Advances in Complex Geometry最新文献

Geometric pluripotential theory on Kähler manifolds Kähler流形的几何多势理论

Advances in Complex Geometry

Pub Date : 2019-02-06 DOI: 10.1090/conm/735/14822

Tam'as Darvas

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Amp`ere type arising in K"ahler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Amp`ere equations. The purpose of this survey is to describe these developments from basic principles.

有限能量多势理论适用于K ahler几何中出现的复Monge-Amp ' ere型方程的变分理论。最近，人们发现许多潜在空间具有丰富的度量几何，有效地将所讨论的变分问题转化为无限维凸优化问题，并给出了底层复杂Monge-Amp ' ere方程解的存在性结果。这次调查的目的是从基本原则出发来描述这些发展。

引用次数: 49

Pluriharmonics in general potential theories 一般势理论中的多谐波

Advances in Complex Geometry

Pub Date : 2017-12-09 DOI: 10.1090/conm/735/14824

F. R. Harvey, H. Lawson, Jr.

The general purpose of this paper is to investigate the notion of "pluriharmonics" for the general potential theory associated to a convex cone $Fsubset {rm Sym}^2({bf R}^n)$. For such $F$ there exists a maximal linear subspace $Esubset F$, called the edge, and $F$ decomposes as $F=E oplus F_0$. The pluriharmonics or edge functions are $u$'s with $D^2u in E$. Many subequations $F$ have the same edge $E$, but there is a unique smallest such subequation. These are the focus of this investigation. Structural results are given. Many examples are described, and a classification of highly symmetric cases is given. Finally, the relevance of edge functions to the solutions of the Dirichlet problem is established.

本文的一般目的是研究与凸锥$F子集{rm Sym}^2({bf R}^n)$相关的一般势理论的“多谐波”概念。对于这样的$F$，存在一个极大的线性子空间$E子集F$，称为边，并且$F$分解为$F=E 0 + F_0$。多谐波或边缘函数是在E$中带有$D^2u 的$u$。许多子方程$F$有相同的边$E$，但有一个唯一的最小子方程。这些是本次调查的重点。给出了结构结果。描述了许多例子，并给出了高度对称情况的分类。最后，建立了边函数与狄利克雷问题解的相关性。

引用次数: 4

The Anomaly flow on unimodular Lie groups 单模李群上的异常流

Advances in Complex Geometry

Pub Date : 2017-05-27 DOI: 10.1090/conm/735/14828

D. Phong, Sebastien Picard, Xiangwen Zhang

The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary points are precisely the solutions of the Hull-Strominger system. Here we examine its long-time behavior on unimodular Lie groups with general unitary Hermitian connections. We find a diverse and intricate behavior, which depends very much on the Lie group and the initial data.

异质弦的超对称真空的Hull-Strominger系统允许具有扭转的一般幺正厄米连接，而不仅仅是陈恩幺正连接。利用这种灵活性的非模李群的解由T. Fei和st .T. Yau得到。异常流是一种静止点恰好是赫尔-施特罗明格系统解的流。本文研究了它在具有一般幺正厄密连接的单模李群上的长时间行为。我们发现了一种多样而复杂的行为，这在很大程度上取决于李群和初始数据。

引用次数: 24

Orbifold regularity of weak Kähler-Einstein metrics 弱Kähler-Einstein指标的轨道正则性

Advances in Complex Geometry

Pub Date : 2015-05-08 DOI: 10.1090/conm/735/14825

Chi Li, G. Tian

In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.

