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Lemma 14.71. Let \( \lambda \in \mathbb{R} \) . Then the Hankel function of the first kind \( {H}_{\lambda }^{\left( 1\right) } \) is of class \( {C}^{\infty } \) in \( \left( {0,\infty }\right) \) and the following properties hold for each \( r > 0 \) : | (1) \( {H}_{-\lambda }^{\left( 1\right) }\left( r\right) = {\mathrm{e}}^{\mathrm{i}{\pi \lambda }}{H}_{\lambda }^{\left( 1\right) }\left( r\right) \)\n\n(2) \( \frac{\mathrm{d}}{\mathrm{d}r}\left\lbrack {{r}^{\lambda }{H}_{\lambda }^{\left( 1\right) }\left( r\right) }\right\rbrack = {r}^{\lambda }{H}_{\lambda - 1}^{\le... | Yes |
Lemma 14.72. The following formulas are valid:\n\n\[ \mathop{\lim }\limits_{{r \rightarrow {0}^{ + }}}\frac{{H}_{0}^{\left( 1\right) }\left( r\right) }{\frac{2\mathrm{i}}{\pi }\ln \left( r\right) } = 1 \] | Proof. The limits in (14.10.5) and (14.10.6) may be found in [60, 10.7.2 and 10.7.7]. If \( \lambda \in \left( {-\infty ,0}\right) \), then item (1) in Lemma 14.71 and (14.10.6) imply (14.10.7). | No |
Proposition 14.74. Let \( \lambda \in \mathbb{R}, k \in \left( {0,\infty }\right) \), and fix a multi-index \( \beta \in {\mathbb{N}}_{0}^{n} \) with \( \left| \beta \right| > 0 \) . Then the following asymptotic expansions hold:\n\n\[{\partial }^{\beta }\left\lbrack {{H}_{\lambda }^{\left( 1\right) }\left( {k\left| x\... | Proof. Fix a multi-index \( \beta = \left( {{\beta }_{1},\ldots ,{\beta }_{n}}\right) \in {\mathbb{N}}_{0}^{n} \) of positive length. For starters observe that repeated applications of the Chain Rule give that, for \( x \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) ,\n\n\[{\partial }^{\beta }\left\lbrack {{H}_{\lambda ... | Yes |
\[ {y}^{\prime } = - {2y} \] | Here \( D = {\mathbb{R}}^{2} \) . Using the procedure in (5) one obtains \[ \frac{dy}{y} = - {2dx} \Leftrightarrow \ln \left| y\right| = - {2x} + C \Leftrightarrow \left| y\right| = {\mathrm{e}}^{C - {2x}}. \] The general solution (with \( \pm {\mathrm{e}}^{C} \) replaced with \( C \) ) is \[ y\left( {x;C}\right) = C{\... | Yes |
\[ {y}^{\prime } = \sqrt{\left| y\right| } \] | Again \( D = {\mathbb{R}}^{2} \) . Since the direction field is symmetric, it follows that if \( y\left( x\right) \) is a solution, then \( z\left( x\right) = - y\left( {-x}\right) \) is also a solution. Indeed, we have\n\n\[ {z}^{\prime }\left( x\right) = {y}^{\prime }\left( {-x}\right) = \sqrt{\left| y\left( -x\right... | Yes |
\[ {y}^{\prime } = - x\left( {\operatorname{sgn}y}\right) \sqrt{\left| y\right| } = \left\{ \begin{array}{lll} - x\sqrt{y} & \text{ for } & y \geq 0 \\ x\sqrt{-y} & \text{ for } & y < 0 \end{array}\right. \] | The direction field is symmetric to the \( x \) -axis; i.e., if \( y\left( x\right) \) is a solution, then so is \( - y\left( x\right) \) . Thus it is sufficient to calculate the positive solutions. From \[ \int \frac{dy}{\sqrt{y}} = - 2\sqrt{y} = - \int {xdx} = \frac{1}{2}\left( {C - {x}^{2}}\right) \] it follows that... | Yes |
\[ {y}^{\prime } = {\mathrm{e}}^{y}\sin x. \] | The direction field is symmetric with respect to the \( y \) -axis and periodic in \( x \) of period \( {2\pi } \), i.e., if \( y\left( x\right) \) is a solution, then so are \( u\left( x\right) = y\left( {-x}\right) \) and \( v\left( x\right) = y\left( {x + {2k\pi }}\right) \) . By separation of variables (7) one obta... | Yes |
Theorem 1. Let \( \mathop{\lim }\limits_{{t \rightarrow \infty }}B\left( t\right) = \infty \) . If \( u \) is a positive solution, then\n\n\[ \mathop{\lim }\limits_{{t \rightarrow \infty }}u\left( t\right) = \mathop{\lim }\limits_{{t \rightarrow \infty }}\frac{b\left( t\right) }{c\left( t\right) } \]\n\nprovided that t... | Proof. This theorem is a substantial generalization of 1.XIII.(a). It can be proved by writing \( y \) as the quotient \( Z\left( t\right) /N\left( t\right) \) with \( N\left( t\right) = {\mathrm{e}}^{B\left( t\right) } \) . The result then follows using l’Hospital’s rule; since both \( B\left( t\right) \) and \( N\lef... | Yes |
Theorem 2. If the coefficients \( b \) and \( c \) are \( T \) -periodic, then there exists exactly one positive \( T \) -periodic solution of (14). | Proof. It is sufficient to show that there is exactly one solution with \( u\left( 0\right) = \) \( u\left( T\right) > 0 \) . Under this assumption \( v\left( t\right) \mathrel{\text{:=}} u\left( {t + T}\right) \) is a solution of (14) with \( v\left( 0\right) = u\left( 0\right) \) . Then \( y = 1/u \) and \( z = 1/v \... | Yes |
Theorem 3. Let the coefficients \( b, c \) be positively bounded. Then equation (13) has exactly one positively bounded solution \( {u}^{ * } \) on \( \mathbb{R} \) ; and if \( u \) is any positive solution, then \( u\left( t\right) - {u}^{ * }\left( t\right) \rightarrow 0 \) as \( t \rightarrow \infty \) . | Proof. Let \( \alpha ,\beta ,\gamma ,\delta \) be positive constants with \( \alpha < b < \beta ,\gamma < c/b < \delta \) in \( \mathbb{R} \) . The first set of these inequalities leads to the estimates\n\n\[ \n{\alpha t} < B\left( t\right) < {\beta t}\text{for}t > 0,{\alpha t} > B\left( t\right) > {\beta t}\text{for}t... | Yes |
The inequality \( {Pv} \geq 0 \) holds for \( v = {\mathrm{e}}^{-x} \), and the inequality \( {Pw} \leq 0 \) is satisfied by the function \[ w\left( x\right) = \left\{ \begin{array}{lll} 2 - x & \text{ for } & 0 \leq x \leq 1 \\ 1/x & \text{ for } & x > 1 \end{array}\right. \] Thus there exists a global solution \( \ph... | The reader should show that \( {v}_{1} = 1/\left( {x + {x}^{-2}}\right) \) is also a lower bound. | No |
One can chose \( v = - \left( {x + 1}\right), w = - x \), as can be easily seen. Thus there exists a global solution \( \phi \) that satisfies the inequality \( - \left( {x + 1}\right) < \) \( \phi \left( x\right) < - x \) . | The reader should show that \( {v}_{1} = - x - 1/3{x}^{2} \) is a better lower bound. | No |
Here the global solutions \( \phi ,\psi \) are bounded; that is, there exists \( L > 0 \) such that \( 0 < \phi < \psi < L \) holds in \( \lbrack 0,\infty ) \) and hence \( {f}_{y}\left( {x, y}\right) = 1/{y}^{2} > \) \( 1/{L}^{2} \mathrel{\text{:=}} \alpha \) . | \[ {u}^{\prime } \geq {\alpha u},\;\text{ which implies that }\;u\left( x\right) \geq u\left( 0\right) {\mathrm{e}}^{\alpha x}. \] But \( u \) is bounded by assumption. This contradiction proves the assertion made at the beginning that there is only one bounded global solution. The estimate \( {v}_{1} < \phi < 1/x \) (... | Yes |
Let \( y \) be a solution and \( y\left( a\right) \geq - a \) for some \( a \geq 0 \) . It is easy to see from the differential equation that there exists \( b > a \) with \( y\left( b\right) > 0 \) . Since the solution of the initial value problem \( {v}^{\prime } = {v}^{3}, v\left( b\right) = y\left( b\right) \) is a... | If \( \phi \) and \( \psi \) are global solutions with \( \phi < \psi \), then accordingly, \( \psi \left( x\right) < - x \) . Thus in (14) we have \( {f}_{y}\left( {x,{y}^{ * }}\right) = 3{y}^{*2} > 3{x}^{2} \), and hence \( u = \psi - \phi \geq \delta \exp \left( {x}^{3}\right) \) , where \( \delta = u\left( 0\right)... | Yes |
