 143--184 (2013; Zbl 1276.14008)] and EPW cubes [\textit{A. Iliev} et al., J. Reine Angew. Math. 748, 241--268 (2019; Zbl 1423.14220)].
\end{itemize} double covers; reflexive sheaves; hypersurfaces; quadratic degeneracy loci; Lagrangian intersection loci; EPW varieties; hyperkähler varieties Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves, \(4\)-folds, \(n\)-folds (\(n>4\)), Hypersurfaces and algebraic geometry, Special varieties Double covers of quadratic degeneracy and Lagrangian intersection loci | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This Ph.D. thesis is devoted to the problem of the computation of the fundamental group of a surface that is obtained by taking the Galois closure of a generic covering. The author restricts itself to ``good'' generic coverings of the projective plane. The main difference between this definition and the standard definition of a generic covering is in some strong ampleness assumption on the line bundle pull-back of \({\mathcal{O}}(1)\). The main result is for the affine case, i.e., restricting to a general affine plane \(\mathbb{A}^2 \subset \mathbb{P}^2\). To express it we need some notation. We write \(X^{\text{aff}}\) for the preimage of \(\mathbb{A}^2\) for a good generic covering, and \(X^{\text{aff}}_{\text{gal}}\) for the preimage of the same affine plane for its Galois closure. Given a group \(G\), the author denotes by \({\mathcal K}(G,n)\) the kernel of the natural map \(G^n \rightarrow G^{ab}\) associating to each \(n-\)tuple of elements of \(G\) the class of their product. The author constructs a surjective map
\[
\pi_1(X^{\text{aff}}_{\text{gal}}) \rightarrow \tilde{\mathcal K}(\pi_1(X^{\text{aff}}),n),
\]
where \(\tilde{\mathcal K}(G,n)\) is a central extension
\[
1 \rightarrow H_2(G,\mathbb{Z}) \rightarrow \tilde{\mathcal K}(G,n)\rightarrow {\mathcal K}(G,n) \rightarrow 1.
\]
He notes moreover that the constructed surjective map is an isomorphism in all known examples except one related to the Veronese surface. He proves that the map is an isomorphism under some connectivity hypotesis for the pull-back of the ramification divisor on the universal covering of \(X^{\text{aff}}_{\text{gal}}\). Christian Liedtke, On fundamental groups of Galois closures of generic projections, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 367, Universität Bonn, Mathematisches Institut, Bonn, 2004. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2004. Coverings in algebraic geometry On fundamental groups of Galois closures of generic projections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider abelian coverings \(f: (X,0)\to (Y,0)\) of n-dimensional complex analytic singularities, with normal total spaces (X,0). Under quite general assumptions on the base space (Y,0) and on the corresponding branching set (D,0)\(\subset (Y,0)\), we show that these coverings can be described as pull-backs of coverings with base space \(({\mathbb{C}}^ m,0)\) and branching set the union of the coordinate hyperplanes, where m is the number of irreducible components of (D,0). - These special coverings over \(({\mathbb{C}}^ m,0)\) are related to abelian quotient singularities in a similar way as the well-known case of Hirzebruch-Jung singularities \((m=2).\)
As an application, we describe the global abelian coverings \(\phi: V\to W\) among projective varieties (for large classes of base spaces W) in terms of finite abelian quotients of weighted projective varieties. abelian coverings; abelian quotient singularities; Hirzebruch-Jung singularities Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Normal analytic spaces On analytic abelian coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In der vorliegenden Arbeit wird eine Methode zur Konstruktion einer Familie verzweigter Galoisüberlagerungen über \(\mathbb{P}_2(\mathbb{C})\) mit beliebiger endlicher Galoisgruppe eingeführt. Mithilfe dieses Verfahrens werden Flächen von allgemeinem Typ erzeugt. Die Methode wird angewendet, um eine Familie von \(A_5\) -Überlagerungen über \(\mathbb{P}_2(\mathbb{C})\) zu konstruieren. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Coverings in algebraic geometry, Surfaces of general type Families of ramified Galois coverings over \(\mathbb{P}_2(\mathbb{C})\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \((X,\mathbb{V}, F,S)\) be a polarized variation of Hodge structure with immersive period mapping.
We prove the following facts, under the assumption that \(X\) is compact:
\(\bullet\) \(X\) satisfies a variant of Gromov's Kähler-hyperbolicity condition [cf. \textit{M. Gromov}, J. Differ. Geom. 33, 263-291 (1992; Zbl 0719.53042)].
\(\bullet\) The universal covering space of \(X\) carries no \(L^2\) harmonic forms with coefficients in \(\mathbb{V}\) outside the middle degree.
\(\bullet\) Let \(\mathbb{V}= \oplus_{{p+q =w \atop p,q \geq 0}} H^{p,q}\) be the Hodge decomposition. One has:
\[
(-1)^{m-p} \left(\sum^w_{i=0} (-1)^ich (H^{w-i,i} \otimes \Omega_X^{p+i}) \right) \cdot \text{Todd} (T_X) \cdot [X]\geq 0
\]
\(\bullet\) There are equality cases to this inequality arising from VHS (variation of Hodge structure) carried by quotients of bounded symmetric domains.
\(\bullet\) Under good circumstances, these locally symmetric equality cases are the only ones. This leads to characterizations of certain totally geodesic subspaces of quotients of bounded symmetric domains. variation of Hodge structure; period mapping; Kähler-hyperbolicity condition P. Eyssidieux La characteristique d'Euler du complexe de Gauss-Manin, J. reine angew. Math. 490 (1997), 155-212. Period matrices, variation of Hodge structure; degenerations, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Hodge theory in global analysis, Variation of Hodge structures (algebro-geometric aspects), Coverings in algebraic geometry The Euler characteristic of the Gauss-Manin complex | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{O. Chisini's} conjecture [Ist. Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur., III. Ser. 8(77), 339--356 (1944; Zbl 0061.35305)] claims that a generic covering of \(\mathbb{P}^2\) of degree \(\geq 5\) is determined by its branch curve. In the paper under review the author considers a generalization of this to the case of normal surfaces. This is checked in two cases: when the maximum of degrees of two generic coverings is \(\geq 2\) or when it is \(\leq 4\). cusp singularity; ramified covering; monodromy; generic covering Kulikov, V.S.: The generalized Chisini conjecture. Proc. Steklov Inst. Math. 241(2), 110--119 (2003) Coverings in algebraic geometry, Ramification problems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Coverings of curves, fundamental group Generalized Chisini's conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be an abelian variety of dimension \(g\) with a projective embedding \(\varphi: A \rightarrow \mathbb P^N\). Let \(W \subset \mathbb P^N\) be a linear subspce of dimension \(N -g -1\) disjoint from the image of \(\varphi\) and \(W_0 \subset \mathbb P^N\) a linear subspace of dimension \(g\) disjoint from \(W\). Consider the linear projection \(\pi_W: \mathbb P^N \dashrightarrow W_0\). The embedding \(\varphi\) is called Galois, if the composition \(\pi_W \varphi: A \rightarrow W_0\) is a Galois covering. \textit{H. Yoshihara} showed in [Rend. Semin. Mat. Univ. Padova 117, 69--85 (2007; Zbl 1136.14041)] that if \(A\) admits a Galois embedding, then it must contain an elliptic curve and moreover, if an abelian surface admits a Galois embedding, it is isogenous to a self-product of an elliptic curve. The main result of this paper is the following generalization: Let \(A\) be an abelian variety of dimension \(g\). If \(A\) admits a Galois embedding, then it is isogenous to a self-product of an elliptic curve. If \(A\) equals a self-product of an elliptic curve, then \(A\) possesses infinitely many Galois embeddings and there are Galois embeddings with arbitrarily large Galois group. Two proofs of the second mentioned result are given. projective embeddings of abelian varieties; Products of elliptic curves Auffarth, R., A note on Galois embeddings of abelian varieties, Manuscr. Math., 154, 279-284, (2017) Algebraic theory of abelian varieties, Coverings in algebraic geometry A note on Galois embeddings of abelian varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Sei \(X\to Y\) eine Galoissche Überlagerung von Kurven über einem algebraisch abgeschlossenen Körper k; deren Galoisgruppe G sei eine Untergruppe von PGL(2,k), der Automorphismengruppe der projektiven Geraden \({\mathbb{P}}^ 1_ k\). Es wird gezeigt, daß \(X\to Y\) gleich dem ``pull back'' einer Galoisschen Überlagerung \({\mathbb{P}}^ 1_ k\to {\mathbb{P}}^ 1_ k\) mit derselben Gruppe G ist. coverings of curves; projective curve; automorphism group; linear fractional transformation; algebraic function field; finite Galois extension; Galois group Arithmetic theory of algebraic function fields, Coverings in algebraic geometry, Galois theory Galois groups acting as linear fractional transformations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal M}_{g,n}\) denote the moduli space of smooth \(n\)-pointed curves of genus \(g\) and \(\overline{\mathcal M}_{g,n}\) its Deligne-Mumford compactification, the moduli space of stable \(n\)-pointed curves. Let \(A^*(\overline{\mathcal M}_{g,n})\) denote the Chow ring and \(R^*(\overline{\mathcal M}_{g,n})\) its subring, the tautological ring of \(\overline{\mathcal M}_{g,n}\). \textit{C. Faber} and \textit{R. Pandharipande} [Mich. Math. J. 48, Spec. Vol., 215--252 (2000; Zbl 1090.14005)], in analogy with a previous conjecture of Faber on \({\mathcal M}_{g}\), stated a conjecture (called by them a speculation) on \(R^*(\overline{\mathcal M}_{g,n})\), saying that it is a Gorenstein ring with socle in codimension \(3g-3\).
The first step to prove this conjecture is to check that the tautological ring has rank 1 in maximal codimension \(3g-3+n\). This is proved in the paper under review. The essential ingredient of the proof is a formula of \textit{T. Ekedahl, S. Lando, M. Shapiro} and \textit{A. Vainshtein} [Invent. Math. 146, No. 2, 297--327 (2001; Zbl 1073.14041)], for the Hurwitz number \(H^g_{\alpha_1\ldots\alpha_n}\) of genus \(g\) irreducible branched covers of \({\mathbb P}^1\) of degree \(\sum\alpha_i\), with simple branching above \(r\) fixed points, branching with monodromy type \((\alpha_1, \ldots, \alpha_n)\) above \(\infty\) and no other branching. stable curve; tautological ring; irreducible branched covers; moduli space of smooth \(n\)-pointed curves; Chow ring Tom Graber and Ravi Vakil, On the tautological ring of \overline{\Cal M}_{\?,\?}, Turkish J. Math. 25 (2001), no. 1, 237 -- 243. Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), (Equivariant) Chow groups and rings; motives, Coverings in algebraic geometry On the tautological ring of \(\overline{\mathcal M}_{g,n}\). | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X_{1}\) and \(X_{2}\) be two compact strongly pseudoconvex CR manifolds of dimension \(2n-1 \geq 5\) which bound complex varieties \(V_{1}\) and \(V_{2}\) with only isolated normal singularities in \(\mathbb C^{N1}\) and \(\mathbb C^{N2}\) respectively. Let \(S_{1}\) and \(S_{2}\) be the singular sets of \(V_{1}\) and \(V_{2}\) respectively and \(S_{2}\) is nonempty. If \(2n - N_{2} - 1 \geq 1\) and the cardinality of \(S_{1}\) is less than 2 times the cardinality of \(S_{2}\), then we prove that any non-constant CR morphism from \(X_{1}\) to \(X_{2}\) is necessarily a CR biholomorphism. On the other hand, let \(X\) be a compact strongly pseudoconvex CR manifold of dimension 3 which bounds a complex variety \(V\) with only isolated normal non-quotient singularities. Assume that the singular set of \(V\) is nonempty. Then we prove that any non-constant CR morphism from \(X\) to \(X\) is necessarily a CR biholomorphism. strongly pseudoconvex CR manifold; rigidity of CR morphism; geometric genus of compact embeddable CR manifold Yau, S.S.-T., Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds, J. Eur. Math. Soc., 13, 175-184, (2011) CR manifolds, Global theory of complex singularities; cohomological properties, Coverings in algebraic geometry Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The first part of the paper focuses on finding a local detection property for projectivity of modules for Frobenius kernels over an algebraically closed field \(k\) of prime characteristic \(p\). For an example of this, consider a finite group \(G\). \textit{L.~Chouinard} [J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)] showed that a \(kG\)-module \(M\) (even infinite dimensional) is projective if and only if it is projective upon restriction to \(kE\) for all elementary Abelian \(p\)-subgroups \(E\) of \(G\). For a `finite' dimensional module, Chouinard's Theorem was later seen to be a consequence of the extensive theory of cohomological support varieties for finite groups. However, Chouinard's Theorem is actually needed to deduce key properties in a general theory of support varieties for arbitrary modules.