在本文中，我们证明了Kahler-Einstein流形的任何非坍缩紧的Gromov-Hausdorff极限在至少为3的复余维的子簇外要么是光滑的，要么是轨道的。

引用次数: 11

Pseudoconcave decompositions in complex manifolds 复杂流形中的伪凹分解

Advances in Complex Geometry

Pub Date : 1900-01-01 DOI: 10.1090/conm/735/14829

Z. Slodkowski

引用次数: 10

Local singularities of plurisubharmonic functions 多次谐波函数的局部奇异性

Advances in Complex Geometry

Pub Date : 1900-01-01 DOI: 10.1090/conm/735/14823

S. Dinew

引用次数: 0

On orthogonal Ricci curvature 在正交里奇曲率上

Advances in Complex Geometry

Pub Date : 1900-01-01 DOI: 10.1090/conm/735/14827

Lei Ni, F. Zheng

In this paper we discuss some recent progresses in the study of compact Kähler manifolds with positive orthogonal Ricci curvature, a curvature condition defined as the difference between Ricci curvature and holomorphic sectional curvature. In the recent works by authors and the joint work of authors with Q. Wang the comparison theorems, vanishing theorems, and structural theorems for such manifolds have been proved. We also constructed examples of this type of manifolds, and give some classification results in low dimensions. 1. Orthogonal Ricci curvature Let (M, g) be a Kähler manifold of complex dimension n. Its orthogonal Ricci curvature Ric⊥ is defined by (cf. [21]): Ric⊥ XX = Ric(X,X)−R(X,X,X,X)/|X|, where X is a non-zero type (1, 0) tangent vector at a point x ∈ M. This curvature arises in the study of the comparison theorem for Kähler manifolds and the previous study of manifolds with so-called nonnegative quadratic orthogonal bisectional curvature (cf. [4], [26], [16], [5]). We refer the readers to [21] for a more detailed account on this topic. Clearly this curvature is closely related to Ricci curvature Ric and holomorphic sectional curvature H. It is natural to ask, what is the relationship between Ric⊥ and Ric or H (other than the obvious one that Ric⊥ + H = Ric for unit length tangent vectors), and what kind of compact complex manifolds M can admit Kähler metrics with Ric⊥ > 0 (or ≥ 0, or ≤ 0, or < 0, or ≡ 0) everywhere? In this paper, we will focus on the curvature condition Ric⊥ and pay particular attention to the class of compact Kähler manifolds with Ric⊥ > 0 everywhere, except in Section 2 where complete noncompact Kähler manifolds are also considered. Throughout this paper, we will assume that the complex dimension n ≥ 2 unless stated otherwise, since Ric⊥ ≡ 0 when n = 1. We start with the following observation. At a point x ∈ M, let us denote by S2n−1 x the unit sphere of all type (1, 0) tangent vector at x of unit length. By a classic result of Berger, The research of LN is partially supported by NSF grant DMS-1401500 and the “Capacity Building for Sci-Tech Innovation-Fundamental Research Funds”. The research of FZ is partially supported by a Simons Collaboration Grant 355557.

本文讨论了具有正正交Ricci曲率的紧致Kähler流形的一些最新研究进展，正正交Ricci曲率是定义为Ricci曲率与全纯截面曲率之差的曲率条件。在作者最近的工作和与王q .的合作工作中，证明了这种流形的比较定理、消失定理和结构定理。我们还构造了这类流形的例子，并给出了一些低维的分类结果。1. 设(M, g)是复维数n的Kähler流形。它的正交里奇曲率Ric⊥由(cf.[21])定义:Ric⊥XX = Ric(X,X)−R(X,X,X,X)/|X|，其中X是点X∈M处的非零型(1,0)切向量。这个曲率出现在Kähler流形的比较定理研究中，以及之前对具有所谓非负二次正交对分曲率的流形的研究中(cf.[4]，[26]，[16]，[5])。我们请读者参阅[21]以获得关于该主题的更详细的说明。显然，这个曲率与Ricci曲率Ric和全纯截面曲率H密切相关，很自然地要问，Ric⊥与Ric或H之间的关系是什么(除了对于单位长度切向量的明显的Ric⊥+ H = Ric)，以及什么样的紧复流形M可以允许Kähler处处存在Ric⊥> 0(或≥0，或≤0，或< 0，或≡0)的度量?在本文中，我们将专注于曲率条件Ric⊥，并特别关注Ric⊥> 0的紧态Kähler流形的类别，除了在第2节中也考虑了完全非紧态Kähler流形。在本文中，除非另有说明，否则我们将假设复维数n≥2，因为当n = 1时，Ric⊥≡0。我们从以下观察开始。在点x∈M处，用S2n−1x表示x处单位长度的所有类型(1,0)切向量的单位球。根据Berger的经典结果，LN的研究得到了美国国家科学基金DMS-1401500和“科技创新能力建设-基础研究基金”的部分支持。FZ的研究得到了Simons合作基金355557的部分支持。

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引用次数: 15

Analysis of the Laplacian on the moduli space of polarized Calabi-Yau manifolds 偏振Calabi-Yau流形模空间上的拉普拉斯分析

Advances in Complex Geometry

Pub Date : 1900-01-01 DOI: 10.1090/conm/735/14826

Zhiqin Lu, Hangjun Xu

引用次数: 0