Corollary 1. If the sets \( A \) and \( B \) are homeomorphic and if \( A \) has the fixed point property, then \( B \) also has the fixed point property. | The proof is very simple. Let \( h : A \rightarrow B \) be a homeomorphism and \( f \) : \( B \rightarrow B \) a continuous mapping. Then \( F = {h}^{-1} \circ f \circ h \) is a continuous mapping of \( A \) to itself. If \( x \) is a fixed point of \( F \), then the image point \( \xi = h\left( x\right) \) is a fixed ... | Yes |
Corollary 2. Let the set \( A \subset {\mathbb{R}}^{n} \) be compact, and let there exist a continuous mapping \( P : {\mathbb{R}}^{n} \rightarrow A \) with \( {\left. P\right| }_{A} = {\operatorname{id}}_{A} \), i.e., \( P\left( x\right) = x \) for \( x \in A \) . Then \( A \) has the fixed point property. | For the proof let \( B \supset A \) be a closed ball and \( f : A \rightarrow A \) continuous. Then \( F = f \circ P \) is a continuous mapping of \( B \) into itself. By the Brouwer fixed point theorem, \( F \) has a fixed point \( \xi \), and because \( F\left( B\right) \subset A \), this fixed point belongs to \( A ... | Yes |
Corollary 3. A nonempty, convex, and compact set \( A \subset {\mathbb{R}}^{n} \) has the fixed point property. | Proof. For every \( x \in {\mathbb{R}}^{n} \) there exists, since \( A \) is convex and compact, exactly one \ | No |
Floquet 表示定理: 考虑 Floquet 系统 \( {y}^{\prime } = A\left( x\right) y \) ,假设 \( A\left( x\right) \) 是 \( \omega \) 周期连续的矩阵函数,则(i) 系统的每个基本解矩阵 \( \Phi \left( \mathrm{x}\right) \) 均可表示 \[ \Phi \left( \mathsf{x}\right) = \mathsf{P}\left( \mathsf{x}\right) {\mathsf{e}}^{\mathsf{B}\mathsf{x}},\;\left( *\right) \] 其中 \( P\left( x... | 证: 由对数矩阵存在定理可知,存在常数矩阵 \( \mathrm{B} \) ,使得 \( {\mathrm{e}}^{\omega \mathrm{B}} = \mathrm{C} \) . 要证存在矩阵 \( P\left( x\right) \) 满足等式 \( \left( *\right) \) ,即 \( \Phi \left( x\right) = P\left( x\right) {e}^{Bx} \) ,只要证明 \( P\left( x\right) \mathrel{\text{:=}} \Phi \left( x\right) {\mathrm{e}}^{-{Bx}} \) 满足要求即可. 显然 \( P\l... | Yes |
定理: 考虑 \( {y}^{\prime } = f\left( {x, y}\right) \) ,这里函数 \( f\left( {x, y}\right) \) 在平面开域 \( \Omega \) 上满足标准假设. 设 \( y = \phi \left( x\right) \) 是饱和解,它的最大存在区间为 \( \left( {\alpha ,\beta }\right) \) . 对右端点 \( \beta \) ,必然发生以下三种情况之一:\n(i) \( \beta = + \infty \) ;\n(ii) \( \beta < + \infty ,\phi \left( \mathrm{x}\right) \... | 证: 只证 \( {\beta }_{1} \) 的存在性. 关于 \( {\alpha }_{1} \) 的存在性完全类似.\n\n情形一: \( \beta = + \infty \) . 由于紧集 \( {\Omega }_{1} \) 有界,故存在 \( A > 0, B > 0 \) ,使得 \( {\Omega }_{1} \) 包含在开矩形 \( \left| x\right| < A,\left| y\right| < B \) 之中. 取 \( {\beta }_{1} = A + 1 \) 即可使得 \( \left( {\mathrm{x},\phi \left( \mathrm{x}\right) }\rig... | Yes |
Problem 1 Are there scalar-flat non Ricci-flat ALE spaces with vanishing ADM mass? | This seemed to the author a very natural guess, a sort of Liouville-type Theorem for scalar-flat metrics. In fact this turns out to be largely false (see Le Brun [22], Section 6, p. 244, and [24], Example 2, Section 6.7) and motivated a beautiful work by Hein-Lebrun [18] which will be precisely quoted and commented in ... | No |
Proposition 3.2 Let \( \\left( {M, g,\\omega }\\right) \) be a compact extremal Kähler manifold with extremal vector field \( {X}_{s} \), with \( T \) -invariant metric \( g \) and \( {\\mu }_{\\omega } \) a normalized moment map for the action of \( G \). Let \( f \\in {C}^{\\infty }{\\left( M\\right) }^{T} \) such th... | \[ \\mathcal{E}\\left( {f, c + \\frac{1}{\\operatorname{vol}\\left( M\\right) }{\\int }_{M}{s}_{\\omega }d{\\mu }_{\\omega },{X}_{s} + X}\\right) = - \\frac{1}{2}{\\mathbb{L}}_{\\omega }\\left( f\\right) - \\frac{1}{2}\\left\\langle {\\nabla {s}_{\\omega },\\nabla f}\\right\\rangle - \\left\\langle {{\\mu }_{\\omega },... | Yes |
Proposition 3.4 Let \( \\left( {X,\\eta }\\right) \) be a scalar flat ALE resolution of \( {\\mathbb{C}}^{m}/\\Gamma \), then the centraliser of \( \\Gamma \) in \( U\\left( m\\right) \) satisfies\n\n\[ \n{C}_{U\\left( m\\right) }\\left( \\Gamma \\right) \\subset {\\operatorname{Iso}}_{0}\\left( {X,\\eta }\\right) .\n\... | We point out that a consequence of Proposition 3.4 is that \( \\eta \) is invariant for the action of any torus in \( {C}_{U\\left( m\\right) }\\left( \\Gamma \\right) \) in particular for the action of the special torus \( \\widetilde{T} \) we chose at the beginning. Moreover, as explained in the proof, given a vector... | No |
Theorem 6.1 Let \( \Gamma \vartriangleleft U\left( m\right) \) finite acting freely on \( {\mathbb{S}}^{{2m} - 1} \), let \( \left( {X,\omega }\right) \) be a scalar-flat ALE Kähler orbifold such that there is a compact \( K \subset X \) such that\n\n\[ X \smallsetminus K \simeq \left( {{\mathbb{C}}^{m} \smallsetminus ... | The proof of previous Theorem is nothing more than a check that the compact analysis passes to the relevant weighted analysis on non compact ALE spaces. | No |
Let \( \left( {X,\omega }\right) \) be a Kähler manifold and pr \( : X \rightarrow \Delta \) be a proper holomorphic map to the ball \( \Delta \subset {\mathbb{C}}^{1} \) centered at 0 of radius \( R \) . Let \( \left( {L, h}\right) \) be a holomorphic line bundle over \( X \) equipped with a hermitian metric (maybe si... | restriction to \( {X}_{0} \) is equal to \( f \), and such that the following optimal estimate holds\n\n\[ \n\frac{1}{\pi {R}^{2}}{\int }_{X}{\left| F\right| }_{\omega, h}^{2}d{V}_{X,\omega } \leq {\int }_{{X}_{0}}{\left| f\right| }_{\omega, h}^{2}d{V}_{{X}_{0},\omega }. \n\]\n\n(1)\n\nWe note that the volume form \( {... | Yes |
Corollary 3.1 Let \( \\left( {X,\\omega }\\right) \) be a Kähler manifold with a proper map \( \\operatorname{pr} : X \\rightarrow \\Delta \) to a ball \( \\Delta \\subset {\\mathbb{C}}^{1} \) centered at 0 of radius \( R \) . Let \( \\left( {L, h}\\right) \) be a holomorphic line bundle over \( X \) equipped with a he... | \[ \\frac{1}{\\pi {R}^{2}}{\\int }_{X}{\\left| F\\right| }_{\\omega, h}^{2}d{V}_{X,\\omega } \\leq {\\int }_{{X}_{0}}{\\left| f\\right| }_{\\omega, h}^{2}d{V}_{{X}_{0},\\omega } \] (34) and \( {\\left. F\\right| }_{{X}_{0}} = {\\operatorname{pr}}^{ * }\\left( {dt}\\right) \\land f \), where \( t \) is the standard coor... | Yes |
Corollary 3.2 Let \( \left( {X,\omega }\right) \) be a Kähler manifold and \( \operatorname{pr} : X \rightarrow \Delta \) be a proper map to the ball \( \Delta \subset {\mathbb{C}}^{1} \) centered at 0 of radius \( R \) . Let \( L \) be a holomorphic line bundle over \( X \) equipped with a hermitian metric (maybe sing... | Proof The proof given here follows closely [4, A.1]. Set \[ {C}_{1} \mathrel{\text{:=}} {\int }_{{X}_{0}}{\left| f\right| }_{\omega, h}^{\frac{2}{m}}d{V}_{{X}_{0},\omega }\;\text{ and }\;{C}_{2} \mathrel{\text{:=}} \frac{1}{\pi {R}^{2}}{\int }_{X}{\left| F\right| }_{\omega, h}^{\frac{2}{m}}d{V}_{X,\omega }. \] If \( {C... | Yes |