The author considers Frobenius kernels of a smooth algebraic group \(G\) over \(k\) to which a theory of support varieties exists for finite dimensional modules. For such modules, it follows that projectivity can be detected by subgroup schemes that are Frobenius kernels of the additive group \(\mathbb{G}_a\). The natural question arises as to whether or not this property can be deduced directly and further for arbitrary modules. Building upon work of the reviewer [Proc. Am. Math. Soc. 129, No. 3, 671-676 (2001; Zbl 0990.20028)] for unipotent algebraic group schemes, the author shows that the detection property holds in general. The author then proceeds to develop a theory of `support cones' for arbitrary modules and uses the above detection result to derive desired properties of these cones. Lastly, the author gives a description of support cones in terms of Rickard idempotent modules as done for finite groups by \textit{D. Benson, J. Carlson}, and \textit{J. Rickard} [Math. Proc. Camb. Philos. Soc. 120, No. 4, 597-615 (1996; Zbl 0888.20003)]. algebraic groups; Frobenius kernels; projectivity; cohomological support varieties; support cones; Rickard idempotent modules; group schemes Pevtsova, Julia, Infinite dimensional modules for Frobenius kernels, J. Pure Appl. Algebra, 0022-4049, 173, 1, 59\textendash86 pp., (2002) Representation theory for linear algebraic groups, Group schemes, Modular Lie (super)algebras, Modular representations and characters, Group rings of infinite groups and their modules (group-theoretic aspects), Group rings Infinite dimensional modules for Frobenius kernels | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We build on preceding work of \textit{J.-P. Serre} [Comment. Math. Helv. 59, No. 4, 651--676 (1984; Zbl 0565.12014), C. R. Acad. Sci., Paris, Sér. I Math. 311, No. 9, 547--552 (1990; Zbl 0742.14030)], \textit{H. Esnault, B. Kahn} and \textit{E. Viehweg} [J. Reine Angew. Math. 441, 145--188 (1993; Zbl 0772.57028)] and \textit{B. Kahn} [J. Pure Appl. Algebra 97, No. 2, 163--188 (1994; Zbl 0861.55022)] to establish a relation between invariants, in modulo 2 étale cohomology, attached to a tamely ramified covering of schemes with odd ramification indices. The first type of invariant is constructed using a natural quadratic form obtained from the covering. In the case of an extension of Dedekind domains, mains, this form is the square root of the inverse different equipped with the trace form. In the case of a covering of Riemann surfaces, it arises from a theta characteristic. The second type of invariant is constructed using the representation of the tame fundamental group, which corresponds to the covering. Our formula is valid in arbitrary dimension. For unramified coverings the result was proved by the above authors.
The two main contributions of our work consist in (1) showing how to eliminate ramification to reduce to the unramified case, in such a way that the reduction is possible in arbitrary dimension, and, (2) getting around the difficulties, caused by the presence of crossings in the ramification divisor, by introducing what we call ``normalisation along a divisor''. Our approach relies on a detailed analysis of the local structure of tame coverings.
We include a review of the relevant material from the theory of quadratic forms on schemes and of the basic simplicial techniques needed for our purposes. Cassou-Noguès, P.; Erez, B.; Taylor, M. J.: Invariants of a quadratic form attached to a tame covering of schemes. J. th. Des nombres de Bordeaux 12, 597-660 (2000) Quadratic forms over global rings and fields, Algebraic theory of quadratic forms; Witt groups and rings, Galois cohomology, Coverings in algebraic geometry Invariants of a quadratic form attached to a tame covering of schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb F\) be the finite field of \(q\) elements and let \(\mathcal P(n,q)\) denote the projective space of dimension \(n-1\) over \(\mathbb F\). We construct a family \(H^n_{k,i}\) of combinatorial homology modules associated to \(\mathcal P(n,q)\) for coefficient fields of positive characteristic co-prime to \(q\). As \(F\text{GL}(n,q)\)-representations these modules are obtained from the permutation action of \(\text{GL}(n,q)\) on the Grassmannians of \(\mathbb F^n\). We prove a branching rule for \(H^n_{k,i}\) and use this to determine the homology representations completely. Our result include a duality theorem and the characterization of \(H^n_{k,i}\) through the standard irreducibles of \(\text{GL}(n,q)\) over \(\mathbb F\). finite projective spaces; representations; homology decompositions; dualities; Grassmannians; posets J. Siemons and D. Smith, Homology representations of GL(n, q) from Grassmannians in cross-characteristics, to appear. Modular representations and characters, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Combinatorial structures in finite projective spaces, Other homology theories in algebraic topology, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Chain complexes (category-theoretic aspects), dg categories, Duality in applied homological algebra and category theory (aspects of algebraic topology), Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups Some homology representations for Grassmannians in cross-characteristics. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Es wird die absolute Galoisgruppe \(\Lambda =Gal({\bar {\mathbb{Q}}}/{\mathbb{Q}})\) der algebraisch abgeschlossenen Hülle \({\bar {\mathbb{Q}}}\) des Körpers \({\mathbb{Q}}\) der rationalen Zahlen studiert durch ihre Einbettung in die arithmetische Fundamentalgruppe \(\Gamma\) und ihre Operation via Konjugation auf der zugehörigen algebraischen Fundamentalgruppe \(\Pi\) bezüglich eines aufgeschlossenen algebraischen Funktionskörpers \(K/{\mathbb{Q}}\) vom Geschlecht g und einer endlichen Teilmenge S der Primdivisoren von \({\mathbb{Q}}(t):\) nach Satz 1.2 ist \(\Gamma\) semidirektes Produkt des Normalteilers \(\Pi\) mit zu \(\Lambda\) isomorphen Komplementen. Dabei entstehen \(\Gamma\) und \(\Pi\) aus K und S wie folgt: Es sei \(\bar K={\bar {\mathbb{Q}}}(t)\), \(\bar S\) die Menge der Primdivisoren von \(\bar K,\) die als Primteiler eines \(p\in S\) auftreten, und \(\bar M\) der maximale außerhalb \(\bar S\) unverzweigte Erweiterungskörper von \(\bar K.\) Dann ist \(\bar M/\)K galoissch (1.1) und \(\Gamma:=Gal(\bar M/K)\), \(\Pi:=Gal(\bar M/\bar K)\). Hier ist die Struktur von \(\Pi\) als freie proendliche Gruppe \(\Phi_ r\) vom Rang \(r=2g+s-1\) (proendliche Komplettierung der Fundamentalgruppe einer an \(s=| \bar S|\) Stellen gelochten topologischen bzw. Riemannschen Fläche vom Geschlecht g) bekannt.
Der erste Aspekt ist die konkrete Einbettung von \(\Lambda\) in Out \(\Phi_ r\). Der zweite Aspekt, die explizite Beschreibung der Operation von \(\Lambda\) auf der Kommutatorfaktorgruppe \(\Phi_ r/\Phi'_ r\) (Satz 2.2), erlaubt die Herleitung von Bedingungen, unter denen sich endliche Faktorgruppen von \(\Phi_ r\) (mit Erzeugendensystem \(Z=(\rho_ 1,...,\rho_{2g},\sigma_ 1,...,\sigma_ s)\), das die Relation \([\rho_ 1,\rho_ 2]...[\rho_{2g-1},\rho_{2g}]\sigma_ 1...\sigma_ s=1\) von \(\Phi_ r\) erfüllt) als Galoisgruppen über K nachweisen lassen.
Für eine Galoiserweiterung \(\bar N/\bar K\) mit Galoisgruppe G werden Definitionskörper K'\(\geq K\) gesucht, also algebraische Funktionskörper \(K'/{\mathbb{Q}}\) mit genauem Konstantenkörper \({\mathbb{Q}}\) und einer regulären Körpererweiterung \(N/K'\) mit \(K'\otimes_{{\mathbb{Q}}} {\mathbb{Q}}'=\bar K\), \(N \otimes_{{\mathbb{Q}}} {\bar {\mathbb{Q}}}=\bar N\). Der Autor findet obere Schranken für den Grad [K':K], die rein gruppentheoretisch aus G und Z zu berechnen sind. Insofern werden die Ergebnisse seiner Habilitationsschrift [J. Reine Angew. Math. 349, 179-220 (1984; Zbl 0555.12005)] verfeinert.
Als Anwendungen diskutiert der Autor die Realisierung der Gruppen \(SL_ 2({\mathbb{F}}_ q)\) und \(PSL_ 2({\mathbb{F}}_ q)\), \(q=p^ f\), als Galoisgruppen über Kreisteilungskörpern bzw. \({\mathbb{Q}}\) und zeigt, daß die Mathieugruppen \(M_{12}\) und \(M_{22}\) als Galoisgruppen über \({\mathbb{Q}}\) realisiert werden können. Mit diesen Methoden wurden auch die Jankogruppen \(J_ 1\) und \(J_ 2\) [\textit{G. Hoyden- Siedersleben}, J. Algebra 97, 17-22 (1985; Zbl 0574.12011)] und \(M_{11}\) [Verf. und \textit{A. Zeh-Marschke}, J. Number Theory 23, 195-202 (1986)] über \({\mathbb{Q}}\) realisiert. Bezüglich weiterer einfacher Gruppen siehe die neun Artikel im Kapitel ''Rigidity und Galois Groups'', Proc. Rutgers Group Theory Year, 1983-1984 (1984)] und Verf. [Manuscr. Math. 51, 253-265 (1985; Zbl 0574.12010)], zum Einbettungsproblem bei zusammengesetzten Gruppen siehe Verf. [Invent. Math. 80, 365-374 (1985; Zbl 0567.12015)]. inverse problem of Galois theory, algebraic function field; arithmetic fundamental group; algebraic fundamental group; special linear group; SL(2,q); Mathieu group; \(M_{12}\); \(M_{22}\) Matzat, B.H.: Zwei Aspekte konstruktiver Galoistheorie,J. Algebra 96 (1985), 499--531 Galois theory, Representations of groups as automorphism groups of algebraic systems, Arithmetic theory of algebraic function fields, Finite simple groups and their classification, Simple groups: sporadic groups, Coverings in algebraic geometry Zwei Aspekte konstruktiver Galoistheorie. (Two aspects of constructive Galois theory) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies a (local) analytic cover \(X\to {\mathbb{C}}^ 2\) which is Galois with a dihedral group \(D_ m\) as its group and whose ramification locus is a cusp \(u^ p=v^ q\). An example of such cover is given by \(h_ m: (x,y)\to (u,v)=((x^ m+y^ m)/2,xy),\) with \(p=2\). It is proved that any such cover is induced from \(h_ m\). analytic cover; dihedral group; ramification locus G. Teodosiu, Some analytic dihedral coverings, Stud. Cerc. Mat. 40 (1988), 155-159. Zbl0646.14013 MR952368 Coverings in algebraic geometry, Complex singularities, Singularities in algebraic geometry Some analytic dihedral coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a reduced connected \(k\)-scheme pointed at a rational point \(x\in X(k)\). By using tannakian techniques we construct the Galois closure of an essentially finite \(k\)-morphism \(f:Y\to X\) satisfying the condition \(H^{0}(Y,\mathcal O_{Y})=k\); this Galois closure is a torsor \(p: \hat X_Y \to X\) dominating \(f\) by an \(X\)-morphism \(\lambda: \hat X_Y \to Y\) and universal for this property. Moreover, we show that \(\lambda : \hat X_Y \to Y\) is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over \(Y\) is still an essentially finite vector bundle over \(X\). We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor \(f:Y\to X\) under a finite group scheme satisfying the condition \(H^{0}(Y,\mathcal O_{Y})=k, Y\) has a fundamental group scheme \(\pi _{1}(Y,y)\) fitting in a short exact sequence with \(\pi _{1}(X,x)\). essentially finite; finite group scheme; Galois closure Antei, M.; Emsalem, M., Galois closure of essentially finite morphisms, J. Pure Appl. Algebra, 215, 11, 2567-2585, (2011) Coverings in algebraic geometry, Group schemes, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Separable extensions, Galois theory Galois closure of essentially finite morphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E\) be an elementary Abelian \(p\)-group of rank \(r\) and let \(k\) be an algebraically closed field of characteristic \(p\). We prove that if \(M\) is a \(kE\)-module of stable constant Jordan type \([a_1]\dots [a_t]\) with \(\sum_ja_j\leq\min(r-1,p-2)\) then \(a_1=\cdots=a_t=1\). The proof uses the theory of Chern classes of vector bundles on projective space. modular representations; elementary Abelian groups; modules of constant Jordan type; Chern characters; vector bundles; Chern classes Benson, D, Modules of constant Jordan type with small non-projective part, Algebr. Represent. Theory, 1, 29-33, (2013) Modular representations and characters, Group rings of finite groups and their modules (group-theoretic aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli Modules of constant Jordan type with small non-projective part. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study certain deformations of bidouble Galois covers of the product of two complex projective lines in connection with examples of simply connected algebraic surfaces which are diffeomorphic but not deformation equivalent. The discriminant surface of the deformation space is shown to be a quartic hypersurface isomorphic to the discriminant of the space of degree three polynomials on the projective line. Several nice properties of this hypersurface are established. The local braid monodromy of the deformed degree four coverings is determined. The local deformed branch curves that appear are the classical three-cuspidal plane quartics with a bitangent at infinity. braid monodromy; branched coverings F Catanese, B Wajnryb, The \(3\)-cuspidal quartic and braid monodromy of degree 4 coverings (editors C Ciliberto, A V Geramita, B Harbourne, R M Miró-Roig, K Ranestad), de Gruyter (2005) 113 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Varieties of low degree, Low-dimensional topology of special (e.g., branched) coverings, Topological properties of mappings on manifolds, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Coverings in algebraic geometry The 3-cuspidal quartic and braid monodromy of degree 4 coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, a finite ramified covering is a generically finite morphism of complete non-singular varieties over the complex number field \({\mathbb{C}}\). If S is a hypersurface in \({\mathbb{P}}^ n\) with only normal crossings and if \(S_ 1,S_ 2,...,S_ m\) are the irreducible components of S, one has \(\pi_ 1({\mathbb{P}}^ n-S)\cong {\mathbb{Z}}\alpha_ 1\oplus {\mathbb{Z}}\alpha_ 2\oplus...\oplus {\mathbb{Z}}\alpha_ m/(n_ 1\alpha_ 1+n_ 2\alpha_ 2+...+n_ m\alpha_ m)\), where \(\alpha_ i\) is a small loop around \(S_ i\) and \(n_ i\) is the degree of \(S_ i\) [cf. \textit{M. Oka}, Q. J. Math., Oxf., II. Ser. 28, 229-242 (1977; Zbl 0352.14015)]. Let \({\mathbb{H}}_{\nu}\) \((\nu =1,2,...)\) be the subgroup of \(\pi_ 1({\mathbb{P}}^ n-S)\) generated by the elements \(\nu n_ 1\alpha_ 1,\nu n_ 2\alpha_ 2,...,\nu n_ m\alpha_ m\). We study a certain finite ramified covering \(f_{\nu}: K_{\nu}\to {\mathbb{P}}^ n\) such that the induced morphism \(K_{\nu}-f_{\nu}^{-1}(S)\to {\mathbb{P}}^ n-S\) is the unramified covering associated with \({\mathbb{H}}_{\nu}\). This covering was first constructed by \textit{S. Kawai} [Comment. Math. Univ. St. Pauli 30, 87-103 (1981; Zbl 0496.14012)]. So we call it the \(\nu\)-th Kawai covering of \({\mathbb{P}}^ n\) associated with S. Let S be a hypersurface in \({\mathbb{P}}^ n\) with only normal crossings and let K be the \(\nu\)-th Kawai covering space of \({\mathbb{P}}^ n\) associated with S. The author gives two tables where are calculated in which cases the canonical bundle \(\omega_ K\) of \(K\) and the dual line bundle \(\omega^*_ K\) of \(\omega_ K\) are ample, trivial, semiample and there are also calculated the Kodaira dimension \(\kappa(K)\) of \(K\) and \(\kappa (\omega^*_ K,K)\). The different cases are given by inequalities among: \(\nu\), \(m=\) number of irreducible components of \(S\), \(d=\) degree of \(S\). fundamental group; finite ramified covering; Kawai covering space; canonical bundle; dual line bundle; Kodaira dimension Coverings in algebraic geometry, Ramification problems in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the canonical line bundles of some finite ramified covering spaces of \(P^ n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let f: \(Y\to X\) be a finite locally abelian Galois covering of complex algebraic surfaces having at most Hirzebruch-Jung singularities. The author continues in this paper his investigations on the proportionality relation \(C(Y)=(\deg (f))C(X)\), where C(X and C(Y) are the fundamental numbers of X and Y, computed from the corresponding Chern numbers and some contributions from the ramification divisors of f (in X and Y) and from the singularities occuring in this situation. Galois covering; Hirzebruch-Jung singularities; fundamental numbers; ramification divisors R.-P. Holzapfel, Chern number relations for locally abelian Galois coverings of algebraic surfaces, Math. Nachr. 138 (1988), 263 -- 292. Coverings in algebraic geometry, Characteristic classes and numbers in differential topology, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Chern number relations for locally abelian Galois coverings of algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors give a new singularity-theory-proof of a formula conjectured by \textit{A. Hurwitz} [Math. Ann. 39, 1-61 (1891; JFM 23.0429.01)] on the number of topological types of rational functions on \({\mathbb C}\) with fixed critical values, fixed orders of poles, and which are Morse off of their poles. (Combinatorial proofs of this formula were found by \textit{M. Crescimanno} and \textit{W. Taylor} [Nucl. Phys. B 437, 3-24 (1995; Zbl 0972.81165)] (in a special case), by \textit{I. P. Goulden} and \textit{D. M. Jackson} [Proc. Am. Math. Soc. 125, 51-60 (1997; Zbl 0861.05006)], and by \textit{V. Strehl} [Sémin. Lothar. Comb. 37, 12 pp. (1996; Zbl 0886.05006)].) The proof in the present paper is inspired by \textit{V. I. Arnold}'s work [Funct. Anal. Appl. 30, 1-14 (1996; Zbl 0898.32019)] on Laurent polynomials and uses the geometry of the moduli space of ordered tuples of points on the line and properties of the corresponding Lyashko-Looijenga mapping.
Also, it is shown that the variety of topological types of Morse functions as above is an Eilenberg-MacLane \(K(\pi,1)\)-space, where \(\pi\) is a subgroup of the braid group of specified index. topological types of rational functions; Lyashko-Looijenga mapping; variety of topological types of Morse functions; Eilenberg-MacLane \(K(\pi,1)\)-space V. V. Goryunov and S. K. Lando, On enumeration of meromorphic functions of the line, The Arnoldfest (Toronto, 1997), Fields Inst. Commun. 24, A.M.S., Providence, R.I., 1999. CMP 2000:08 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Automorphisms of infinite groups, Enumeration in graph theory, Coverings in algebraic geometry On enumeration of meromorphic functions on the line | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f\colon Y \to {\mathbb P}^n\) be an abelian cover, namely \(Y\) is a normal complex variety such that there exists a finite abelian group \(G\) that acts faithfully on \(Y\), the quotient \(Y/G\) is isomorphic to \({\mathbb P}^n\) and \(f\) is the quotient map. Such a map \(f\) is determined by the components of the branch locus together with some combinatorial data related to the \(G\)-action above each component (an algebraic treatment of the general theory of abelian covers can be found in [Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 14, No. 5, 695--705 (2007; Zbl 0721.14009)]). Here the author determines necessary and sufficient conditions on a collection of irreducible hypersurfaces of \({\mathbb P}^n\) and the corresponding combinatorial data for the existence of a smooth \(G\)-cover of \(f\colon Y \to {\mathbb P}^n\) associated with it. In addition, he studies in detail the case in which \(Y\) is a Calabi--Yau manifold, listing all the possibilities for small values of \(n\). abelian cover; projective space; Calabi--Yau manifold DOI: 10.1007/s00229-007-0112-4 Coverings in algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects), Calabi-Yau theory (complex-analytic aspects), Group actions on varieties or schemes (quotients) Smooth finite abelian uniformizations of projective spaces and Calabi-Yau orbifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems To any finite covering \(f:Y\to X\) of degree \(d\) between smooth complex projective manifolds, one associates a vector bundle \(E_f\) of rank \(d-1\) on \(X\) whose total space contains \(Y\). It is known that \(E_f\) is ample when \(X\) is a projective space [\textit{R. Lazarsfeld}, Math. Ann. 249, 153--162 (1980; Zbl 0434.32013)], a Grassmannian [\textit{L. Manivel}, Invent. Math. 127, No.2, 401--416 (1997; Zbl 0906.14011)], or a Lagrangian Grassmannian \textit{M. Kim} and \textit{L. Manivel}, Topology 38, No. 5, 1141--1160 (1999; Zbl 0935.14008)]. We show an analogous result when \(X\) is a simple abelian variety and \(f\) does not factor through any nontrivial isogeny \(X'\to X\). This result is obtained by showing that \(E_f\) is \(M\)-regular in the sense of \textit{G. Pareschi} and \textit{M. Popa} [J. Am. Math. Soc. 16, No. 2, 285--302 (2003; Zbl 1022.14012)], and that any \(M\)-regular sheaf is ample. O. Debarre, On coverings of simple abelian varieties, Bull. Soc. Math. France 134 (2006), 253-260. Coverings in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Isogeny, Algebraic theory of abelian varieties, Subvarieties of abelian varieties On coverings of simple abelian varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, a space is either a rigid analytic space or the Berkovich space associated to this. The notion of étale covering map is introduced. This notion generalizes both finite étale coverings and topological coverings. On this notion the definition of the étale fundamental group is build. This group has a topology, is prodiscrete and maps to the topological and the algebraic fundamental group. For curves, i.e., spaces of dimension 1, some results on this étale fundamental group are given. The example of the étale covering map \(\log: \{z\in\mathbb{C}_p \mid|z-1 |<1\} \to\mathbb{C}_p\) seems to show that, even for the affine line, the étale fundamental group is too big to write down explicitly. Local \(\mathbb{Q}_l\)-systems are introduced and the connection with \(\mathbb{Q}_l\)-representations of the étale fundamental group is proven. Up to this point, the spaces were Berkovich spaces. The theory is then translated to the case of rigid analytic spaces. The motivation for developing the theory of étale fundamental groups comes from the work of Rapoport and Zink on moduli spaces for certain \(p\)-divisible groups and \(p\)-adic period maps. It is proved here that (a modification of) the period map is an étale covering map. Further results on the \(F\)-crystals and local systems, connected with the \(p\)-adic period map, are given. In particular, the \(p\)-adic period map provides an étale covering map of a projective space over, say, \(\mathbb{C}_p\). Thus the étale fundamental group of this projective space is not trivial! \(F\)-crystals; \(p\)-divisible groups; \(p\)-adic period maps; rigid analytic space; Berkovich space; étale covering map; étale fundamental group De Jong, A. J., Étale fundamental groups of non-Archimedean analytic spaces, Compositio Math., 97, 1-2, 89-118, (1995) Local ground fields in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Formal groups, \(p\)-divisible groups Étale fundamental groups of non-Archimedean analytic spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite group scheme over an algebraically closed field \(k\) of characteristic \(p>0\). That is, \(G\) is a group scheme whose coordinate algebra \(k[G]\) is finite-dimensional. Let \(kG:=\Hom_k(k[G],k)\) be the associated group algebra, a finite-dimensional cocommutative Hopf algebra. Associated to a block \(B\) of \(kG\) are various support spaces whose relationships are studied in this paper.
For a finite-dimensional \(G\)-module \(M\), let \(V_G(M)\) denote its cohomological support variety. For a block \(B\) of \(kG\), if \(B\) is considered as a \(kG\)-module via the left regular representation, then it is projective and its support variety is trivial. To obtain a more interesting variety, let \(V_G(\mathcal B)\) be the union of the support varieties of all modules in \(B\). This has been seen for example by work of \textit{R. Farnsteiner} [Bull. Lond. Math. Soc. 39, No. 1, 63-70 (2007; Zbl 1121.16015)] to give information on the representation type of \(B\).
In the first main result of the paper, it is shown that if \(B\) is instead considered as a \(kG\)-module via the adjoint representation, then \(V_G(\mathcal B)=V_G(B)=V_G(M)\) where \(M\) is an indecomposable summand of \(B\).
The variety \(V_G(\mathcal B)\) is not an intrinsic invariant of \(B\) since it depends on the underlying group \(G\). Considering the block \(B\) as an algebra in its own right, one can consider an alternate ``support variety'' by using Hochschild cohomology \(HH^*(B)\). Let \(X_B\) denote the maximal spectrum of \(HH^*(B)\), a variety which is an intrinsic invariant of \(B\). The author shows that there is a finite surjective map of varieties \(X_B\to V_G(\mathcal B)\). In the case of ordinary finite groups and the principle block \(B_0\), \textit{M. Linckelmann} [J. Lond. Math. Soc., II. Ser. 81, No. 2, 389-411 (2010; Zbl 1201.20005)] showed that \(X_{B_0}\cong V_G(k)\) by showing that \(HH^*(B_0)\cong H^*(G,k)\) modulo nilpotents. In the general setting of finite group schemes, the author shows that this holds at least under the assumption that either the principle block \(B_0\) has a single simple module or \(V_G(k)\) is one-dimensional.
