复几何中的双截曲率和全纯截面曲率

Complex manifolds are basic objects in the study of several complex variables and complex geometry. A central topic in the study of complex manifolds is the interaction between topology, curvature, and complex structure. For example, one may consider the bisectional curvature and holomorphic sectional curvature defined by the Chern connection on Hermitian manifolds. A natural question is to study Hermitian manifolds with positive (nonnegative) or negative (nonpositive) bisectional curvature (holomorphic sectional curvature), In general Hermitian manifolds whose curvatures satisfy some special algebraic properties are objects of interest. Recall that Kähler manifolds belong to an important class of Hermitian manifolds. We will study the following topics:.1. Function theory on complete noncompact Kähler manifolds with nonnegative bisectional curvature..2. Compact Kähler manifolds and Hermitian manifolds with positive (nonnegative) holomorphic sectional curvature..3. The classification problem of the space of orthogonal complex structures on a certain class of Riemannian manifolds..4. Complex geometry of bounded pseudoconvex domains in complex Euclidean spaces, and the study of complete Kähler manifolds with negative (nonpositive) bisectional curvature.

复流形是多复变函数和复几何的基本研究对象。复流形研究的一个核心问题是复流形的复结构、拓扑、和曲率的制约关系。比如考虑Hermitian流形上用陈联络定义的双截曲率和全纯截面曲率，一个自然的问题是研究具正(非负)、负(非正)曲率、或者曲率满足某种特殊性质的Hermitian流形的几何，凯勒流形是Hermitian流形一个重要例子。本项目研究课题：.1. 具有非负双截曲率的完备非紧凯勒流形的全纯函数。.2. 具正(非负)全纯截面曲率的紧致凯勒流形和Hermitian流形。.3. 一类给定的黎曼流形上的正交复结构的分类问题。.4. 复欧式空间中的有界拟凸域的复几何，以及具有负(非正)双截曲率的完备凯勒流形的研究。

本项目关注复流形上的两个重要曲率：双截曲率和全纯截面曲率。受经典的黎曼曲面的单值化定理的启发，本项目着眼于非负或者非正的双截曲率或全纯截面曲率的高维复流形的分类。项目的主要研究课题包括非负双截曲率的凯勒流形上的全纯函数理论，正全纯截面曲率的凯勒流形的构造，以及Hermitian流形上的相应问题等。本项目取得的成果包括：在某类具有非负双截曲率的凯勒流形上，我们利用偏微分方程和L2方法等工具证明了多项式增长的全纯函数的定量估计；我们证明了完备非紧的正全纯截面曲率的凯勒度量的存在性和不存在性结果。这些结果对于我们了解高维复流形上的曲率有积极意义。