Theorem 3.3 ([4, Thm 0.1]) Let \( p : X \rightarrow Y \) be a fibration between two projective manifolds, and let \( \omega \) be a Kähler metric on \( X \) . Let \( L \rightarrow X \) be a line bundle endowed with a metric (maybe singular) \( h \) such that \( i{\Theta }_{h}\left( L\right) \geq 0 \) . Suppose that the... | An alternative proof of Theorem 3.3 is given by using the optimal extension proved in [13, Thm 2.1, Cor 3.7]. We should remark that, if \( {\varphi }_{L} \) has arbitrary singularity, the proof of in \( \left\lbrack {4\text{, Thm 0.1}}\right\rbrack \) uses the existence of ample line on \( X \) . Therefore the assumpti... | Yes |
Theorem 3.5 Let \( p : X \rightarrow Y \) be a proper fibration between two Kähler manifolds and let \( \omega \) be a Kähler metric on \( X \) . Let \( L \rightarrow X \) be a line bundle endowed with a metric (maybe singular) \( h = {h}_{0} \cdot {e}^{-\varphi } \) such that \( i{\Theta }_{h}\left( L\right) \geq 0 \)... | Proof By Lemma 3.4, \( {p}_{ * }\left( {{\left( {K}_{X/Y}\right) }^{m} \otimes L \otimes {\mathcal{I}}_{m}\left( \varphi \right) }\right) \) is coherent. Using [12] (cf. also [5, Thm 10.7, p. 47]), there exists a subvariety \( Z \) of \( Y \) of codimension at least 1 such that \( p \) is smooth on \( Y \smallsetminus ... | Yes |
Theorem 6 Given an abstract Euclidean crystallographic group there is a unique class of affine realization, for each field \( K \supset \mathbb{Z} \) . | Proof \( {Ad} : \Gamma \rightarrow {GL}\left( \Lambda \right) \) makes \( \Lambda \) a \( \Gamma \) -module, a trivial \( \Lambda \) module, hence also a \( G \) -module.\n\nWe have seen in Remark 5,(a), that an affine realization is given by a cocycle \( {u}_{\gamma } \) in \( {Z}^{1}\left( {\Gamma ,{V}_{K}}\right) \)... | Yes |
Proposition 8 Suppose that \( \Gamma \) is an Euclidean crystallographic group, which by Theorem 6 is a subgroup of \( \operatorname{Aff}\left( {V}_{\mathbb{Q}}\right) \) . Consider a lattice \( {\Lambda }^{\prime } \subset \Lambda { \otimes }_{\mathbb{Z}}\mathbb{Q} \), such that \( \Lambda \subset {\Lambda }^{\prime }... | Proof The group \( {\Gamma }^{\prime } \) is obtained from \( \Gamma \) by adding some translations, which lie in the kernel of \( {Ad} : {Aff}\left( {V}_{\mathbb{Q}}\right) \rightarrow {GL}\left( {V}_{\mathbb{Q}}\right) \), so the image \( {Ad}\left( {\Gamma }^{\prime }\right) \) coincides with \( G = {Ad}\left( \Gamm... | Yes |
Theorem 1 Let \( \{ \theta \} \) be a big cohomology class and \( \varphi \in \mathcal{E}\left( {X,\theta }\right) \) . Then \( \psi \in \mathcal{E}\left( {X,\theta }\right) \) if and only if \( {P}_{\left\lbrack \theta ,\varphi \right\rbrack }\left( \psi \right) = \psi \) . | We refer to [8, Theorem 1.2] for a proof. | No |
Theorem 2 Assume that \( \theta \) is a smooth closed \( \left( {1,1}\right) \) -form such that \( \{ \theta \} \) is big. Let \( {V}_{\theta } \) be the envelope of \( \theta \) . Then we have the following:\n\n(i) for any \( \varphi \in \mathcal{E}\left( {X,\theta }\right) \) we have\n\n\[ v\left( {\varphi, x}\right)... | Proof We first argue (i). From Theorem 1 it follows that \( {P}_{\left\lbrack \theta ,\varphi \right\rbrack }\left( {V}_{\theta }\right) = {V}_{\theta } \) . Take any \( x \in X \) . Then trivially \( v\left( {\varphi, x}\right) \geq v\left( {{V}_{\theta }, x}\right) \) . We will argue by contradiction. Assume that \( ... | Yes |
Corollary 1 Let \( \left\{ {\theta }_{1}\right\} ,\left\{ {\theta }_{2}\right\} \) be big and nef classes. Then for any \( {\varphi }_{1} \in \operatorname{PSH}\left( {X,{\theta }_{1}}\right) \) and \( {\varphi }_{2} \in \operatorname{PSH}\left( {X,{\theta }_{2}}\right) \) we have \[ {\varphi }_{1} + {\varphi }_{2} \in... | Proof The implication \( \left( \Rightarrow \right) \) is proved in [10, Theorem B]. For the reverse implication, fix a Kähler form \( \omega \) such that \( {\theta }_{j} \leq \omega, j = 1,2 \) . It follows from part (ii) of Theorem 2 that \( {\varphi }_{j} \in \mathcal{E}\left( {X,\omega }\right) ,\forall j = 1,2 \)... | Yes |
We want to show that \( T + \omega \notin \mathcal{E}\left( {X,{\alpha }_{1} + {\alpha }_{2}}\right) \). | Now, from the multilinearity of the non-pluripolar product we get\n\n\[ \n{\int }_{X}\left\langle {\left( T + \omega \right) }^{2}\right\rangle = {\int }_{X}\left\langle {\left( {\pi }^{ \star }{\omega }_{FS} + \left\lbrack E\right\rbrack + \omega \right) }^{2}\right\rangle = {\int }_{X}\left\langle {\left( {\pi }^{ \s... | Yes |
Proposition 1 Let \( \alpha \) be a big and nef cohomology class and \( T \in \mathcal{E}\left( {X,\alpha }\right) \), then\n\n\[ T = \langle T\rangle \]\n\nMoreover, when \( \alpha \) is merely big and \( {\dim }_{\mathbb{C}}X = 2 \) we have that given \( T \in \mathcal{E}\left( {X,\alpha }\right) \), \n\n\[ \langle T... | Proof First we prove that, given \( {T}_{\min } \in \alpha \) then \( {T}_{\min } = \left\langle {T}_{\min }\right\rangle \) . Observe that \( {T}_{\min } - \left\langle {T}_{\min }\right\rangle \) is a \( \left( {1,1}\right) \) -currents that is supported on the non-Kähler locus \( {E}_{nk}\left( \alpha \right) = \) \... | Yes |
Proposition 1.2 Let \( \left( {X,\omega }\right) \) be a compact Kähler manifold such that \( {\mathrm{{HSC}}}_{\omega } \leq 0 \) , and suppose there exists a direction \( \left\lbrack v\right\rbrack \in P\left( {T}_{X,{x}_{0}}\right) \) such that \( {\operatorname{HSC}}_{\omega }\left( {{x}_{0},\left\lbrack v\right\r... | Sketch of the Proof A computation shows (see for instance [1], or [6, Section 2.1] for a more general computation) that, up to a positive constant multiple (which depends only on \( \dim X \) ), we have for all \( x \in X \)\n\n\[ \n{s}_{\omega }\left( x\right) \simeq {\int }_{P\left( {T}_{X, x}\right) }{\operatorname{... | Yes |
Lemma 2.1 (Exercise 8, page 219 of [3]) Let \( X \) be a smooth projective variety of general type which contains no rational curves. Then, \( {K}_{X} \) is ample. | Proof Since there are no rational curves on \( X \), Mori’s theorem implies as above that \( {K}_{X} \) is nef. Since \( {K}_{X} \) is big and nef, the Base Point Free theorem tells us that \( {K}_{X} \) is semi-ample. If \( {K}_{X} \) were not ample, then the morphism defined by (some multiple of) \( {K}_{X} \) would ... | Yes |
Lemma 3.1 The function \( {T}_{\varepsilon } \) satisfies the following inequality:\n\n\[ \n{T}_{\varepsilon } > - \frac{{u}_{\varepsilon }}{n} \n\]\n\nIn particular, if \( \left\{ {u}_{\varepsilon }\right\} \) converges uniformly to \( - \infty \) on \( X \), then \( {T}_{\varepsilon } \) converges uniformly to \( + \... | Proof Let \( 0 < {\lambda }_{1} \leq \cdots \leq {\lambda }_{n} \) be the eigenvalues of \( {\omega }_{\varepsilon } \) with respect to \( \omega \), so that \( 0 < 1/{\lambda }_{n} \leq \cdots \leq 1/{\lambda }_{1} \) are the eigenvalues of \( \omega \) with respect to \( {\omega }_{\varepsilon } \) . Then,\n\n\[ \n{e... | Yes |