\textit{E. M. Friedlander} and \textit{J. Pevtsova} [Am. J. Math. 127, No. 2, 379-420 (2005; Zbl 1072.20009)] gave a non-cohomological or purely representation-theoretic characterization of \(V_G(M)\) in terms of a space \(P(G)_M\) of equivalence classes of so-called \(p\)-points. A \(p\)-point associated to \(M\) is a certain map from \(k[t]/(t^p)\to kG\) such that the restriction of \(M\) to \(k[t]/(t^p)\) (via this map) is not projective. As above, let \(P(G)_{\mathcal B}\) denote the union of all \(P(G)_M\) for \(M\) in \(B\). Again, \(P(G)_{\mathcal B}\) depends not only on \(B\) but also on \(G\). On the other hand, following \textit{R. Farnsteiner} [Contemp. Math. 442, 61-87 (2007; Zbl 1185.16023)], the author defines a space \(F(B)\) of equivalence classes of flat maps \(k[t]/(t^p)\to B\) which is intrinsic to \(B\). It is shown that the projection of \(kG\to B\) induces an injective and continuous map \(P(G)_{\mathcal B}\to F(B)\). Parallel to the above result, it is shown that if the principle block contains a single simple module, then the map \(P(G)_{\mathcal B_0}\to F(B_0)\) is a homeomorphism. A key ingredient to getting that homeomorphism is showing that if \(G\) is a unipotent finite group scheme, then every flat map is equivalent to a \(p\)-point. This latter result is interesting in its own right in that it allows one to potentially reformulate the definition of a \(p\)-point. finite group schemes; blocks; cohomological support varieties; \(p\)-points; Hochschild cohomology; flat maps; simple modules; group algebras; cocommutative Hopf algebras (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Group schemes, Modular representations and characters, Hopf algebras and their applications, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Cohomology of groups Module invariants and blocks of finite group schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This note answers the following question which arose in a paper by \textit{J. de Jong} and \textit{M. van der Put} [Doc. Math., J. DMV 1, 1--56 (1996; Zbl 0922.14012); remark (2.5.11)]: Let \(X\) be a connected one-dimensional and separated rigid space over some complete, non-archimedean valued field \(k\). Does \(X\) have an admissible covering by affinoid subsets \(\{X_i\}\), such that each \(X_i\) meets only finitely many \(X_j\)?
In the terminology of the paper cited above, such a covering is called locally finite and an \(X\) having such a covering is called paracompact. We note that if \(X\) is connected and has a locally finite admissible covering by affinoids, then this covering is at most countable. The analogue of the question over the field of complex numbers is Radon's theorem which states that a connected Riemann surface is a countable union of open subsets isomorphic to the open unit disk in \(\mathbb C\). We will prove that the question above has an affirmative answer and we will show the following stronger statement:
There exists an admissible formal scheme \({\mathcal X}\) over the valuation ring \(R\) of \(k\) (i.e. \({\mathcal X}\) is flat over \(R\) and locally topologically of finite type) which is separated, such that \({\mathcal X}\) has a locally finite covering by affine subsets and such that its `generic fibre' \({\mathcal X}\otimes k\) coincides with \(X\).
Special cases of the statement above have already been proved. In section 3, after proving a Mittag-Leffler decomposition theorem, we show that \(X\) is a quasi-Stein space if no irreducible component of \(X\) is complete. Counterexamples for higher dimensional rigid spaces complete this note. one-dimensional rigid space; non-archimedean valued ground field; covering by affinoid subsets Q Liu and M. Van Der Put , On one dimesional separated rigid spaces . To appear in Indagationes Mathematicae. Local ground fields in algebraic geometry, Coverings in algebraic geometry, Arithmetic ground fields for curves On one-dimensional separated rigid spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective surface over \(\mathbb{C}\). Suppose \(i \colon C \hookrightarrow X\) is a smooth curve and \(A\) a complete, base point free linear series over \(C\). Then the \textit{(dual) Lazarsfeld-Mukai (LM) bundle} is the bundle associated to the locally free sheaf \(F\) in the sequence
\[ 0 \ \to \ F \ \to \ H^0 ( C, A ) \otimes {\mathcal O}_X \ \to \ i_* A \ \to \ 0 . \]
LM bundles are much studied and continue to have a wide variety of applications to the study of linear series on curves and elsewhere, as outlined in the introduction to the present paper.
The author defines two types of natural parabolic structure arising from LM bundles. Firstly, let \(a_1, a_2\) be weights with \(0 \le a_1 < a_2 < 1\). A parabolic structure on \(F\) along \(C\) is given by
\[ {\mathfrak F}_C: F|_C \ \supset_{a_1} \ A \otimes {\mathcal O}_X ( - C )|_C \ \supset_{a_2} \ 0 . \]
The parabolic bundle \(F_* := ( F, {\mathfrak F}_C, a_1, a_2 )\) is called a \textit{dual parabolic LM bundle}.
The second type of parabolic structure is defined using blowups. Let \(\pi \colon \tilde{X} \to X\) be the blowup at a point \(q \in i(C)\), and \(E\) the exceptional divisor. For weights \(b_1, b_2\) with \(0 \le b_1 < b_2 < 1\), the author considers the parabolic structure on \(\tilde{F} := \pi^* F\) along \(E\) given by the filtration by trivial bundles
\[ \tilde{\mathfrak F}_C: \tilde{F}|_E \ \supset_{b_1} \ M \otimes {\mathcal O}_E \ \supset_{b_2} \ 0 \]
where \(M = \mathrm{Ker} \left( F(q) \to H^0 (C, A) \right)\). The parabolic bundle \(\tilde{F}_* := ( \tilde{F}, \tilde{\mathfrak F}_C, b_1, b_2 )\) is called the \textit{dual blown up parabolic LM bundle}.
In Theorems 1.1 and 1.2, the author gives sufficient conditions for the parabolic stability of \(F_*\) and \(\tilde{F}_*\) with respect to certain ample line bundles on \(X\) and \(\tilde{X}\). These are used in conjunction with results of Lelli-Chiesa [Zbl 1278.14053] and results of the author [Commun. Algebra 46, No. 4, 1698--1708 (2018; Zbl 1458.14012)] to give examples of stable parabolic bundles on \(\mathbb{P}^2\) and on \(K3\) surfaces respectively.
Next, recall that an \textit{orbifold bundle} on a connected smooth projective variety \(Y\) is a bundle \(E\) over \(Y\) admitting a lifting of the action of a subgroup \(G \subseteq \mathrm{Aut}(Y)\). Moreover, to the pair \((X, C)\) one may associate a \textit{Kawamata cover} \(p\colon Y \to X\) such that \(p^* C\) is a multiple \(NC'\) of a smooth curve \(C' \subset Y\) (see [\textit{R. Lazarsfeld}, Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Berlin: Springer (2004; Zbl 1093.14501); Prop 4.1.12]). \textit{I. Biswas} [Duke Math. J. 88, No. 2, 305--325 (1997; Zbl 0955.14010)] established a correspondence between orbifold bundles on Kawamata covers of \(X\) and certain parabolic bundles on \(X\). The author shows (Theorem 1.3) that for \(1 \le m < N\), the orbifold bundle corresponding to the dual parabolic LM bundle \(\left( F, {\mathfrak F}_C, 0 , \frac{N-m}{N} \right)\) is itself a dual LM bundle on \(Y\); namely, the bundle \(F'\) defined by the sequence
\[ 0 \ \to \ F' \ \to \ H^0 ( C, A ) \otimes {\mathcal O}_Y \ \to \ (j_Y^m)_* A_m \ \to \ 0 , \]
where \(A_m\) is the pullback of \(A\) to the nonreduced curve \(mC' \subset Y\), and \(j_Y^m\) is the inclusion \(mC' \hookrightarrow Y\). This is applied in Theorem 1.4 to give new examples of semistable LM bundles on covers of \(\mathbb{P}^2\) and of \(K3\) surfaces. parabolic bundle; semistability; Lazarsfeld-Mukai bundle; surface Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Semistability of Lazarsfeld-Mukai bundles via parabolic structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In previous papers, ideas of Deligne are used to prove the factoriality of the surface \(Z^p = f(X,Y)\) for a generic choice of the polynomial \(f(X,Y)\) of arbitrary degree \(\geq 4\) (with \(p \geq 3)\). In this paper we study the class group of the surface \(Z^n = f(X,Y)\) for arbitrary positive integer \(n\). The above mentioned calculation leads us naturally to conjecture that the class group of \(Z^n = f(X,Y)\) is factorial for a generic choice of \(f\). To be more precise, let \(f = \sum T_{ij} X^iY^j\) be a generic polynomial with indeterminate coefficients and let \(A_n = K[X,Y,Z]/(Z^n - f)\) where \(K\) is the algebraic closure of \(\mathbb{F}_p (T_{ij})\) with \(\mathbb{F}_p\) the prime field of \(p\) elements \((p \geq 3)\). Assume the degree of \(f\) is at least 4. Then we conjecture:
\[
\text{for all }n \in \mathbb{F}^+,\;A_n \text{ is factorial}. \tag{*}
\]
In this paper we prove that (*) reduces to the case \(\text{g.c.d.} (p,n) = 1\). In section 1 and 2, descent techniques are used to study the class group of arbitrary surfaces \(Z^n = f\). In section 3 the reduction of (*) to the case \(\text{g.c.d.} (p,n) = 1\) is accomplished by analyzing the action of \({\mathcal G} = \text{Gal} (K, \mathbb{F}_p (T_{ij}))\) on the divisor class group of \(Z^{pm} = f\). simple coverings of the affine plane; class group of surface; divisor class group Coverings in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Class groups Generic simple coverings of the affine plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X_{p,q}\) be elliptic surfaces over \({\mathbb{P}}_ 1\) with at most 2 multiple fibres of multiplicities p and q. It is known that \(\pi_ 1(X_{p,q})={\mathbb{Z}}/k\) where \(k=g.c.d.(p,q)\). In the special case where \(k=1\) and \(p_ g=0\), one has the following theorem:
The surfaces \(X_{2,q}\) with \(q=2n+1\) are pairwise differentiably inequivalent [see e.g. \textit{C. Okonek} and \textit{A. Van de Ven}, Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)].
This result is interesting for two reasons; first of all, the surfaces \(X_{p,q}\) are all homeomorphic to \({\mathbb{P}}_ 2\) with 9 points blown up [see \textit{M. H. Friedman}, J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)]. Furthermore, this theorem is in sharp contrast with a recent result by \textit{M. Ue} [Invent. Math. 84, 633-643 (1986; Zbl 0595.14028)] who shows that for elliptic surfaces with at least 3 multiple fibres or elliptic surfaces over base curves S with \(g(S)>1\), their diffeomorphism type is completely determined by their homeomorphism type.
In this paper, using their previous work and by computing Donaldson's invariants, the authors generalize the previous theorem to the cases where \(k>1\). Also notice that, at least for odd k, the \(X_{p,q}\) are homeomorphic in view of recent results by Hambleton and Kreck. cyclic fundamental group; elliptic surfaces; diffeomorphism type; homeomorphism type Lübke, M.; Okonek, C.: Differentiable structures on elliptic surfaces with finite fundamental group. Compositio math. 63, 217-222 (1987) Elliptic surfaces, elliptic or Calabi-Yau fibrations, Coverings in algebraic geometry, Differential topological aspects of diffeomorphisms Differentiable structures of elliptic surfaces with cyclic fundamental group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This expository article, which is intended to address a wide audience from geometry in general, is a short introduction to the so-called Shafarevich conjecture on the universal covers of algebraic manifolds and describes the current state of the conjecture. The exposition is lucid and accessible to non-specialists, but the reader should be alerted that there are many typographical errors.
The author starts with general principles of birational classification theory and connects the principles with fundamental groups, thereby stating his famous conjecture to the effect that the universal covering of a smooth complex projective variety should be holomorphically convex in the sense of the theory of analytic functions of several variables. A more geometric interpretation of the conjecture is as follows.
