Extension Phenomena for Holomorphic Geometric Structures
Benjamin McKay
最常见的复解析G-结构和嘉当几何的类型不能有复余维2或更多的奇点。
The most commonly encountered types of complex analytic G-structures and Cartan geometries cannot have singularities of complex codimension 2 or more.
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Holomorphic geometric structures on Hopf manifolds
本文在不依赖 Poincaré-Dulac 正规形式的情况下，于所有复 Hopf 流形上构造了可积全纯 G-结构与平坦全纯 Cartan 几何，并借助适配坐标图给出了该定理的新证明。
Geometric properties and related results for holomorphic Segre preserving maps
本文研究了$\mathbb{C}^{n+1}$中实超曲面复化之间的全纯Segre保持映射。特别地，我们找到了几个充分条件，确保Segre横截性和总Segre非退化的地图必须持有。
On geometric properties of holomorphic isometries between bounded symmetric domains
本文研究有界对称域间关于Bergman度量及典范Kähler度量的全纯等距映射，证明仿射线性截面的像与仿射线性子空间相交，并构造满全纯浸没使等距映射成为其整体全纯截面。
Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi-Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In particular, if the geometric structure is rigid in Gromov's sense, then the fundamental group of the manifold must be infinite. We also prove that compact complex manifolds of algebraic dimension one bearing a holomorphic Riemannian metric must have infinite fundamental group.
Holomorphic geometric structures on Kaehler-Einstein manifolds
证明了具有第一类非负的全纯抛物几何的紧致Kaehler流形是复环面上有理齐次簇的平坦丛。我们还证明了具有负第一陈类的紧致Kaehler流形的局部Hermitian对称簇是全纯共微几何的完备的Kaehler流形。
For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class $\mathcal C$, whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class $\m…
Holomorphic geometric structures on Oeljeklaus-Toma manifolds
本文研究了 Oeljeklaus-Toma 流形上的全纯几何结构，证明了仿射型全纯几何结构的局部齐性，并针对局部共形 Kähler 情形推广至所有全纯几何结构与全纯 Cartan 几何。
Non-Kähler Calabi-Yau manifolds and holomorphic geometric structures
本文研究了非Kaehler紧复流形上全纯几何结构的性质，证明了Vaisman Calabi-Yau流形上所有全纯类型全纯几何结构都是局部同构的，并且如果几何结构是刚性的，则Vaisman流形必须是一个Kodaira流形。
We prove that any holomorphic locally homogeneous geometric structure on a complex torus, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true is any dimension. In higher dimension we prove it here for nilpotent models. We also prove that in any dimension the translation invariant $(X, G)$-structures form a union of connected components in the deformation space of $(X, G)$-structures.
Locally Homogeneous Holomorphic Geometric Structures on Projective Varieties
本文研究带齐次代数簇模型的光滑射影簇上的全纯局部齐次结构，并刻画所有能展开到模型的子簇。
A Geometric Proof of Removal of Boundary Singularities of Pseudo-Holomorphic Curves
证明了伪全纯曲线边界上奇点去除的两个定理。在一个定理中，我们不需要关于曲线面积的先验假设。证明使用了一个加倍的参数，目的是将有边界的曲线转换为无边界的曲线。我们的方法是新的和几何的，它不需要索伯列夫空间和偏微分方程。
Meromorphic almost rigid geometric structures
研究了复流形上亚纯几乎刚性几何结构的局部Killing李代数。这导致了紧复流形轴承全纯刚性几何结构的分类结果。
Twistors, Holomorphic Disks, and Riemann Surfaces with Boundary
复流形Z中的全纯圆盘模空间，其边界被约束在极大全实子流形P中，最近被发现是许多几何上丰富的扭量对应的基础.本文的目的是开发一个通用的Fredholm正则性准则的全纯曲线与边界，然后显示如何适用于，特别是各种模问题的扭理论的兴趣。
This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed.