Problem 1.5 Let \( n > 1 \) . Does there exist a surjective holomorphic map \( {\mathbb{C}}^{n} \rightarrow \) \( {\mathbb{C}}^{n} \smallsetminus {\overline{\mathbb{B}}}^{n}? \) | In this connection, we mention that Dixon and Esterle (see [6, Theorem 8.13, p. 182]) constructed for every \( \epsilon > 0 \) a finitely sheeted holomorphic map \( f : {\mathbb{C}}^{2} \rightarrow {\mathbb{C}}^{2} \) whose image avoids the closed unit ball \( {\overline{\mathbb{B}}}^{2} \) but contains the complement ... | Yes |
Theorem 1.6 Assume that \( X \) is a compact algebraically subelliptic manifold and \( S \) is an affine algebraic manifold such that \( \dim S \geq \dim X \) . Then, every algebraic map \( S \rightarrow X \) is homotopic (through algebraic maps) to a surjective strongly dominating algebraic map \( S \rightarrow X \) .... | The proof of Theorem 1.6 is based on Theorem 4.1 which is taken from [11]. It says in particular that, given an affine algebraic manifold \( S \) and an algebraically subelliptic manifold \( X \), a holomorphic map \( S \rightarrow X \) that is homotopic to an algebraic map through a family of holomorphic maps can be a... | No |
Let \( X \) be an Oka manifold, and let \( f : S \rightarrow X \) be a continuous map which is holomorphic on a neighborhood of the set \( K = { \cup }_{j = 1}^{\infty }{K}_{j} \) . Let dist be a distance function on \( X \) inducing the manifold topology. Given a sequence \( {\epsilon }_{j} > 0\left( {j \in \mathbb{N}... | Proof We may assume that dist is a complete metric on \( X \) and that \( \mathop{\sum }\limits_{j}{\epsilon }_{j} < \infty \) . Let \( {\left( {a}_{j}\right) }_{j \in \mathbb{N}} \) be the sequence of real numbers in condition (c). Set\n\n\[{S}_{j} \mathrel{\text{:=}} \left\{ {p \in S : \rho \left( p\right) \leq {a}_{... | No |
Theorem 4.1 Assume that \( S \) is an affine algebraic manifold and \( X \) is an algebraically subelliptic manifold. Given an algebraic map \( f : S \rightarrow X \), a compact \( \mathcal{O}\left( S\right) \) -convex subset \( K \) of \( S \), an open set \( U \subset S \) containing \( K \), and a homotopy \( {f}_{t... | Proof of Theorem 1.6 The proof uses Theorem 4.1 and is similar to that of Theorem 1.1. The main difference is that the initial map \( f : S \rightarrow X \) must be algebraic. For the sake of simplicity, we present the details only in the special case when \( S = {\mathbb{C}}^{n} \) with \( n = \dim X \) .\n\nFix a poi... | Yes |
Theorem 3.1 (Hind-Lisi, [13], Hutchings, [15], Christianson-Nelson, [2]) If \( 1 \leq x \leq 2 \) then \( {p}_{2}\left( x\right) = 1 + x \) . | To be precise, Hind-Lisi established the case \( x = 2 \), Hutchings dealt with all \( x \leq {12}/5 \) and Christianson-Nelson completed the theorem as stated. | Yes |
For every \( D \in {\mathcal{D}}^{k}\left( M\right) \) there exist unique forms \( \Phi \in {\Lambda }^{k}M \otimes {TM},\Psi \in {\Lambda }^{k + 1}M \otimes \) TM such that\n\n\[ D = {\mathcal{L}}_{\Phi } + {\mathcal{I}}_{\Psi } \] | so\n\n\[ {\mathcal{D}}^{ * }\left( M\right) = \mathcal{L}\left( M\right) \oplus \mathcal{I}\left( M\right) \]\n\nWe denote \( {\mathcal{L}}_{\Phi } = \mathcal{L}\left( D\right) \) and \( {\mathcal{I}}_{\Psi } = \mathcal{I}\left( D\right) \) | No |
Theorem 1 Let \( \Phi \in {\Lambda }^{1}M \otimes {TM} \) such that \( {R}_{\Phi } = I{d}_{T\left( M\right) } + \Phi \) is invertible. Then: (i) \( {e}_{\Phi } \) is a solution of the Maurer-Cartan equation in \( \left( {{\mathcal{D}}^{ * }\left( M\right) ,\neg ,\left\lbrack {\cdot , \cdot }\right\rbrack }\right) \) . ... | \[ {e}_{\Phi } = {\mathcal{L}}_{\Phi } + {\mathcal{I}}_{b\left( \Phi \right) } \] | No |
Theorem 4 Let \( \Phi \in {\Lambda }^{1}M \otimes {TM} \) such that \( {R}_{\Phi } = I{d}_{T\left( M\right) } + \Phi \) is invertible. Then \[ {T}_{{e}_{\Phi }}\left( {\mathfrak{{MC}}\left( M\right) }\right) = \left\{ {{\mathcal{L}}_{\Psi } + {\mathcal{I}}_{\omega \left( \Psi \right) } : \Psi \in {\Lambda }^{1}M \otime... | Proof Let \( t \mapsto {e}_{\sigma \left( t\right) } \) a \( \mathfrak{{MC}}\left( M\right) \) -valued curve through \( {e}_{\Phi } \), with \( \sigma \left( t\right) = \Phi + {t\Psi } + o\left( t\right) \) , \( \Psi \in {\Lambda }^{1}M \otimes {TM} \) . We have \[ {e}_{\sigma \left( t\right) } = {\mathcal{L}}_{\sigma ... | Yes |
Theorem 5 Let \( \Phi \in {\Lambda }^{1}M \otimes {TM} \) such that \( {R}_{\Phi } = I{d}_{T\left( M\right) } + \Phi \) is invertible.\n\n(1) Suppose \( {e}_{\Phi } \in {\mathfrak{{MC}}}_{0}\left( M\right) \). Then\n\n\[ \n{T}_{{e}_{\Phi }}\left( {\mathfrak{M}{\mathfrak{C}}_{0}\left( M\right) }\right) \subset \left\{ {... | Proof\n\n(1) Let \( t \mapsto {e}_{\sigma \left( t\right) } \) a \( \mathfrak{M}{\mathfrak{C}}_{0}\left( M\right) \) -valued curve through \( {e}_{\Phi } \), with \( \sigma \left( t\right) = \Phi + {t\Psi } + o\left( t\right) \),\n\n\( \Psi \in {\Lambda }^{1}M \otimes {TM} \) and \( \left\lbrack {\sigma \left( t\right)... | Yes |
Corollary 2 \( {T}_{0}\left( {\mathfrak{M}{\mathfrak{C}}_{0}\left( M\right) }\right) = \left\{ {{\mathcal{L}}_{\Psi } : \Psi \in {\Lambda }^{1}M \otimes {TM},\left\lbrack {\Psi ,\Psi }\right\rbrack = 0}\right\} \) . | Proof By Theorem 5, \( {T}_{0}\left( {\mathfrak{M}{\mathfrak{C}}_{0}\left( M\right) }\right) \subset \left\{ {{\mathcal{L}}_{\Psi } : \Psi \in {\Lambda }^{1}M \otimes {TM},\left\lbrack {\Psi ,\Psi }\right\rbrack = 0}\right\} \) . Conversely, if \( \Psi \in {\Lambda }^{1}M \otimes {TM} \) and \( \left\lbrack {\Psi ,\Psi... | Yes |
To begin with, let us consider the Teichmüller space of \( {\mathbb{S}}^{1} \times {\mathbb{S}}^{1} \) . By Riemann’s uniformization theorem, every Riemann surface diffeomorphic to \( {\mathbb{S}}^{1} \times {\mathbb{S}}^{1} \) is a compact complex torus, that is the quotient of \( \mathbb{C} \) by a lattice \( \mathbb... | A classical computation shows that the lattice can be assumed to be of the form \( \mathbb{Z} \oplus \mathbb{Z}\tau \) with \( \tau \) belonging to the upper half-plane \( \mathbb{H} \) . Moreover, two such lattices give rise to biholomorphic tori if and only they are related through the formula\n\n\[ \n{\tau }^{\prime... | Yes |
It was proved by Kodaira in [5] that any complex surface homeomorphic to \( {\mathbb{S}}^{3} \times {\mathbb{S}}^{1} \) is a Hopf surface, that is the quotient of \( {\mathbb{C}}^{2} \smallsetminus \{ \left( {0,0}\right) \} \) by the group generated by a holomorphic contraction of \( {\mathbb{C}}^{2} \) . | Besides, every such contraction can be either linearized and then diagonalized with eigenvalues of modulus strictly less than 1 , or reduced to the following resonant normal form\n\n\[ \left( {z, w}\right) \mapsto {g}_{\lambda, p}\left( {z, w}\right) \mathrel{\text{:=}} \left( {{\lambda z} + {w}^{p},{\lambda }^{p}w}\ri... | Yes |