Let \(X\) be a smooth complex projective variety and \(\pi:U(X)\to X\) the universal covering. Then there should be a surjective holomorphic mapping (the Remmert reduction) \(\varphi:U(X)\to Y\) onto a Stein space \(Y\) with fibres connected and compact. If this is the case, the action of the fundamental group \(\pi_1(X)\) on \(U(X)\) induces a commutative diagram
\[
\begin{tikzcd} U(X) \ar[r,"\varphi"] \ar[d,"\pi" '] & Y \ar[d] \\ X \ar[r,"\rho" '] & R \end{tikzcd}
\]
with \(R\) compact and \(\rho\) surjective, the fibres of \(\varphi\) and \(\rho\) being identical. An alternative description of \(R\) would be that it is a sort of universal family of subvarieties \(Z\subset X\) such that the images of \(\pi_1(Z)\) in \(\pi_1(X)\) are finite. Though the conjecture itself is still far from being proved, a construction of \(R\) which satisfies the universal property above (with some minor modifications) was constructed by \textit{J. Kollár} [``Shafarevich maps and automorphic forms'' (Princeton 1995; Zbl 0871.14015)]. The article ends with a list of applications of the construction of \(R\) and new questions. Shafarevich conjecture; universal covering; Remmert reduction Shafarevic, I. R.: Basic algebraic geometry. (1977) Coverings in algebraic geometry, Holomorphically convex complex spaces, reduction theory, Homotopy theory and fundamental groups in algebraic geometry Classification, fundamental groups and universal covering spaces of algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(g\geq 3\), \(l\) a prime with exponent \(\nu\) in \(2g-2\). The author proves that there are infinitely many proper smooth geometrically connected genus \(g\) curves \(C\) over number fields \(K\) such that for any \(L\)-rational point \(x\) with \([L:K]\) finite and not divisible by \(l^\nu\), the group
\[
\rho_{A,C,x}(G_L)\cap \text{Inn}\,\Pi_g^{(l)}
\]
is infinite, where \(G_L\) is the absolute Galois group of \(L\), \(\Pi_g^{(l)}\) the pro-\(l\) completion of the geometric fundamental group of \(C\), \(\rho_{A,C,x}:G_K\to \Aut\, \Pi_g^{(l)}\) the automorphism representation. number field; Galois representation M. Matsumoto: Difference between Galois representations in automorphism and outer- automorphism groups of a fundamental group, Proc. Amer. Math. Soc. 139 (2011), 1215-1220. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Coverings in algebraic geometry, Coverings of curves, fundamental group Difference between Galois representations in automorphism and outer-automorphism groups of a fundamental group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a complete valuation ring with equal characteristic \(p>0\), with fraction field \(K\) and residue field \(k\). Let \(X\) be a formal \(R\)-scheme of finite type, which is normal, connected and flat over \(R\) such that both the generic fiber \(X_K\) and the special fiber \(X_k\) over \(\text{Spec} R\) are geometrically integral. In this paper studied is how an étale \(\mathbb{Z}/p^n\mathbb{Z}\)-torsor over the generic fiber \(X_K\) extends to a torsor over \(X\) under a suitable group scheme for \(n=1,2\) under certain condition. In the case \(n=1\), the group schemes that arise as the group of the torsor are determined. In the case \(n=2\), the degeneration of étale \(\mathbb{Z}/p^2\mathbb{Z}\)-torsors is classified with locally given ``explicit canonical integral equations'' that describe the reduction of \(p^2\)-cyclic covers in equal characteristic \(p>0\). Artin-Schreier-Witt theory; torsors; degeneration; group schemes Mohamed Saïdi, On the degeneration of étale \?/\?\? and \?/\?²\?-torsors in equal characteristic \?>0, Hiroshima Math. J. 37 (2007), no. 2, 315 -- 341. Coverings of curves, fundamental group, Coverings in algebraic geometry, Fibrations, degenerations in algebraic geometry, Formal methods and deformations in algebraic geometry On the degeneration of étale \(\mathbb Z/p\mathbb Z\) and \(\mathbb Z/p^2\mathbb Z\)-torsors in equal characteristic \(p>0\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a connected complex manifold \(M\), a finite branched covering of \(M\) is a finite proper holomorphic mapping \(f:X\to M\) of an irreducible normal complex space \(X\) onto \(M\). The first named author of this paper studied the moduli space of equivalence classes of finite branched coverings of the projective line \(P^1\) in his previous paper [\textit{M. Namba}, `Families of meromorphic functions on compact Riemann surfaces', Springer, Berlin (1979; Zbl 0417.32008)]. To compactify the moduli space of equivalence classes of finite branched coverings, we have to consider degeneration of branched coverings.
The authors study degenerate families of finite branched coverings of \(P^1\) and the \(m\)-dimensional complex projective space \(P^m\). The authors first introduce a picture which topologically represents a finite branched covering of the complex projective line. The authors call such a picture a Klein picture. The authors prove, among other things, that the topological structure of a degenerating family of finite brached coverings of \(P^1\) can be determined by the permutation monodromy of the general fiber and the braid monodromy of the family. finite branched covering; Klein picture M. Namba and M. Takai, Degenerating families of finite branched coverings, Osaka J. Math. 40 (2003), no. 1, 139-170. Complex-analytic moduli problems, Fibrations, degenerations in algebraic geometry, Coverings in algebraic geometry, Moduli, classification: analytic theory; relations with modular forms Degenerating families of finite branched coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We note that the incidence cycles of a seminormal curve \(X\) intervene in the calculation of the arithmetic genus \(p_ a(X)\), of the algebraic fundamental group \(\pi_ 1^{\text{alg}}(X)\) and of the Picard group \(\text{Pic}(X)\) of \(X\). Really we do not consider only seminormal curves, but more generally varieties obtained from a smooth variety by glueing a finite set of points.
Let \(X\) be a connected variety (by a variety we mean a reduced quasi- projective scheme over an algebraically closed field \(k\)) of pure dimension \(r\) whose singular locus \(\text{Sing}(X)\) consists of a finite set of points \(P_ 1,\dots,P_ m\), such that the normalization \(\overline{X}\) of \(X\) is a smooth variety having \(n\) connected components, every one of them of dimension \(r\), \(\overline{X}_ 1,\dots,\overline{X}_ n\) and the normalisation morphism \(\pi: \overline{X} \to X\) is the composition of a finite number of glueing morphisms, that is every singular point of \(X\) is the glueing of a finite number of points of \(\overline{X}\). Let \(M\) be the number of the points of \(\pi^{-1}(\text{Sing}(X))\); we define \(\nu(X) = M - m - n+1\). We associate to \(X\) the graph \(\Gamma\) whose vertices are \(P_ 1,\dots,P_ m,X_ 1,\dots,X_ n\) and whose edges represent the \(M\) branches of \(X\) in this way: if \(x_ r\) is a branch over \(P_ i\) and \(x_ r \in \overline{X}_ j\), an edge joining \(P_ i\) and \(X_ j\) is constructed. Any cycle of the graph \(\Gamma\) associated to \(X\) is said to be an incidence cycle of \(X\). \(\nu(X)\) is the number of the independent incidence cycles of \(X\) [see the author, J. Pure Appl. Algebra 35, 77-83 (1985; Zbl 0579.13003)]. We prove the following results:
(1) If \(X\) is projective, we have \(p_ a(X) = p_ a(\overline{X}_ 1)+\dots + p_ a(\overline{X}_ n) + (-1)^{r-1}\nu(X)\).
(2) We have \(\pi_ 1^{\text{alg}}(X) \cong (\pi_ 1^{\text{alg}}(\overline{X}_ 1) * \dots * \pi_ 1^{\text{alg}}(\overline{X}_ n) * L_{\nu(X)})^ \wedge\), where \(L_ \nu\) denotes the free group with \(\nu\) generators, \(*\) denotes the free product of groups and \(\wedge\) denotes the completion of the group.
(3) We have \(\text{Pic}(X)\cong \text{Pic}(\overline{X}_ 1) \oplus \dots \oplus \text{Pic}(\overline{X}_ n)\oplus \nu(X)k^*\), where \(k^*\) is the multiplicative group \(k-\{0\}\) and \(\nu k^*\) denotes the direct sum of \(\nu\) copies of \(k^*\). seminormal curve; arithmetic genus; algebraic fundamental group; Picard group Coverings of curves, fundamental group, Special algebraic curves and curves of low genus, Coverings in algebraic geometry, Picard groups On the incidence cycles of a curve: Some geometric interpretations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In these papers the author develops further his theory concerning the behaviour of Chern numbers by finite coverings of algebraic surfaces [see, for example, his previous paper in Ann. Global Anal. Geom. 4, 1-70 (1986; Zbl 0609.14022)]. Here he considers the general case \(f:\quad Y\to X,\) where f is finite and Y, X are normal projective surfaces. In fact, after some birational transformations, he reduces the situation to the case when Y and X have only Hirzebruch-Jung singularities and f is ``regular''. For simplicity, let us say that f is regular if the irreducible one-dimensional components of the branch locus of f are smooth and they cross each other ``nicely'' after taking the minimal desingularisation of Y. Let us assume this, and try to explain the main result. For a compact surface Z, denote by \(c_ 2(Z)\) the Euler number \(\sum (-1)^ i\dim (H^ i(Z,{\mathbb{C}})) \). To define the signature \(\tau\) (Z) the author considers a resolution of singularities \(\tilde Z,\) takes the orthogonal \(<E_ Z>^{\perp}\) of the subspace \(<E_ Z>\) generated in \(H_ 2(\tilde Z,{\mathbb{R}})\) by the components of the exceptional divisor \(E_ Z\) and puts \(\tau\) (Z) \(= the\) signature of the quadratic space \(<E_ Z>^{\perp}\). For a smooth curve D on Y denote by \(e(D)=2- 2g(D)\) its Euler number and by \((D^ 2)\) its selfintersection number on the minimal resolution of Y. Now take a point Q on Y. For regular coverings there are at most two curves through Q with non-trivial ramification indices, denote these indices by \(v_ Q\) and \(v_ Q'\). If Q is a singular point of Y, denote by \(<d_ Q,e_ Q>\) its type. The main result is:
\[
\left( \begin{matrix} c_ 2(X)\\ \tau (X)\end{matrix} \right)=B\left( \begin{matrix} c_ 2(Y)\\ \tau (Y)\end{matrix} \right)+\sum_{D}B_ D\left( \begin{matrix} e\quad (D)\\ (D^ 2)\end{matrix} \right) +\sum_{Q}B_ Q\left( \begin{matrix} (v_ q'+1)/d_ Q-2\\ 2(v_ Q+1)e\quad_ Q/d_ Q\end{matrix} \right),
\]
where \(B=\left( \begin{matrix} 1/| f| \\ 0\end{matrix} \begin{matrix} 0\\ 1/| f| \end{matrix} \right), \) \(B_ D=\frac{1}{| f|}\left( \begin{matrix} v_ D-1\\ 0\end{matrix} \begin{matrix} 0\\ 1/3(v_ D^ 2-1)\end{matrix} \right), \) \(B_ Q=\frac{1}{2| f|}\left( \begin{matrix} v_ Q-1\\ 0\end{matrix} \begin{matrix} 0\\ v_ Q^ 1-1\end{matrix} \right) \) \((| f| =\deg (f)).\)
The author emphasises the \(GL_ 2\)-form of the formula and its diagonalizing feature. Several other going-down and going-up formulas are also given. As the author says, to establish all these he needs to work in a larger category of surfaces, the so called category of arranged surfaces.
Concerning the word ``discrete'' appearing in the title of this paper, let us quote the author's explanation: ``We consider a covering \(f:\quad Y\to X\) as a ``discrete motion'', which happens immediately without time difference. \(| f| =\deg (f)\) sheets of the surface Y jump together to a surface X, curves on Y jump together to curves on X and points on Y jump also together. We asked for a numerical description of such jumps and found \(GL_ 2\)-representations''. Finally, let us mention the paper of \textit{B. Iversen} [Am. J. Math. 92, 968-996 (1970; Zbl 0232.14013)] concerning the case of coverings of smooth surfaces. behaviour of Chern numbers; Hirzebruch-Jung singularities; category of arranged surfaces; coverings of smooth surfaces Coverings in algebraic geometry, Surfaces and higher-dimensional varieties, Characteristic classes and numbers in differential topology, Singularities of surfaces or higher-dimensional varieties, Topological properties in algebraic geometry Discrete analysis of surface coverings. I.; II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this note is to fix an imprecision in Theorem 3.3 of the original paper [the author, ibid. 11, No. 1, 13--25 (2018; Zbl 1390.14102)], which affects the correctness of the statement and leads to a mistake in Corollary 3.5. Families, moduli, classification: algebraic theory, Special surfaces, Rational and ruled surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(K3\) surfaces and Enriques surfaces, Surfaces of general type, Hypersurfaces and algebraic geometry, Rational and birational maps, Coverings in algebraic geometry, Rational and unirational varieties, Rationally connected varieties Correction to: ``Irrationality issues for projective surfaces'' | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A connected analytic space over a nonarchimedean field K is called simply connected if it has no nontrivial connected analytic coverings. The author proves the following two criteria for a quasi-compact K-analytic space X to be simply connected: (i) there is a morphism with simply connected fibres from X to a simply connected analytic space; (ii) X has irreducible and smooth reduction.
These criteria are applied to show that in all known examples of p-adic uniformization (Mumford curves, certain abelian varieties, Mumford surfaces) the uniformization is indeed by the universal covering.