Holomorphic geometric models for representations of $C^*$-algebras
在全纯齐性向量丛的截面空间上实现了$C^*$ -代数的表示.利用再生核的概念研究了相应的截空间，再生核适用于处理定义在丛的底空间上的对合反同态。应用这种技术的膨胀理论的完全正映射的探讨和复杂的齐性空间的关键作用与Stinesspring膨胀指出。一般的结果进一步说明了讨论的几个具体议题，包括相似轨道的表示顺从巴拿赫代数，相似轨道的条件期望，几何模型的表示的Cuntz代数，自同态的关系${\mathcal B}({\mathcal H})$，和非交换随机分析。
Holomorphic Extension from Weakly Pseudoconcave CR Manifolds
设M是局部可嵌入的光滑CR流形，具有一定的CR维数m和CR余维数d.我们发现了一个改进的局部几何条件，保证在M上的一个点p，CR分布的芽是光滑函数，并有扩展到全纯函数芽的全周围邻域的p。我们的条件是一种形式的弱伪随机，密切相关的本质伪随机[HN 1]。应用CR亚纯函数和映射。给出了满足我们新条件的显式例子，但它们不是强意义上的伪凹。这些结果表明，对于余维d > 1，存在当d = 1时不可见的额外现象。
A geometrical proof of the Hartogs extension theorem
100年前，也就是1906年，Hartogs发表了一个著名的扩张现象（多复变数的诞生），其整体对应物后来被Osgood（1929）以完全一般性的方式陈述：包含在C^n（n >= 2）中的连通边界bD的连通邻域V（bD）中的全纯函数确实全纯且唯一地扩张到区域D。这是一个长期存在的开放性问题，以获得一个证明只使用分析光盘，因为没有赫尔维茨（1897年），Hartogs（1906年）和E. E。Levi（1911）在一些特殊的模型案例中。非常出乎意料的是，福奈斯在1998年展示了C^2中的一个拓扑上奇怪的（非伪凸的）区域D^F，它不能被全纯圆盘填充，当我们附加要求圆盘必须完全位于D^F内时。然而，应该指出的是，标准的，不受限制的光盘方法通常允许光盘去搜索域（只要想想Levi pseudo-random）。重申这些相当古老的期望，我们表明，全球Hartogs扩张定理可以建立在这样一个自然的方式。
Characteristic forms of complex Cartan geometries III: G-structures
本文研究复Cartan几何中G-结构的特征形式，提出从结构群表示论直接计算Dolbeault上同调中的特征类关系，并扩展至无限型几何结构。
Differential geometry of holomorphic vector bundles on a curve
这些笔记是基于在2009年维拉德莱瓦暑期学校的量子场论的几何和拓扑方法的五个系列讲座。讲座的目的是给一个介绍微分几何方法的研究全纯向量丛上的一个紧凑的连接黎曼曲面。
A holomorphic Engel structure determines a flag of distributions $\mathcal{W}\subset \mathcal{D}\subset \mathcal{E}$. We construct examples of Engel structures on $\mathbf{C}^4$ such that each of these distributions is hyperbolic in the sense that it has no tangent copies of $\mathbf{C}$. We also construct two infinite families of pairwise non-isomorphic Engel structures on $\mathbf{C}^4$ by controlling the curves $f:\mathbf{C}\to \mathbf{C}^4$ tangent to $\mathcal{W}$. The first is characterised by the topology of the set of points in $\mathbf{C}^4$ admitting $\mathcal{W}$-lines, and the seco…
Geometric Zeta Functions, $L^2$-Theory, and Compact Shimura Manifolds
我们定义了局部对称空间的几何zeta函数作为Ruelle和Selberg zeta函数的推广。作为零处的一个特殊值，我们得到流形的Reidemeister挠率。对于埃尔米特空间，这些zeta函数具有Ray和Singer的全纯挠率和全纯$L^2$ -挠率的商作为特殊值，其中后者是通过Atiyah的$L^2$ -理论定义的。对于更高的基本秩扭曲扭转数出现。
The analytic moduli space of holomorphic submersions
本研究构建了参数化紧凯勒底空间上稳定紧合全纯淹没的解析模空间，结合几何不变理论与最优辛联络的几何PDE研究，证明其为豪斯多夫复空间且带Weil-Petersson型凯勒度量，特例为紧凯勒流形向量丛模空间。
We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and sho…
On the geometry of holomorphic curves and complex surface
本文通过复双线性对称形式定义了全纯曲线和复表面的几何不变量，并应用奇点理论得到实情况下的经典结果类比。
We discuss some geometric aspects of PSL(2,C), SL(2,C), and the space G of the geodesics of H^3 equipped with some suitable structures of Riemannian holomorphic manifolds of constant sectional curvature. We also observe that G is a symmetric space for the group PSL(2,C) and use it to deduce some correlations between their holomorphic Riemannian metrics.
Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold extends uniquely to the envelope of holomorphy of the domain. This result completes the open problems of my earlier paper on extension of holomorphic geometric structures on complex manifolds. We use this result to classify the Hopf manifolds which admit holomorphic reductive geometries, and to classify the Hopf manifolds which admit holomorphic parabolic geometries. Every Hopf manifold which admits a holomorphic parabolic geometry with a given model admits a flat one. We classify flat holomorphic par…
Geometric decompositions of almost contact manifolds
这些笔记的目的是介绍使用近似全纯技术在几乎接触和接触几何。我们发展了近似全纯几何的体系。一旦完成，我们素描的存在性的两个主要的几何分解可用于几乎接触或接触流形：开放的书籍和莱夫谢茨铅笔。文中还讨论了这两种分解在求解切触结构存在性问题中的应用。
Given a compact Kähler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying Kähler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain d…
We give a dynamical description, in terms of a Weil-type zeta function, to the holomorphic torsion with coefficients for certain compact Hermitian locally symmetric manifolds, whose connected group G of isometries of the universal cover has only one conjugacy class of cuspidal maximal parabolic subgroup and satisfies a technical Ansatz relative to the given coefficients. A distinguishing feature of our zeta function is that its construction involves in an essential way the geometry of a standard compactification of the universal cover. The two senior authors are indebted to their junior collea…
We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form, we prove that they satisfy a similar version of the classic Gauss-Codazzi equations, and conversely that solutions of Gauss-Codazzi equations are immersion data of some equivariant map. This study has some interesting geometric consequences: 1) it provides a formalism to study immersions of surfaces into SL(2,C) and into the space of geodesics of H^3; 2) it generalizes the classical theory of…
Complex surfaces and interpolation on pseudo-holomorphic cylinders
格罗莫夫已经证明了如何构造全纯映射的平面上的一个复流形与规定的价值观的一个格子。本文证明了从柱面S到殆复流形（M，J）的伪全纯映射的一个类似插值定理。导出了从S到（M，J）的伪全纯映射空间的性质，特别是这个空间大小的下界（用平均维数表示）.当M是曲线的模空间时，这给出了非紧复曲面的构造。这些方法涉及无数次手术。在此基础上，得到了两条拟全纯曲线的胶合过程的一个改进，建立了两条胶合拟全纯曲线的几何性质。
Geometry of $K$-trivial Moishezon manifolds : decomposition theorem and holomorphic geometric structures
本文利用 MMP 与叶状结构理论证明 $K$-平凡 Moishezon 或低维 Fujiki 流形的 Beauville–Bogomolov 型分解，并刻画其全纯几何结构、环面商及基本群性质。
This paper introduces a complex representation for spacelike surfaces in the Lorentz-Minkowski space $L^4$, based in two complex valued functions which can be assumed to be holomorphic or anti-holomorphic. When the immersion is contained in quadrics of $L^4$, the representation then allows us to obtain interesting partial differential equations with holomorphic or anti-holomorphic parameters, within which we find the partial Riccati Equation. Using then theory of holomorphic complex functions we construct explicitly new local solutions for those PDEs together with its associated geometric solu…
Holomorphic projective connections on surfaces and osculating spaces
该研究从曲面第二邻域与积分曲线密切性态两方面研究射影n维空间中曲面上的复解析射影联络，还分析了特殊有理变换的作用并给出相关群联合不变量的几何解释。
We prove a Bochner type vanishing theorem for compact complex manifolds $Y$ in Fujiki class $\mathcal C$, with vanishing first Chern class, that admit a cohomology class $[α] \in H^{1,1}(Y,\mathbb R)$ which is numerically effective (nef) and has positive self-intersection (meaning $\int_Y α^n \,>\, 0$, where $n\,=\,\dim_{\mathbb C} Y$). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold $Y$ are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group o…
The purpose of this article is to show a geometric version of Zabrodin-Wiegmann conjecture for an integer Quantum Hall state. Given an effective reduced divisor on a compact connected Riemann surface, using the canonical holomorphic section of the associated canonical line bundle as well as certain initial data and local normalisation data, we construct a canonical non-zero element in the determinant line of the cohomology of the $p$-tensor power of the line bundle. When endowed with proper metric data, the square of the $ L^{2} $-norm of our canonical element is the partition function associa…
Holomorphic shadows in the eyes of model theory
定义了殆复流形（M，J）的一个子集为全纯影，如果它是紧致复流形的J-全纯映射的像.注意J-全纯曲线是全纯阴影，紧复流形的复子簇也是。证明了在一定条件下，流形M上的殆复结构J上的全纯阴影在（M，J）的笛卡儿积中形成Zapriki型结构.检查这导致非平凡的几何问题和结果。然后，我们将Hrushovski和Zilber的工作应用于Zurki型结构。我们还重述了Gromov和McDuff关于辛几何中J-全纯曲线的结果。
In the present paper, we describe two geometric notions, holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups, and prove that on Hom-Lie groups in the left invariant setting, these structures are related to each other. We study Kähler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex struc…
We introduce an axiomatic definition for the Kodaira dimension and classify Thurston geometries in dimensions $\leq 5$ in terms of this Kodaira dimension. We show that the Kodaira dimension is monotone with respect to the partial order defined by maps of non-zero degree between 5-manifolds. We study the compatibility of our definition with traditional notions of Kodaira dimension, especially the highest possible Kodaira dimension. To this end, we establish a connection between the simplicial volume and the holomorphic Kodaira dimension, which in particular implies that any smooth Kähler 3-fold…
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