Proposition 1 The Teichmüller space \( \mathcal{T}\left( {{\mathbb{S}}^{3} \times {\mathbb{S}}^{1}}\right) \) cannot be endowed with the structure neither of an analytic space nor of a non-Hausdorff analytic space. | Proof An analytic space, even non-Hausdorff, is locally Hausdorff since it is locally modelled onto the zero set of some holomorphic functions in \( {\mathbb{C}}^{n} \) . This contradicts (13) and the subsequent discussion. | Yes |
Example 4 An easy but yet interesting example is that of a linear foliation of a torus. Consider the trivial foliation of \( {\mathbb{R}}^{2} \) given by parallel straight lines making a fixed angle \( \alpha \) with the horizontal. It descends on the torus \( {\mathbb{R}}^{2}/{\mathbb{Z}}^{2} \) as a foliation by curv... | 1. If \( \alpha = p/q \) is rational then the leaves of the foliation are closed curves diffeomorphic to \( {\mathbb{S}}^{1} \) that make \( p \) turns in the vertical direction and \( q \) in the horizontal one. The foliation has only compact leaves.\n\n2. If \( \alpha \) is irrational then the leaves are diffeomorphi... | Yes |
Theorem 2 Let \( {X}_{0} = \left( {M,{J}_{0}}\right) \) be a compact complex manifold whose underlying smooth structure is \( M \) . Assume (20). Then there exists a finite-dimensional analytic space \( {K}_{0} \) such that the space \( \mathcal{I}\left( M\right) \) is locally isomorphic to \( {K}_{0} \times {\operator... | Of course this local isomorphism preserves locally the action, so this gives really a foliated chart for the action. The plaques are open neighborhoods of the identity in the Fréchet manifold \( {\operatorname{Diff}}^{0}\left( M\right) \) and a transverse local section is given by the analytic space \( {K}_{0} \) . So ... | No |
Theorem 3 Let \( {X}_{0} = \left( {M,{J}_{0}}\right) \) be a compact complex manifold whose underlying smooth structure is \( M \) . Let \( {\operatorname{Aut}}^{0}\left( {X}_{0}\right) \) be the connected component of the identity in the automorphism group of \( {X}_{0} \) . Then\n\n1. A neighborhood of the identity i... | As in the previous case, this local isomorphism preserves locally the action, but this time this does not give a foliated chart for the action. The problem is that the plaques are now modelled onto \( \left( {{\operatorname{Diff}}^{0}\left( M\right) /{\operatorname{Aut}}^{0}\left( {X}_{0}\right) }\right) \), i.e. depen... | Yes |
Lemma 2.2 Let \( M \) be a complex Hermitian manifold, \( \mathcal{F} \subset {H}^{0}\left( {\mathcal{O}}_{M}\right) \) a normal family, and \( K \subset M \) a compact subset. Then there exists a number \( {A}_{K} > 0 \) such that \( \mathop{\sup }\limits_{K}\left| {f}^{\prime }\right| \leq {A}_{K} \) | Proof By contradiction, suppose there exists \( x \in K, v \in {T}_{x}M \), and a sequence \( {f}_{i} \in \mathcal{F} \) such that \( \mathop{\lim }\limits_{i}\left| {{D}_{v}{f}_{i}}\right| = \infty \) . We choose a disk \( \Delta \overset{j}{ \hookrightarrow }M \) with compact closure in \( M \), tangent to \( v \) in... | Yes |
Theorem 2.7 (Montel) Let \( M \) be a complex manifold and \( \mathcal{F} \subset {H}^{0}\left( {\mathcal{O}}_{M}\right) \) a normal family of functions. Denote by \( \overline{\mathcal{F}} \) its closure in the \( {\mathcal{C}}^{0} \) -topology. Then \( \overline{\mathcal{F}} \) is compact and contained in \( {H}^{0}\... | Proof Let \( \left\{ {f}_{i}\right\} \) be a sequence of functions in \( \mathcal{F} \) . By Tychonoff’s theorem, for each compact \( K \), there exists a subsequence of \( \left\{ {f}_{i}\right\} \) which converges pointwise on a dense countable subset \( Z \subset K \) . Taking a diagonal subsequence, we find a subse... | Yes |
Theorem 2.8 Let \( M \) be a complex manifold, and \( {H}_{b}^{0}\left( {\mathcal{O}}_{M}\right) \) the space of all bounded holomorphic functions, equipped with the sup-norm \( {\left| f\right| }_{\text{sup }} \mathrel{\text{:=}} \mathop{\sup }\limits_{M}\left| f\right| \) . Then \( {H}_{b}^{0}\left( {\mathcal{O}}_{M}... | Proof Let \( \left\{ {f}_{i}\right\} \in {H}_{b}^{0}\left( {\mathcal{O}}_{M}\right) \) be a Cauchy sequence in the sup-norm. Then \( \left\{ {f}_{i}\right\} \) converges to a continuous function \( f \) in the sup-topology.\n\nSince \( \left\{ {f}_{i}\right\} \) is a normal family, it has a subsequence which converges ... | Yes |
Theorem 2.14 Let \( X \) be a complex variety, and let \( \gamma : X \rightarrow X \) be a holomorphic contraction such that \( \gamma \left( X\right) \) is precompact. Consider the Banach space \( V = {H}_{b}^{0}\left( {\mathcal{O}}_{X}\right) \) with the sup-metric. Then \( {\gamma }^{ * } : V \rightarrow V \) is com... | Proof Let \( {B}_{C} \mathrel{\text{:=}} \left\{ {v \in V\left| \right| v{\left. \right| }_{\sup } \leq C}\right\} \) . Then\n\n\[ \n{\left| {\gamma }^{ * }f\right| }_{\sup } = \mathop{\sup }\limits_{{x \in \overline{\gamma \left( X\right) }}}\left| {f\left( x\right) }\right| .\n\]\n\nTherefore, for any sequence \( \le... | Yes |
Proposition 4.3 Fix a precompact subset \( {\widetilde{M}}_{c}^{a} \mathrel{\text{:=}} {\phi }^{-1}\left( {\lbrack 0, a\lbrack }\right) \), where \( \phi : {\widetilde{M}}_{c} : \) \( \rightarrow {\mathbb{R}}^{ > 0} \) is the Kähler potential. Let \( A \) be the ring of bounded holomorphic functions on \( {\widetilde{M... | Proof Since \( {P}_{k}\left( t\right) \) is a minimal polynomial of \( {\gamma }^{ * } \) on \( A/{\mathfrak{m}}^{k} \), the endomorphism \( {P}_{k}\left( {\gamma }^{ * }\right) \) acts trivially on \( A/{\mathfrak{m}}^{k} \), by Cayley-Hamilton theorem, hence it maps \( A \) to \( {\mathfrak{m}}^{k} \) .\n\nFrom Riesz... | Yes |
Proposition 4.4 Let \( {H}^{0}{\left( {\mathcal{O}}_{{\widetilde{M}}_{c}}\right) }_{\text{fin }} \subset {H}^{0}\left( {\mathcal{O}}_{{\widetilde{M}}_{c}}\right) \) be the set of \( {\gamma }^{ * } \) -finite functions and \( \mathfrak{m} \) the maximal ideal of the origin in \( {\widetilde{M}}_{c} \) . Then \( {H}^{0}... | Proof A subspace \( V \subset A \) is dense in \( \mathfrak{m} \) -adic topology in \( A \Leftrightarrow \) the quotient \( V/v \cap {\mathfrak{m}}^{k} \) surjects to \( A/{\mathfrak{m}}^{k} \) . This is proven in Proposition 4.3 for the ring of bounded holomorphic functions on \( {\widetilde{M}}_{c}^{a} \) . However, ... | Yes |
Theorem 5.6 Assume that the spherical shell conjecture is true. Then Theorems 1.1 and 1.3 are true in dimension 2. | Proof The only part of the proof missing for dimension 2 is Rossi-Andreotti-Siu Theorem (3.2), see also Remark 1.4. We used it to prove the following result (which is stated here as a conjecture, because we don’t know how to prove it for class \( {\mathrm{{VII}}}_{0} \) non-Kato surfaces).\n\nConjecture 5.7 Let \( M \)... | No |