The proofs rely on the author's previous paper [Compos. Math. 45, 165-198 (1982; Zbl 0491.14014)]. (Note that in Lemma 4.1 the additional hypothesis \(``\dim Z\geq 2''\) has to be inserted.) simply connected rigid analytic space; constant sheaf; p-adic uniformization M. van der PUT . '' A note on p-adic uniformization ''. Proc. Nederl. Akad. Wetensch. A . 90 ( 3 ), 313 - 318 ( 1987 ). MR 914089 | Zbl 0624.32018 Non-Archimedean analysis, Local ground fields in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Coverings in algebraic geometry A note on p-adic uniformization | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective variety over an algebraically closed field \(k\) of characteristic \(p>0\) of \(\dim X\geq 4\) and Picard number \(\rho(X)=1\). Suppose that \(X \) satisfies \(H^i(X, F_X^{m\ast}({\Omega}_X^j)\otimes \mathcal L^{-1})=0\) for any ample line bundle \(\mathcal L\) on \(X \), and any nonnegative integers \(m\), \(i\), \(j\) with \(0\leq i+j<\dim X\), where \(F_{X}:X\to X\) is the absolute Frobenius morphism. Let \(Y \) be a smooth variety obtained from \(X \) by taking hyperplane sections of \(\dim \geq 3\) and cyclic covers along smooth divisors. If the canonical bundle \(\omega_{Y}\) is ample (resp. nef), then we prove that \(\Omega_{Y}\) is strongly stable (resp. strongly semistable) with respect to any polarization. Li, L; Shentu, J, Strong stability of cotangent bundles of cyclic covers, C. R. Math. Acad. Sci. Paris, 352, 639-644, (2014) Positive characteristic ground fields in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Coverings in algebraic geometry Strong stability of cotangent bundles of cyclic covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The conjecture that a generic morphism of degree at least 5 is completely determined by its branch curve was proposed by \textit{O. Chisini} [Ist. Lombardo, Sci. Lett., Rend., Cl. Sci. Mat. Natur., III. Ser. 8(77), 339-356 (1944; Zbl 0061.35305)]. In this note, we establish the following consequence of Kulikov's results on the Chisini conjecture [\textit{Vik. S. Kulikov}, Izv. Math. 63, No. 6, 1139-1170 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 6, 83-116 (1999; Zbl 0962.14005)].
Theorem 1. A generic ramified covering \(f:S\to \mathbb{P}^2\) of degree at least 12 is uniquely determined by its branch curve in \(\mathbb{P}^2\). In other words, the Chisini conjecture holds for generic morphisms of degree \(\geq 12\). ramified covering; branch curve; Chisini conjecture S. Yu. Nemirovskiĭ, On Kulikov's theorem on the Chisini conjecture, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), 77-80 Zbl1012.14005 MR1829404 Ramification problems in algebraic geometry, Coverings in algebraic geometry Kulikov's theorem on the Chisini conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \((Y,\mathcal{G})\) be a Riccati foliation on \(Y\) and let \(\pi:(X,\mathcal{F})\rightarrow(Y,\mathcal{G})\) be a double cover ramified over some normal-crossing curves. We will determine the minimal model of \(\mathcal{F}\) and compute its Chern numbers \(c_1^2(\mathcal{F})\), \(c_2(\mathcal{F})\), and \(\chi(\mathcal{F})=(c_1^2(\mathcal{F})+c_2(\mathcal{F}))/12\). We will prove that the slope \(\lambda(\mathcal{F})=c_1^2(\mathcal{F})/\chi(\mathcal{F})\) satisfies \(4\leq\lambda(\mathcal{F})<12\). Chern number; Riccati foliation; double cover; slope inequality Singularities of holomorphic vector fields and foliations, Coverings in algebraic geometry, Fibrations, degenerations in algebraic geometry, Dynamical aspects of holomorphic foliations and vector fields On the slope of non-algebraic holomorphic foliations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems As is well known, the Lefschetz theorems for the étale fundamental group of quasi-projective varieties do not hold. We fill a small gap in the literature showing they do for the tame fundamental group. Let \(X\) be a regular projective variety over a field \(k\), and let \(D\hookrightarrow X\) be a strict normal crossings divisor. Then, if \(Y\) is an ample regular hyperplane intersecting \(D\) transversally, the restriction functor from tame étale coverings of \(X\setminus D\) to those of \(Y\setminus D\cap Y\) is an equivalence if dimension \(X \geq 3\), and is fully faithful if dimension \(X=2\). The method is dictated by work of \textit{A. Grothendieck} and \textit{J. P. Murre} [The tame fundamental groups of a formal neighbourhood of a divisor with normal crossings on a scheme. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0216.33001)]. They showed that one can lift tame coverings from \(Y\setminus D\cap Y\) to the complement of \(D\cap Y\) in the formal completion of \(X\) along \(Y\). One has then to further lift to \(X\setminus D\). Esnault, H; Kindler, L, Lefschetz theorems for tamely ramified coverings, Proc. Am. Math. Soc., 144, 5071-5080, (2016) Coverings in algebraic geometry, Ramification problems in algebraic geometry Lefschetz theorems for tamely ramified coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From abstract and summary: It is a classical problem, first studied by {O. Zariski} [Ann. Math., II. Ser. 32, 485-511 (1931; Zbl 0001.40301)] and van Kampen, to understand how the type and relative position of the singularities of a plane projective curve are reflected in the fundamental group of its complement. In this paper the author outlines a relationship between the fundamental group of the complement of a reducible plane curve, and certain geometric invariants depending on the local type and configuration of the singularities on the curve. This generalizes the relationship between the Alexander polynomial of plane curves and the position of singularities. Complete details will appear elsewhere. Zariski problem; fundamental group of complement of plane curve; configuration of plane curve singularities Libgober A. Abelian branched covers of projective plane. In: Singularity Theory. London Math Soc Lecture Note Ser, vol. 263. Cambridge: Cambridge Univ Press, 1999, 281--289 Homotopy theory and fundamental groups in algebraic geometry, Singularities of curves, local rings, Coverings in algebraic geometry, Plane and space curves Abelian branched covers of projective plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G_1\) and \(G_2\) be hermitian algebraic groups with associated symmetric domains \(X_1\) and \(X_2\), respectively. A holomorphic embedding \(\tau : X_1 \to X_2\) is called weakly equivariant if there exists a morphism of algebraic groups \(\rho : G_1 \to G_2\) compatible with \(\tau\). It is called strongly equivariant if, in addition, the image of \(X_1\) is totally geodesic in \(X_2\). If the groups \(G_i\) are defined over \(\mathbb{Q}\), then we can define the conjugate \(\tau^\sigma : X^\sigma_1 \to X_2^\sigma\) of \(\tau\), for an automorphism \(\sigma\) of \(\mathbb{C}\). The main theorem of this paper is that if \(\tau\) is strongly equivariant, then so is \(\tau^\sigma\). This was conjectured by \textit{Min Ho Lee} who proved the analogous statement for weakly equivariant maps [Pac. J. Math. 149, No. 1, 127-144 (1991; Zbl 0782.32026)]. hermitian algebraic groups DOI: 10.2140/pjm.1994.165.207 Classical groups (algebro-geometric aspects), Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Coverings in algebraic geometry, Group actions on varieties or schemes (quotients), Modular and Shimura varieties Conjugates of strongly equivariant maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give a brief survey of many of the highlights of our present understanding of the young subject of quantum metric spaces, and of the quantum Gromov-Hausdorff distance between them. We include examples. M. A. Rieffel, ''Compact quantum metric spaces,'' in Operator Algebras, Quantization, and Noncommutative Geometry, Contemp. Math. 365, pp. 315--330 (Amer. Math. Soc., 2004). Noncommutative differential geometry, Coverings in algebraic geometry, Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces, Noncommutative geometry (à la Connes) Compact quantum metric spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a minimal surface of general type with generically finite canonical map \(\varphi:S \dashrightarrow \mathbb P^{p_g(S)-1}\) of degree \(d\). By a classical result of \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)] the maximum possible value of the degree \(d\) is \(36\), but which positive integers do actually occur as such degrees is still an open question. Surfaces with \(d=1,2,\ldots, 9\) are easy to construct, but only few surfaces with \(d>9\) are known and nowadays we know examples with \(d=12,16,24,27,32,36\).
In the paper under review the author uses the theory of abelian covers developed by \textit{R. Pardini} [J. Reine Angew. Math. 417, 191--213 (1991; Zbl 0721.14009)] to construct two minimal surfaces of general type having degree of the canonical map \(d=20\). These surfaces are obtained as ramified \(\mathbb Z^4_2\)-covers of a Del Pezzo surface of degree 5 and have invariants \(p_g=3\), \(q=0\), \(K^2=20\) resp. \(24\); in the former case he canonical linear system is base point free, while in the latter case it has a non-trivial fixed part. abelian covers; canonical map; surfaces of general type Surfaces of general type, Rational and birational maps, Coverings in algebraic geometry Some examples of algebraic surfaces with canonical map of degree 20 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [This review concerns also the preceding item Zbl 0738.14021.]
Let \(k\) be an algebraically closed field of characteristic \(p>0\), let \(m\) be a nonnegative integer and let \(X_ m\) be the Zariski surface defined by an equation of the form \(z^{p^ m}=G(x,y)\) with \(G\in k[x,y]\). Let \(cl(X_ m)\) denote the divisor class group of the coordinate ring of \(X_ m\). Supplementing the work of P. Blass, P. Deligne and J. Lang, and using techniques of the fundamental group and purely inseparable descent, the two papers under review carry forward the computation of \(cl(X_ m)\) for a sufficiently general \(G\) (i.e. all \(G\) with coefficients in a dense Zariski open set) as also for a generic \(G\) (i.e. when the coefficients of \(G\) are indeterminates). The case \(m=1\) is completely solved. The following two theorems summarise the main final results:
Theorem 1. For a generic or sufficiently general \(G\), \(cl(X_ 1)=0\) if \(\deg(G)\geq4\) and \(p>2\), while \(cl(X_ 1)=\mathbb{Z}/2\mathbb{Z}\) if \(\deg(G)\geq5\) and \(p=2\).
Theorem 2. For a generic of sufficiently general \(G\), \(cl(X_ m)=0\) if \(\deg(G)\geq4\) and \(p\geq5\).
Theorem 2 is extended also to the case of a higher dimensional hypersurface defined by \(z^{p^ m}=G(x_ 1,\ldots,x_ r)\). characteristic p; Zariski surface; divisor class group of the coordinate ring; fundamental group; Zariski open set purely inseparable descent Blass, P.; Lang, J., Generic Zariski surfaces, Compos. Math., 73, 345-361, (1990) Special surfaces, Coverings in algebraic geometry, Finite ground fields in algebraic geometry, Special algebraic curves and curves of low genus Generic Zariski surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give an overview on the series of our articles [``Filtered moment graph sheaves'', Preprint, \url{arXiv:1508.05579}; ``Sheaves on the alcoves. I: Projectivity and wall crossing functors'', Preprint; ``Periodic structures on affine moment graphs. II: Multiplicities and modular representations'' (in preparation)] that aims at introducing a new approach towards the ``combinatorial'' category introduced by \textit{H. H. Andersen} et al. in their work [Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Paris: Société Mathématique de France (1994; Zbl 0802.17009)] on Lusztig's conjecture on the irreducible highest weight characters of modular algebraic groups. modular representation theory; Lusztig's conjecture; Andersen-Jantzen-Soergel category Fiebig, P., Lanini, M.: Periodic structures on affine moment graphs II: multiplicities and modular representations (\textbf{in preparation}) Modular representations and characters, Representation theory for linear algebraic groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Modular Lie (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The combinatorial category of Andersen, Jantzen and Soergel and filtered moment graph sheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Author's abstract: We consider surjective endomorphism \(f\) of degree \(>1\) on the projective \(n\)-space \(\mathbb{P}^n\) with \(n=3\), and \(f^{-1}\)-stable hypersurfaces \(V\). We show that \(V\) is a hyperplane (i.e., \(\deg(V)=1\)) but with four possible exceptions; It is conjectured that \(\deg (V)=1\) for any \(n\geq2\) [\textit{J. E. Fornaess} and \textit{N. Sibony}, Astérisque. 222, 201--231 (1994; Zbl 0813.58030); \textit{J.-Y. Briend, S. Cantat} and \textit{M. Shishikura}, Math. Ann. 330, No. 1, 39--43 (2004; Zbl 1056.32018)]. endomorphism; iteration; projective 3-space Zhang, D.-Q.: Invariant hypersurfaces of endomorphisms of the projective 3-space. In: Masuda, K., Kojima, H., Kishimoto, T. (eds.) \textit{Affine Algebraic Geometry}, pp. 314-330. World Sci. Publ., Hackensack (2013) Fano varieties, Coverings in algebraic geometry, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Invariant hypersurfaces of endomorphisms of the projective 3-space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\widetilde\mathbb{C}^2\) and \(\mathbb{C}^2\) be two copies of the complex plane, and let \(f:\widetilde\mathbb{C}^2\to \mathbb{C}^2\) be a polynomial mapping whose Jacobian is a non-zero constant. The well-known Jacobian conjecture states that \(f\) is polynomially invertible. -- The mapping \(f\) is polynomially invertible if and only if a generic point in \(\mathbb{C}^2\) has exactly one pre-image. We define the topological degree of a mapping to be the number of pre-images of a generic point. It is known that there are no polynomial mappings of degree two or three whose Jacobian is a non-zero constant. The case of degree two is elementary, and the case of degree three was considered by \textit{S. Yu. Orevkov} [Math. USSR, Izv. 29, 587-596 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 6, 1231-1240 (1986; Zbl 0624.13016)]. The main result of this paper is as follows.