Lemma 1.2 Every adjoint orbit \( M \) in \( {\mathfrak{g}}_{u} \) intersects \( {iC}\left( {H}_{0}\right) \) in exactly one point. | Proof Let \( f\left( X\right) = b\left( {X,{H}_{0}}\right) \) . Since \( M \) is compact, \( f \) has stationary points on \( M \) . A stationary point \( {X}_{0} \) is characterized by\n\n\[ \n{df}\left( {X}_{0}\right) = 0 \Leftrightarrow 0 = b\left( {\left\lbrack {X,{X}_{0}}\right\rbrack ,{H}_{0}}\right) = b\left( {X... | Yes |
Theorem 1.4 On a flag manifold \( M \) of \( {\mathbf{G}}_{u} \) it is possible to define a complex structure and a \( {\mathbf{G}}_{u} \) -invariant Kähler structure. | Proof Fix a point \( {p}_{0} \) of \( M \), corresponding to \( {\Upsilon }_{0} \in {\mathfrak{g}}_{u} \) . The stabilizer \( {\mathbf{E}}_{{\mathbf{G}}_{u}}\left( {\Upsilon }_{0}\right) \) contains a maximal torus \( \mathbf{T} \) and therefore there are finitely many parabolic subalgebras \( \mathfrak{q} \) of the co... | No |
Theorem 1.5 There are finitely many orbits in \( {\mathcal{M}}_{ + } \) and in \( {\mathcal{M}}_{ - } \) . | The proof of Theorem 1.5 is done by considering the elements of \( {\mathcal{M}}_{ \pm } \) in the Grassmannian of \( \dim \left( \mathfrak{q}\right) \) -subspaces of \( \mathfrak{g} \) . On each orbit we can pick a \( {\mathfrak{q}}^{\prime } \) containing a \( \theta \) -stable Cartan subalgebra of \( \mathfrak{g} \)... | Yes |
Proposition 1.11 The inclusion \( {M}_{0}\left( p\right) \hookrightarrow {M}_{ - }\left( p\right) \) is a generic CR embedding. A necessary and sufficient condition for \( {M}_{0}\left( p\right) \) to be \( \mathfrak{n} \) -reductive is that | \[ {M}_{0}\left( p\right) = {M}_{ + }\left( p\right) \cap {M}_{ - }\left( p\right) \] | No |
Proposition 2.2 The map (11) is onto and we can find \( r > 0 \) such that its restriction to \( \left\{ {b\left( {T, T}\right) < {r}^{2}}\right\} \) is a diffeomorphism with the image. | Let \( {\mathbf{K}}_{0}{ \times }_{{\mathbf{V}}_{0}}{\mathfrak{m}}_{0} \) be the quotient of \( {\mathbf{K}}_{0} \times {\mathfrak{m}}_{0} \) by \( \left( {{x}_{1},{T}_{1}}\right) \sim \left( {{x}_{2},{T}_{2}}\right) \) iff \( {x}_{2} = {x}_{1} \cdot y \) and \( {T}_{1} = \operatorname{Ad}\left( y\right) \left( {T}_{2}... | Yes |
Proposition 2.5 Assume that \( {M}_{0} \) is q-pseudoconcave and has a CR algebra which is HNR. Then \( {M}_{ - } \) is q-pseudoconcave and \( \left( {n - q}\right) \) -pseudoconvex and the natural restriction maps\n\n\[ \n{H}_{\bar{\partial }}^{p, j}\left( {U}_{r}\right) \rightarrow {H}_{{\bar{\partial }}_{{M}_{0}}}^{... | Proof The statement follows from [4] and the computation of the signature of the exhaustion function \( \phi \) . | No |
Proposition 7 Let \( V \) be a real vector space and let \( {V}^{ \star } \) be the dual, a complex subspace \( L \subset \left( {V \oplus {V}^{ \star }}\right) \otimes \mathbb{C} \) is the holomorphic space of a pseudo calibrated generalized complex structure on \( V \) if and only if\n\n\[ L = L\left( {F,\epsilon }\r... | Proof Let (,) be the natural symplectic structure of \( V \oplus {V}^{ \star } \) and let \( J \) be a complex structure on \( V \oplus {V}^{ \star }\left( \right) \) ,)-invariant. Let \( L = \{ v - {iJv} \mid v \in V\} \), then \( L \) is a Lagrangian subspace of \( \left( {V \oplus {V}^{ \star }}\right) \otimes \math... | Yes |
Theorem 12 ([20]) For \( \lambda \left( {\lambda + 1}\right) \neq 0 \) the pseudo calibrated generalized complex structure \( \widehat{J} \) is \( \nabla \) -integrable if and only if the following conditions hold: | \[ \left\{ \begin{array}{l} N\left( H\right) = 0 \\ \nabla H = 0 \\ {d}^{\nabla }g = 0. \end{array}\right. \] | Yes |
Corollary 13 If \( H = 0 \) the pseudo calibrated generalized complex structure \( \widehat{J} = \) \( \left( \begin{matrix} 0 & - {g}^{-1} \\ g & 0 \end{matrix}\right) \) is \( \nabla \) -integrable if and only if the following condition holds: | \[ {d}^{\nabla }g = 0 \] | Yes |
Theorem 20 ([7]) Let \( \\left( {M, H, g}\\right) \) be a complex Norden manifold, there exists a unique linear connection \( D \) with torsion \( T \) on \( M \) such that:\n\n\[ \n\\left\\{ \\begin{array}{l} \\left( {{D}_{X}g}\\right) \\left( {Y, Z}\\right) = 0 \\\\ T\\left( {{HX}, Y}\\right) + T\\left( {X,{HY}}\\rig... | Remark 21 D is defined by: \( {D}_{X}Y = {\\nabla }_{X}Y - \\frac{1}{2}H\\left( {{\\nabla }_{X}H}\\right) Y \), where \( \\nabla \) is the Levi Civita connection of \( g \) and satisfies the condition \( \\overline{DH} = 0 \) . | Yes |
For \( \lambda \neq 0 \) the map: \( \psi : T\left( M\right) \otimes \mathbb{C} \rightarrow T\left( M\right) \otimes \mathbb{C} \) defined by: \( \psi \left( Z\right) = \) \( Z + {iHZ} \) is an isomorphism and the following holds\n\n\[ \psi \left( {Z - {iHZ}}\right) - {ig}\left( {\psi \left( Z\right) }\right) = - {\lam... | Proof \( \psi \) is injective if and only if \( i \) is not an eigenvalue of \( H \) . \( ▱ \) | No |
Corollary 30 If \( \lambda \neq 0 \) then:\n\n\[ \n{E}_{\widehat{J}}^{1,0} = \left\{ {-{\lambda Z} - {ig}\left( {Z + {iHZ}}\right) \mid Z \in {C}^{\infty }\left( {T\left( M\right) \otimes \mathbb{C}}\right) }\right\} \]\n\n\[ \n{E}_{\widehat{J}}^{0,1} = \{ - {\lambda Z} + {ig}\left( {Z - {iHZ}}\right) \mid Z \in {C}^{\... | Computing Jacobiator on \( {E}_{\widehat{J}}^{1,0} \) and \( {E}_{\widehat{J}}^{0,1} \) we get the following result: | No |
Proposition 32 ([19]) Let \( \\left( {M, H, g, D}\\right) \) be a complex Norden manifold with the natural canonical connection \( D \), Jacobi identity holds on \( {E}_{\\widehat{j}}^{1,0} \) and \( {E}_{\\widehat{j}}^{0,1} \) if and only if for all \( X, Y, Z \\in {C}^{\\infty }\\left( {T\\left( M\\right) }\\right) \... | \[ \n{R}^{D}\\left( {{HX},{HY}}\\right) - H{R}^{D}\\left( {{HX}, Y}\\right) - H{R}^{D}\\left( {X,{HY}}\\right) - {R}^{D}\\left( {X, Y}\\right) = 0 \n\] \n\[ \n\\left( {{R}^{D}\\left( {{HX}, Y}\\right) - {R}^{D}\\left( {X,{HY}}\\right) }\\right) Z + \\left( {{R}^{D}\\left( {{HZ}, X}\\right) + {R}^{D}\\left( {Z,{HX}}\\ri... | Yes |
Let \( \\left( {M, g}\\right) \) be a pseudo Riemannian manifold and let \( \\nabla \) be a linear connection on \( M \) . Let \( \\widehat{J} = \\left( \\begin{matrix} 0 & - {g}^{-1} \\\\ g & 0 \\end{matrix}\\right) \), then \( \\nabla = T\\left( M\\right) \) and, denoted \( L = {E}_{\\widehat{J}}^{1,0} \), we have \(... | In this case is \( {E}_{\\widehat{J}}^{1,0} = \\left\\{ {Z - {ig}\\left( Z\\right) \\mid Z \\in {C}^{\\infty }\\left( {T\\left( M\\right) \\otimes \\mathbb{C}}\\right) }\\right\\} \) . In particular, denoted by \( p : (T\\left( M\\right) \\oplus {\\left( T\\left( M\\right) \\right) }^{ \\star } \\otimes \\mathbb{C} \\r... | Yes |
Proposition 37 If \( {E}_{\widehat{j}}^{1,0} \) and \( {E}_{\widehat{j}}^{0,1} \) are complex Lie algebroids then \( {\left( {\bar{\partial }}_{\widehat{j}}\right) }^{2} = 0 \) and \( {\left( {\partial }}_{\widehat{J}}\right) }^{2} = 0 \) | Proof It follows from the fact that Jacobi identity holds on \( {E}_{\widehat{j}}^{1,0} \) and \( {E}_{\widehat{j}}^{0,1} \) .\n\nIt turns out that \( {\bar{\partial }}_{\widehat{j}} \) is the exterior derivative \( d \) of the Lie algebroid \( L = {E}_{\widehat{j}}^{1,0} \) and \( {\partial }}_{\widehat{J}} \) is the ... | Yes |
Theorem 1.1 ([44,45]) Each point \( p \in \left( {{M}^{4}, c\text{, or.}}\right) \) belongs to a neighbourhood in which there are zero, one, two or infinitely many distinct (pairs \( \pm J \) of) compatible complex structures. | The last alternative forces \( {W}_{ + } \) to vanish identically and therefore implies a result of Boyer that a hyperhermitian four-manifold is actually anti-self-dual: \( {W}_{ + } = 0 \) , see [13]. | No |
Proposition 1.4 ([6,12]) For the Lee forms \( {\theta }_{ \pm } \) of a bi-Hermitian surface \( \left( {M, c,{J}_{ \pm }}\right) \) the following holds:\n\n1. The differential of the sum of the Lee forms is anti-self-dual: \( d{\left( {\theta }_{ + } + {\theta }_{ - }\right) }_{SD} = 0 \)\n\n2. The pointwise norm satis... | Proof The proof of Salamon's theorem (1.1) shows that there exist severe constrains for the integrability of a compatible almost complex structure \( J \) on a conformal oriented four-manifold \( \left( {M, c,\text{or.}}\right) \) which where first described by Boyer in [12, Lemma 1] as follows: we can think of the sel... | Yes |
Proposition 2.3 ([6, Lemma 4]) When \( \left( {M, c,{J}_{ \pm }}\right) \) is compact bi-Hermitian with even first Betti number the two Lee forms always satisfy the generalized Kähler condition that\n\n\[{\theta }_{ + } + {\theta }_{ - }\;\text{is exact.}\] | Proof Let \( g \) be the \( {J}_{ + } \) -Gauduchon metric, as \( {b}_{1}\left( M\right) \) is even, we have \( {\theta }_{ + } = {\delta \alpha } \) . On the other hand \( d\left( {{\theta }_{ + } + {\theta }_{ - }}\right) = 0 \) so that the orthogonal decomposition of \( {\theta }_{ - } \) into harmonic, coexact and ... | Yes |
Proposition 2.4 ([2]) Let \( \\left( {M, c,{J}_{ \\pm }}\\right) \) be a compact bi-Hermitian surface with \( {b}_{1}\\left( M\\right) \) odd. Then, with respect to either complex structures \( {J}_{ \\pm } \), the degree of the fundamental line bundle satisfies\n\n\[ \n\\deg \\left( F\\right) \\leq 0 \n\]\n\nwith equa... | Proof As \( F \) corresponds to the closed 1 -form \( {\\theta }_{ + } + {\\theta }_{ - } \) its degree with respect to the \( {J}_{ + } \) -Gauduchon metric \( g \) with Lee form \( {\theta }_{ + } \) is given by the following formula in which \( \\langle \) , \( \\rangle {denotestheglobal}{L}^{2} \) -inner product, s... | Yes |
The canonical bundle of a compact bi-Hermitian surface \( S \) of odd first Betti number has negative degree, in particular \( \operatorname{Kod}\left( S\right) = - \infty \) and \( S \) is said to be in class VII of Kodaira classification. | Proof As the degree computes the signed volume of the divisor of a virtual meromorphic section and \( \mathbf{T} \) is an effective or zero divisor we have from the fundamental equation (2): \( 0 \leq \operatorname{vol}\left( \mathbf{T}\right) = \deg \left( F\right) - \deg \left( K\right) \) so that \( \deg \left( K\ri... | Yes |
Theorem 2.6 ([17,20,21,36,41]) For a surface \( S \) in class \( {VI}{I}_{0}^{ + } \) the following conditions are equivalent and they imply that \( S \) is diffeomorphic to \( \left( {{S}^{1} \times {S}^{3}}\right) \# {\overline{\mathbb{{CP}}}}_{2} \) . 1. \( S \) is a Kato surface 2. \( S \) contains \( {b}_{2}\left(... | As already mentioned, a non-trivial divisor of the form \( D = G - {mK} \) with \( {c}_{1}\left( G\right) = 0 \) is called a NAC (numerically anticanonical) divisor in the terminology of Dloussky [17] because its Chern class is a negative multiple of the canonical class. It is known that \( D \) is automatically effect... | Yes |
Proposition 2.8 ([2,17]) A minimal compact bi-Hermitian surface \( S \) with \( {b}_{1} \) -odd is either a Hopf surface when \( {b}_{2} = 0 \), or else is a Kato surface of index 1 when \( b \mathrel{\text{:=}} {b}_{2} > 0 \) . In particular, \( S \) is diffeomorphic to a finite quotient of \( {S}^{1} \times {S}^{3} \... | Proof Suppose \( {b}_{2}\left( S\right) = 0 \) and that \( S \) is not a Hopf surface, then by Bogomolov theorem \( S \) must be an Inoue-Bombieri surface which however is impossible by Corollary 2.5, as already mentioned. The case \( b = {b}_{2}\left( S\right) > 0 \) is handled by Theorem 2.6 (3) because the fundament... | Yes |
Proposition 2.10 ([23, Prop.4.7]) Let \( T \) be the set where the two complex structures of a compact bi-Hermitian surface \( \left( {M, c,{J}_{ \pm }}\right) \) are dependent (2) then \( {b}_{0}\left( T\right) \leq 2 \) . Furthermore, equality holds if and only if \( {b}_{1}\left( M\right) = 1 \) and the metric is ge... | Proof The fact that \( \mathbf{T} \) is disconnected when \( {b}_{1} \) is odd and \( \left( {g,{J}_{ \pm }}\right) \) is generalized Kähler was proved in [6, Prop.4]. Conversely, when \( {b}_{2}\left( S\right) \) is even we know from (9) that the fundamental divisor \( \mathbf{T} = - K \) is anticanonical and by Enriq... | No |
Theorem 3.3 ([25]) A Kato surface with \( {Dl}\left( S\right) \in \{ {2b},{3b}\} \) admits bi-Hermitian metrics with \( {b}_{0}\left( \mathbf{T}\right) = 1 \) if and only if \( S \) is a parabolic Inoue surface of real type. | Proof We know from the Dloussky number and the discussion of the previous section that the only candidates are parabolic or hyperbolic Inoue surfaces. In both cases the anticanonical bundle is effective and disconnected: \( - K = E + C \) . It follows from the fundamental equation (2) that \( F \) must be effective and... | Yes |
Theorem 3.4 ([25]) Every intermediate Kato surface \( S \) of index 1 admits a logarithmic deformation \( {S}_{\lambda } \) which is bi-Hermitian with \( {b}_{0}\left( T\right) = 1 \) . | Idea of Proof Let \( S \) be an intermediate Kato surface and let \( D \mathrel{\text{:=}} {D}_{1} + \cdots + {D}_{b} \) be the union of all rational curves in \( S \) which is in fact its maximal curve; by a logarithmic deformation of \( S \) we mean a deformation of the pair \( \left( {S, D}\right) \) . It is known t... | Yes |
Theorem 2.1 (YTD Conjecture for Fano Manifolds) Let \( X \) be an \( n \) -dimensional smooth Fano manifold. Then\n\n\[ \n\\text{there exists a KE metric in}{2\\pi }{c}_{1}\\left( {K}_{X}^{-1}\\right) \\Leftrightarrow X\\text{is}K\\text{-polystable.} \n\] | The direction \ | No |
Corollary 3.2 ([57]) KE del Pezzo orbifolds with orbifold groups at the singularities contained in \( {SU}\left( 2\right) \) are classified. | For example, KE del Pezzo orbifolds of degree three with such singularities are precisely given by all cubic surfaces in \( {\mathbb{P}}^{3} \) with only nodal (i.e., \( {A}_{1} \) ) singularities plus the toric cubic \( \left\{ {{xyz} = {t}^{3}}\right\} \cong {\mathbb{P}}^{2}/{\mathbb{Z}}_{3} \), since in this case th... | No |