Theorem. Let \(f:\widetilde \mathbb{C}^2\to \mathbb{C}^2\) be a polynomial mapping of topological degree four. Then the Jacobian of \(f\) cannot be a non-zero constant.
This paper is a continuation of part I of this paper [\textit{A. V. Domrina} and \textit{S. Yu. Orevkov}, Math. Notes 64, No. 6, 732-744 (1998); translation from Mat. Zametki 64, No. 6, 847-862 (1998; Zbl 0955.14044)], where the case when \(f\) has one dicritical component was proved to be impossible. Here we consider the remaining cases. Jacobian conjecture; topological degree; polynomial mapping A. V. Domrina, ''On Four-Sheeted Polynomial Mappings of \(\mathbb{C}\)2. II: The General Case,'' Izv. Ross. Akad. Nauk, Ser. Mat. 64(1), 3--36 (2000) [Izv. Math. 64, 1--33 (2000)]. Jacobian problem, Rational and birational maps, Morphisms of commutative rings, Coverings in algebraic geometry, Ramification problems in algebraic geometry On four-sheeted polynomial mappings of \(\mathbb{C}^2\). II: The general case | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In a previous paper [Acta Arithm. 71, 107-137 (1995; Zbl 0840.11015)], \textit{U. Zannier} proved that there are distinct polynomials \(F,G \in {\mathbb{C}}[t]\) having roots of prescribed multiplicities and with \(\deg(F- G)\) as small as predicted by a theorem attributed to \textit{R. C. Mason} that is a predecessor to the abc conjecture.
The author is interested here in the analogous problem in positive characteristic. The method used by the author is good reduction of certain covers of curves of genus \(0\). More precisely, let \(F(t)=\prod_{i=1}^{h}(t-\xi_{i})^{\mu_{i}}\), \(G(t)=\prod_{i=1}^{k}(t- \eta_{i})^{\nu_{i}}\) be distinct polynomials of degree \(n\) with coefficients in a field \(K\) of characteristic zero, with discrete valuation \(\upsilon\) having perfect residue field \(K_0\) of characteristic \(p>0\) and \(\mathcal{O}\) as valuation ring. Then the rational function \(F/G\) defines a cover \(f_{K}:{\mathbb{P}}_{K}^1\to {\mathbb{P}}_{K}^1\). We say that \(f_{K}\) has good reduction if there exists an \(\mathcal O\)-morphism \(f_{\mathcal O}:=F^{*}/G^{*}:{\mathbb{P}}_{\mathcal{O}}^1\to {\mathbb{P}}_{\mathcal{O}}^1\) that extends \(f_{K}\) such that the reduction of \(f_{\mathcal{O}}\) mod \(\upsilon\) gives a separable and generically unramified \(K_{0}\)-morphism \(f_{K_{0}}:{\mathbb{P}}_{K_{0}}^1\to {\mathbb{P}}_{K_{0}}^1\) of the same degree of \(f_{K}\). The cover has potential good reduction if it has good reduction over a finite field extension of \(K\). The main result of the paper gives some sufficient conditions for the cover \(f_{K}\) to have potential good reduction at \(\upsilon\). The proof is mainly by means of \(p\)-adic analytic continuation of Puiseux series as developed by \textit{B. Dwork} and \textit{P. Robba} [Trans. Am. Math. Soc. 256,199-213 (1979; Zbl 0426.12013)]. covers of projective line; good reduction; abc conjecture; positive characteristic; continuation of Puiseux series U. ZANNIER, Good reduction of certain covers P1 3 P1 , Israel J. of Math., 124 (2001), pp. 93-114. Zbl1015.14010 MR1856506 Coverings of curves, fundamental group, Algebraic functions and function fields in algebraic geometry, Arithmetic ground fields for curves, Curves over finite and local fields, Local ground fields in algebraic geometry, Coverings in algebraic geometry Good reduction of certain covers \(\mathbb{P}^1\to\mathbb{P}^1\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is an interesting survey talk on the problem of classification of algebraic surfaces of general type. It is well known that both numerical invariants \(K_S^2\) and \(\chi(\mathcal O_S)\) of a minimal surface \(S\) of general type satisfy \(K_S^2\geq 1\), \(\chi(\mathcal O_S)\geq 1\) and \(K_S^2\leq 9\chi(\mathcal O_S)\). When \(\chi(\mathcal O_S)\) is sufficiently large, it is shown that there exist minimal surfaces \(S\) of general type whose numerical invariants can fill the preceding region except near the upper bound \(K^2<9\chi\). But for smaller \(\chi\), the problem is more complicated. On the other hand, since the canonical model of a surface is an important birational invariant, the problem of finding canonical surfaces becomes attractive. In this paper, the author raises two questions:
1. For which \((x,y)\) in the region described above is there a surface with an almost generic canonical projection (equivalently, when is the canonical system without base points and yielding a birational map)?
2. When is there a surface \(S\) which is embedded by a canonical system?
In fact, the recent advances in homological algebra can be applied to this purpose. This paper starts on basics of homological algebra of polynomial rings. Then the canonical surfaces in low dimensional projective spaces are discussed in detail. Many interesting examples with small numerical invariants are presented. In the last section, applications of homological algebra to the effective construction of algebraic varieties with given invariants are discussed. homological algebra; Euler characteristic; canonical surface; Cohen-Macaulay ring; classification of algebraic surfaces of general type; minimal surface of general type; canonical system Fabrizio Catanese, Homological algebra and algebraic surfaces, Algebraic geometry --- Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 3 -- 56. Surfaces of general type, Homological algebra in category theory, derived categories and functors, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, Coverings in algebraic geometry, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Topological properties in algebraic geometry Homological algebra and algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is a survey of the known results concerning the following problem: Given a smooth projective surface \(X\) and a generic projection \(\pi:X\to {\mathbb{P}}^2\), compute the fundamental group \(\Pi\) of the complement of the branch curve of \(\pi\). The group \(\Pi\) does not change if \(X\) varies in a flat family, therefore if one can degenerate \(X\) to a union of planes such that no three of them have a common line, then the computation is reduced to the case where the branch curve is a union of lines with certain combinatorial properties. In order to describe an algorithm for computing the fundamental group of the complement of a plane curve, the author introduces braid groups and braid monodromy and a suitable form of the Van Kampen theorem and gives some explicit examples.
Notice that one of the main motivations for this kind of computations was the question whether the connected components of the moduli spaces of surfaces of general type correspond to diffeomorphism classes of the underlying \(4\)-manifolds: recently \textit{M. Manetti} [``On the moduli space of diffeomorphic surfaces of general type'' (preprint S.N.S., Pisa 1998)] seems to have given a negative answer to the above question. fundamental group; braid group; algebraic surface; complement of the branch curve; braid monodromy; Van Kampen theorem; diffeomorphism classes M. Teicher, Braid groups, algebraic surfaces and fundamental groups of complements of branch curves, Algebraic geometry --- Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 127 -- 150. Homotopy theory and fundamental groups in algebraic geometry, Ramification problems in algebraic geometry, Coverings in algebraic geometry, Families, moduli, classification: algebraic theory, Braid groups; Artin groups Braid groups, algebraic surfaces and fundamental groups of complements of branch curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author presents the problem for the decomposition numbers for perverse sheaves, obtains some methods for the their computation in some simple cases and computes them explicitly for a simple surface singularity and for the closure of the minimal non-trivial nilpotent orbit in a simple Lie algebra. After recalling the definition of perverse sheaves over \(\mathbb{K}, \mathbb{Q}, \mathbb{F}\) he considers a \(t\)-category having the heart endowed with a torsion theory. He studies the interaction between torsion theories and \(t\)-structures and obtains the failure of commutativity between truncations and modular reduction. The decomposition numbers are obtained in the setting of recollements, they are defined for perverse shaves then it is studied the equivariance. Then the author presents some techniques for computation of the decomposition numbers. In characteristic zero, there is obtained some information from the study of semi-small and small proper separable morphisms. The results are important for the modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra and for reductive algebraic groups schemes using the affine Grassmannian of the Langlands dual group. perverse sheaves; intersection cohomology; integral cohomology; torsion theories; \(t\)-structures; decomposition matrices; simple singularities; minimal nilpotent orbits Juteau, D., Decomposition numbers for perverse sheaves, \textit{Annales de l'Institut Fourier}, 59, 3, 1177-1229, (2009) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Intersection homology and cohomology in algebraic topology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Modular representations and characters Decomposition numbers for perverse sheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field, and let \((X,D)\) be a pair consisting of a smooth variety \(X\) and a rational divisor \(D\in\text{Div}(X)\otimes\mathbb{Q}\) defined over \(k\). Suppose that the support of \(D\) has only normal crossings and, furthermore, that there is an integer \(n\), prime to \(\text{char}(k)\), for which \(nD\) is an integral divisor. In the present paper, the authors provide a construction that associates to such a pair \((X,D)\) a certain Deligne-Mumford stack \(\chi\) over \(X\), which is shown to be the ``minimal covering'' of \(X\) (in the sense of stacks) on which \(D\) becomes an integral divisor \(\widetilde D\). Using the particular properties of this associated stack \(\chi\), it is then shown how the classical vanishing theorem of Kawamata-Viehweg [\textit{H. Esnault} and \textit{E. Viehweg}, Lectures on vanishing theorems (DMV Seminar, Basel, Birkhäuser Verlag (1992; Zbl 0779.14003)] can naturally be reinterpreted as a consequence of a version of the Kodaira vanishing theorem for smooth, proper and tame Deligne-Mumford stacks à la Deligne-Illusie-Raynaud [\textit{P. Deligne} and \textit{L. Illusie}, Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)]. Methodically, the stack \(\chi\) associated to the pair \((X,D)\) is constructed as a moduli space classifying certain log structures on the variety \(X\), using the basic framework of Fontaine-Illusie and its elaboration by \textit{K. Kato} [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191--224 (1989; Zbl 0776.14004)]. The authors' stack-theoretic interpretation of the Kawamata-Viehweg vanishing sheds some new light on the entire ``Minimal Model Program'' in (higher-dimensional) birational geometry, and it might lead to an additional tool in the study of the singularities produced by carrying out this program in dimension greater than two. minimal model program; log structures; divisors; sheaves of differentials; birational maps Kenji Matsuki & Martin Olsson, ``Kawamata-Viehweg vanishing as Kodaira vanishing for stacks'', Math. Res. Lett.12 (2005) no. 2-3, p. 207-217 Vanishing theorems in algebraic geometry, Generalizations (algebraic spaces, stacks), Minimal model program (Mori theory, extremal rays), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry Kawamata-Viehweg vanishing as Kodaira vanishing for stacks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems By classical Hodge theory the abelianization of the fundamental group \(\pi_ 1(X)\) of a smooth projective variety has even rank. The aim of this note is to present a generalization of Deligne's mixed Hodge theory to the case of cohomology with coefficients in locally constant sheaves arising from orthogonal representations of the fundamental group.
As a corollary to the existence of such a theory it is proved that if X is a compact normal complex algebraic variety then \(\pi_ 1(X)\in E^+\) where \(E^+\) is the class of groups defined as follows. One lets \(E^+\) be the class of finitely presented groups G such that for any real orthogonal representation M, \(H^ 1(G,M)\) has even real dimension and then one lets E be the class of groups G all of whose subgroups of finite index lie in E. This leads to examples of finitely generated groups which cannot occur as \(\pi_ 1(X)\) for any normal compact complex variety X. abelianization of the fundamental group; mixed Hodge theory D. Arapura, Hodge theory with local coefficients and fundamental groups of varieties , Bull. Amer. Math. Soc. (N.S.) 20 (1989), no. 2, 169-172. Homotopy theory and fundamental groups in algebraic geometry, Topological properties in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Coverings in algebraic geometry Hodge theory with local coefficients and fundamental groups of varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In previous papers, the authors computed the Poincaré series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincaré series were expressed as the integer parts of certain fractional power series, whose interpretation was not given. In this paper, we show that, up to a simple change of variables, these fractional power series are reductions of the equivariant Poincaré series for filtrations on the ring of germs of functions on the universal Abelian cover of the surface. We compute these equivariant Poincaré series. universal abelian cover; rational surface singularity; Poincaré series Guse\?ın-Zade, S. M., Delgado, F., Campillo, A.: Universal abelian covers of rational sur- face singularities, and multi-index filtrations. Funktsional. Anal. i Prilozhen. 42, no. 2, 3-10 (2008) (in Russian) Singularities of surfaces or higher-dimensional varieties, Coverings in algebraic geometry, Singularities in algebraic geometry Universal abelian covers of rational surface singularities and multi-index filtrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0564.00013.]
In the first part of this paper (see the preceding review) the author outlined a program for studying algebraic surfaces by means of the arithmetic of the braid monodromies of their general projections. In this part the author calculates the corresponding braid invariants for a class of surfaces obtained as double covers of quadrics and embedded into a projective space by a complete linear system of curves of bidegree (m,n). Similarly to the first part the method of computation is based on certain degenerations of the surface into copies of planes and minimal ruled surfaces. arithmetic of the braid monodromies; double covers of quadrics; degenerations Moishezon, B.: Algebraic surfaces and arithmetic of braids. II In: Harper, J.R., Mandelbaum, R. (eds.) Combinatorial methods in topology and algebraic geometry. (Contemp. Math., vol. 44, pp. 311?344) Providence, RI: Am. Math. Soc. 1985 Coverings in algebraic geometry, Braid groups; Artin groups, Moduli, classification: analytic theory; relations with modular forms, Differential topological aspects of diffeomorphisms Algebraic surfaces and the arithmetic of braids. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This comprehensive survey article is an elaborated and considerably expanded version of the author's lectures delivered at the C.I.M.E. Summer School 2003 in Cetraro, Italy. Its main theme is the theory of complex algebraic surfaces of general type, their deformations, and their moduli spaces from the viewpoint of symplectic geometry. As the author points out, this course may be regarded as a continuation of the C.I.M.E. course ``Moduli of algebraic surfaces'' that he had held about 20 years ago [Theory of moduli, Lect. 3rd Sess. Cent. Int. Mat. Estivo, Montecatini Terme/Italy 1985, Lect. Notes Math. 1337, 1--83 (1988; Zbl 0658.14017)], this time with the special focus on some recent developments concerning the differential-geometric aspects of the classification of algebraic surfaces.
The text comprises six lectures (or chapters), each of which is subdivided into several sections.
Lecture 1 briefly recalls the basic facts about projective and Kähler manifolds, symplectic structures, the birational equivalence of algebraic varieties, the Enriques classification of algebraic surfaces of special type, and some fundamental construction techniques for projective varieties.
Lecture 2 provides a more detailed introduction to surfaces of general type, their canonical models, and their singularities. This lecture contains a proof of Bombieri's theorem on pluricanonical embeddings as well as an analysis of deformations and simultaneous resolutions of isolated singularities of complex spaces and the fundamental properties of the deformation equivalence relation for surfaces.
In Lecture 3 it is shown that deformation equivalence implies diffeomorphism equivalence. Then, using symplectic approximations of projective varieties admitting a smoothing component, on the one hand, and J. Moser's well-known theorem on symplectomorphic fibres on the other, it is proved that a surface of general type carries a canonical symplectic structure which is unique up to symplectomorphisms. At the end of this lecture, the author describes some particular, sufficiently mild degenerations of surfaces of general type preserving this canonical symplectic structure. This leads immediately to the central topic of the whole survey, namely to the comparison of the deformation type and the differentiable type of a minimal surface of general type, that is, to the celebrated ``DEF = DIFF speculation'' of \textit{R. Friedman} and \textit{J. W. Morgan} [Smooth four-manifolds and complex surfaces. Berlin: Springer-Verlag (1994; Zbl 0817.14017)]. The first counterexamples to this conjecture were obtained by \textit{M. Manetti} [Invent. Math. 143, No. 1, 29--76 (2001; Zbl 1060.14520)], and several further counterexamples are presented in the following parts of the current notes.
To this end, Lecture 4 discusses first a very special class of algebraic surfaces. These are exactly those which admit an unramified covering by a product of curves, and which are therefore called surfaces isogenous to a product. The study of general varieties isogenous to a product is essentially due to the author himself [Am. J. Math. 122, No. 1, 1--44 (2000; Zbl 0983.14013)], and this approach is used here to exhibit surfaces that are not deformation equivalent to their complex conjugate surface. In particular, the author describes several examples constructed by himself [Ann. Math. (2) 158, No. 2, 577--592 (2003; Zbl 1042.14011)], by \textit{I. Bauer}, \textit{F. Catanese} and \textit{F. Grunewald} [Progr. Math. 235, 1--42 (2005; Zbl 1094.14508)], and by \textit{V. S. Kulikov} and \textit{V. M. Kharlamov} [Izv. Math. 66, No. 1, 133--150 (2002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 1, 133--152 (2002; Zbl 1055.14060)].
In Lecture 5, after explaining some more preparatory material on connected sums, surgeries, fibre sums, braid groups, mapping class groups, Lefschetz pencils, and Lefschetz fibrations, the author touches upon M. Freedman's crucial results on the (differential) topology of simply connected, compact and oriented four-manifolds [\textit{M. H. Freedman} and \textit{F. S. Quinn}, Topology of 4-manifolds, Princeton Mathematical Series, 39. Princeton, NJ: Princeton University Press. viii (1990; Zbl 0705.57001)] and related results by C. T. C. Wall, B. Moishezon, R. Mandelbaum, S. Donaldson, P. Kronheimer, and others. This is used, in the sequel, to study the diffeomorphism type of certain series of bidouble covers of quadrics, which depend on three integer parameters \((a,b,c)\) and are called \(abc\)-surfaces. These surfaces, recently established and investigated by \textit{F. Catanese} and \textit{B. Wajnryb} [J. Differ. Geom. 76, No. 2, 177--213 (2007; Zbl 1127.14039)], give the strongest counterexamples to the ``DEF = DIFF speculation'' of Friedman and Morgan, because they provide the first examples of simply connected surfaces which are diffeomorphic but not deformation equivalent.
The proof of the latter fact is sketched at the end of the paper, chiefly via the deformation theory of \(abc\)-surfaces and the theory of smoothings of singularities. The concluding chapter is meant as an epilogue supplementing the foregoing five lectures. Titled ``Deformation, Diffeomorphism and Symplectomorphism Type of Surfaces of General Type'', this part not only adds the above-mentioned recent results by F. Catanese and B. Wajnryb on \(abc\)-surfaces, but also explains the construction of \textit{M. Manetti's} earlier counterexamples [Zbl 1060.14520] to the ``DEF = DIFF'' problem, just for the sake of comparison and completeness. Furthemore, the link of this context to a classical conjecture of \textit{O. Ohisini} (1944) is briefly discussed. All together, this article surveys some of the most recent and most important developments in this very active area of contemporary research in utmost comprehensive, lucid, versatile, enlightening and stimulating a manner. This masterpiece of expository mathematical writing is enhanced by rich bibliography, with numerous hints to it for further reading. survey (algebraic geometry); surfaces of general type; fibrations; Enriques classification; symplectic geometry; deformations; singularities; fundamental group F. Catanese, Differentiable and deformation type of algebraic surfaces, real and symplectic structures, In: Symplectic \(4\)-Manifolds and Algebraic Surfaces, Lecture Notes in Math., \textbf{1938}, Springer-Verlag, 2008, pp. 55-167. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Surfaces of general type, Moduli, classification: analytic theory; relations with modular forms, Fibrations, degenerations in algebraic geometry, Local deformation theory, Artin approximation, etc., Coverings in algebraic geometry, Singularities in algebraic geometry, Symplectic manifolds (general theory) Differentiable and deformation type of algebraic surfaces, real and symplectic structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective geometrically irreducible curve over a perfect field \(k\) of positive characteristic \(p\). Suppose \(G\) is a finite group acting faithfully on \(X\) such that \(G\) has non-trivial cyclic Sylow \(p\)-subgroups. We show that the decomposition of the space of holomorphic differentials of \(X\) into a direct sum of indecomposable \(k [G]\)-modules is uniquely determined by the lower ramification groups and the fundamental characters of closed points of \(X\) that are ramified in the cover \(X \longrightarrow X / G\). We apply our method to determine the \(\text{PSL}(2, \mathbb{F}_\ell)\)-module structure of the space of holomorphic differentials of the reduction of the modular curve \(\mathcal{X}(\ell)\) modulo \(p\) when \(p\) and \(\mathcal{l}\) are distinct odd primes and the action of \(\text{PSL}(2, \mathbb{F}_\ell)\) on this reduction is not tamely ramified. This provides some non-trivial congruences modulo appropriate maximal ideals containing \(p\) between modular forms arising from isotypic components with respect to the action of \(\text{PSL}(2, \mathbb{F}_\ell)\) on \(\mathcal{X}(\ell)\). holomorphic differentials; projective curves; modular curves Curves over finite and local fields, Modular and Shimura varieties, Algebraic functions and function fields in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Modular representations and characters Galois structure of the holomorphic differentials of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X be a 3-dimensional compact complex manifold which admits a flat holomorphic projective structure. There is associated a representation \(\rho: \pi_ 1(X)\to PGL(4)\). The structure of X is determined under the additional assumption that \(\rho(\pi_ 1(X))\) is infinite cyclic (this is satisfied if \(\pi_ 1(X)={\mathfrak Z})\). It is proved that X is either a Hopf manifold (i.e. its universal covering is \({\mathbb{C}}^ 3-\{0\})\) or \(M_ g\) which is constructed as follows. Let W be \({\mathfrak P}^ 3\) with two skew lines \(\ell_ 1=\{z_ 0=z_ 1=0\}\) and \(\ell_ 2=\{z_ 2=z_ 3=0\}\) being deleted and let \(<g>\) be the group of automorphisms of W generated by the projective transformation
\[
g= \begin{pmatrix} \alpha_ 0 & \lambda_ 0 & 0 & 0\\ 0 & \alpha_ 1 & 0 & 0\\ 0 & 0 & \alpha_ 2 & \lambda_ 2\\ 0 & 0 & 0 & \alpha_ 3 \end{pmatrix}, \qquad (|\alpha_ 0| \leq |\alpha_ 1| < |\alpha_ 2| \leq |\alpha_ 3|).
\]
Then \(M_ g=W/<g>\). \(M_ g\) contains an open set which is biholomorphic to a certain neighborhood of a line in \({\mathfrak P}^ 3\). So \(M_ g\) is of \(class L\) as being defined and studied by the author [Jap. J. Math., New Ser. 11, 1-58 (1985; Zbl 0588.32032)]. complex manifold of class L; 3-dimensional compact complex manifold; flat holomorphic projective structure; Hopf manifold Compact analytic spaces, Coverings in algebraic geometry, \(3\)-folds Compact complex 3-folds with projective structures; the infinite cyclic fundamental group case | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author considers the integer-valued isomorphism invariants of certain classes of discrete groups which satisfy the following property: if \(\varphi\) is such an invariant and \(\Delta \subset \Gamma\) is a subgroup of index \(N\), then \(\varphi (\Delta)= N\varphi (\Gamma)\). These invariants allow one to decide sometimes that two groups are not commensurable (i.e. they do not afford isomorphic subgroups of a finite index). Indeed, if \(\varphi_1\) and \(\varphi_2\) are a pair of such invariants, then \(\Gamma\) and \(\Delta\) commensurable implies that \(\varphi_1 (\Gamma)/ \varphi_2 (\Delta)= \varphi_1 (\Gamma)/ \varphi_2 (\Delta)\) (provided that \(\varphi_2 (\Delta)\neq 0)\). Examples of such invariants for the class of the fundamental groups of closed aspherical manifolds are provided by the Euler characteristic and the signature of the intersection form in the middle-dimensional cohomology of the corresponding aspherical manifold (the latter invariant is defined when the dimension is divisible by 4). The author carries out the calculation of the signature invariant for the 4-dimensional Poincaré duality groups which are the fundamental groups of certain complex surfaces. These surfaces \(X(g, N)\) (first considered by \textit{M. F. Atiyah} [in `Global analysis', Papers in Honor of K. Kodaira, 73-84 (1969; Zbl 0193.52302)] and \textit{K. Kodaira} [J. Anal. Math. 19, 207-215 (1967; Zbl 0172.37901)]) are the cyclic branched coverings of certain products of two curves, each of which is an abelian covering corresponding to the kernel of the reduction modulo \(N\) of the homology of a Riemann surface of genus \(g\). The upshot is that the family of fundamental groups of these complex surfaces contains infinitely many noncommensurable groups provided \(g\geq 2\). The author also poses the question whether the groups provided by this construction for \(g=1\), \(M\neq N\) are commensurable. discrete groups; commensurable; fundamental groups; closed aspherical manifolds; Euler characteristic; signature; intersection form; 4-dimensional Poincaré duality groups; cyclic branched coverings Characteristic classes and numbers in differential topology, Coverings in algebraic geometry, Compact complex surfaces A rational invariant for certain infinite discrete groups | 0 |
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