Theorem 5.2 ([5]) Under the above hypothesis, \( {X}_{0} \) has a natural crepant resolution \( {\widehat{X}}_{0} \) admitting a family of cscK metrics of positive scalar curvature converging to the KE metric on \( {X}_{0} \) in the GH topology, and thus \( {X}_{0} \) is the degenerate variety of a generalized cscK con... | The above theorem is a special case of more general results in [5] (combined with Theorem 4.1), where \( {X}_{0} \) is not assumed to be Fano (e.g., it could have a KE metric of zero or negative Einstein constant), nor smoothable, and the singularities belong to a bigger class. We crucially remark that having the neede... | No |
Lemma 5.3 Let \( \gamma : \mathcal{X} \rightarrow {\mathcal{C}}_{g} \) be a curve of degree \( d \) del Pezzo orbifolds with generically smooth fibers for which \( {K}_{\mathcal{X}/\mathcal{C}}^{-1} \) makes sense. Then \[ {c}_{1}\left( {{\lambda }_{CM}\left( {\mathcal{X} \rightarrow {\mathcal{C}}_{g}}\right) }\right) ... | In fact, by the definition of the CM line bundle for the relative anticanoni-cal polarization and by Grothendieck-Riemann-Roch, we have \( {c}_{1}\left( {{\lambda }_{CM}\left( \mathcal{C}\right) }\right) = \) \( - {\gamma }_{ * }\left( {{c}_{1}^{3}\left( {K}_{\mathcal{X}/\mathcal{C}}^{-1}\right) }\right) \) . Hence, \(... | Yes |
Proposition 5.4 The degree of the CM line bundle on the base of a generic Lefschetz's fibration of (K-stable by Theorem 3.1) cubic surfaces is equal to \[ {c}_{1}\left( {{\lambda }_{CM}\left( {\mathcal{X} \rightarrow {\mathbb{P}}^{1}}\right) }\right) = 8. \] | For this, thanks to the previous lemma, it is sufficient to compute \( {c}_{1}^{3}\left( \mathcal{X}\right) \), where \( \mathcal{X} = B{l}_{{\sum }_{g}}{\mathbb{P}}^{3} \) with \( {\sum }_{g} = {c}_{1} \cap {c}_{2} \) surface of genus \( g = {10} \), by adjunction. Since \( - {K}_{\mathcal{X}} = {4H} - E \), where \( ... | Yes |
Theorem 4 ([6, Theorem 2.1, Remark 2.2]) Let \( X \) be a compact complex manifold of complex dimension \( n \) . Then, for any \( k \in \mathbb{Z} \) , \n\n\[ \mathop{\sum }\limits_{{p + q = k}}{\dim }_{\mathbb{C}}{H}_{A}^{p, q}\left( X\right) \]\n\n\[ \leq \min \{ k + 1,\left( {{2n} - k}\right) + 1\} \cdot \left( {\m... | Proof The proof is essentially algebraic and, for example, the idea behind the first inequality is obtained by thinking that the outgoing corners in a zig-zag contribute to the Aeppli cohomology and the extremal points of a zig-zag to the Dolbeault cohomology and/or its conjugate. Therefore, for any outgoing corners we... | Yes |
Theorem 5 ([6, Theorem 3.1]) A compact complex manifold \( X \) satisfies the \( \partial \bar{\partial } \) - Lemma if and only if, for any \( k \in \mathbb{Z} \), there holds\n\n\[\n\mathop{\sum }\limits_{{p + q = k}}\left( {{\dim }_{\mathbb{C}}{H}_{BC}^{p, q}\left( X\right) - {\dim }_{\mathbb{C}}{H}_{A}^{p, q}\left(... | Proof The first implication is trivial. For the other one notice that, roughly speaking, the vanishing of the numbers \( \mathop{\sum }\limits_{{p + q = k}}\left( {{\dim }_{\mathbb{C}}{H}_{BC}^{p, q}\left( X\right) - {\dim }_{\mathbb{C}}{H}_{A}^{p, q}\left( X\right) }\right) \) means that the number of ingoing corners ... | No |
Theorem 7 ([28, Theorem 4.3]) Let \( \left( {{X}^{2n},\omega }\right) \) be a compact symplectic manifold, then the natural map induced by the identity\n\n\[ \n{H}_{d + {d}^{\Lambda }}^{1}\left( X\right) \rightarrow {H}_{dR}^{1}\left( X\right) \n\]\n\nis an isomorphism. In particular,\n\n\[ \n{\widetilde{\Delta }}^{1} ... | Proof For the sake of completeness we briefly recall here the proof. For the surjectivity, if \( \alpha \) is a \( d \) -closed 1 -form, then it is also \( {d}^{\Lambda } \) -closed, indeed\n\n\[ \n{d}^{\Lambda }\alpha = \left\lbrack {d,\Lambda }\right\rbrack \alpha = - {\Lambda d\alpha } = 0. \n\]\n\nWe need to prove ... | Yes |
Theorem 1.1 Any surface \( X \in {\mathrm{{VII}}}_{0} \) is biholomorphic to either a Hopf surface or to an Inoue surface. | The proof of this theorem [25] uses differential geometric methods, and the renowned Kobayashi-Hitchin correspondence on Gauduchon surfaces \( \left\lbrack {2,{13},{19},{20}}\right\rbrack \) relating Hermite-Einstein connections to polystable holomorphic bundles. | No |
Theorem 1.4 Any surface \( X \in {\mathrm{{VII}}}_{ > 0}^{\min } \) with \( {b}_{2}\left( X\right) \) rational curves is a Kato surface. | This result answers positively a conjecture stated by Kato. | No |
Proposition 2.1 If \( \mathcal{A} \) admits a holomorphic line subbundle \( \mathcal{M} \neq i\left( {\mathcal{K}}_{X}\right) \), then \( X \) has a cycle of curves. | Proof Let \( \mathcal{M} \neq i\left( {\mathcal{K}}_{X}\right) \) be a line subbundle of \( \mathcal{A} \), and let \( j : \mathcal{M} \hookrightarrow \mathcal{A} \) be the embedding defined by the inclusion. The composition \( p \circ j \) is non-zero. Indeed, if this composition vanishes, it will follow \( \mathcal{M... | Yes |
Corollary 2.2 Suppose \( X \in {\mathrm{{VII}}}_{ > 0}^{\min } \), and let \( g \) be a Gauduchon metric on \( X \) such that \( {\deg }_{g}\left( {\mathcal{K}}_{X}\right) < 0 \) . If \( \mathcal{A} \) is not stable, then \( X \) has a cycle of curves. | Proof If \( \mathcal{A} \) is not stable, there will exist a destabilising short exact sequence\n\n\[ 0 \rightarrow \mathcal{M} \hookrightarrow \mathcal{A} \rightarrow {\mathcal{K}}_{X} \otimes {\mathcal{M}}^{ \vee } \otimes {\mathcal{I}}_{Z} \rightarrow 0 \]\n\nwhere \( \mathcal{M} \) is a rank 1, locally free subshea... | Yes |
Corollary 2.3 Suppose \( X \in {\mathrm{{VII}}}_{ > 0}^{\min } \) . If \( \left\lbrack \mathcal{A}\right\rbrack \) coincides with a twisted reduction, then \( X \) has a cycle of curves. | Proof If \( \left\lbrack \mathcal{A}\right\rbrack \) coincides with a twisted reduction, then \( \mathcal{A} \simeq \mathcal{A} \otimes {\mathcal{L}}_{0} \), so \( \mathcal{A} \) contains a line subbundle isomorphic with \( \mathcal{K} \otimes {\mathcal{L}}_{0} \), which cannot coincide with \( i\left( \mathcal{K}\righ... | Yes |
Corollary 2.4 If \( \mathcal{A} \) is the central term of an extension\n\n\[ 0 \rightarrow \mathcal{K} \otimes {\mathcal{L}}^{ \vee }\overset{j}{ \rightarrow }\mathcal{A} \rightarrow \mathcal{L} \rightarrow 0 \]\n\n(1)\n\nwith \( {c}_{1}\left( \mathcal{L}\right) \neq 0 \), than \( X \) has a cycle of curves. | Proof If \( {c}_{1}\left( \mathcal{L}\right) \neq 0 \), then \( {c}_{1}\left( {\mathcal{K} \otimes {\mathcal{L}}^{ \vee }}\right) \neq {c}_{1}\left( {\mathcal{K}}_{X}\right) \), so \( j\left( {\mathcal{K} \otimes {\mathcal{L}}^{ \vee }}\right) \) cannot coincide with \( i\left( \mathcal{K}\right) \), because \( \mathca... | No |
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