Locally Hermitian symmetric spaces, non-positive curvature and complex hyperbolicity

本文主要研究非紧型局部厄米对称空间，并将其视为非正截面曲率的特殊复流形。复双曲可以看作是负曲率的一个弱概念。从解析的角度出发，提出了局部厄米对称空间中矢量束模空间中切束的刚性问题。从均匀化的观点出发，他提出了复二球中算术格的表征。他建议研究Fijita关于这种流形上的多形线束的非常丰度的猜想的类比，这与代数几何有关。他建议理解和估计在与流形自然相关的覆盖塔上贝蒂数的增长，将拓扑与几何联系起来。在数论方向上，他提出研究代数数域上定义的两个球商模型上有理点的有限性问题。他还建议在复杂双曲和凯勒几何领域研究其他几个问题，这些问题与他早期的研究方向有关。在数学中，人们感兴趣的模型一方面是优雅的，相对简单的描述，另一方面显示丰富的数学结构。局部厄米对称空间是一种很好的几何模型，不同的数学学科都适用。本建议的主要目的是了解这些模型的几个几何和算术方面，并解释它们之间的相互关系。对所提出的各种性质和问题的理解将提高本学科和其他相关数学学科的知识。

Abstract for DMS - 0104089The focus of the proposal is on the research of locally Hermitian symmetricspaces of non-compact type, considered as special complex manifolds of non-positive sectional curvature. Complex hyperbolicity can be considered as a weak notion of negative curvature. From analytic point of view, the principal investigator proposes to work on the rigidity of tangent bundles among the moduli space of vector bundles of locally Hermitian symmetric spaces.From the uniformization point of view, he proposes to work on characterization of arithmetic lattices in complex two balls. He proposes to investigate the analogueof Fijita's conjectures concerning very ampleness of pluricanonical line bundles on such manifolds, relating to algebraic geometry. He proposes to understand and estimate the growth of Betti numbers on a tower of coverings naturally associated with the manifolds, relating topology to geometry. In the direction of number theory, he proposes to study the question on finiteness of rational points on models of two ball quotients defined over an algebraic number field. He also proposes to work on several other problems in the area of complex hyperbolicityand Kaehler geometry, relating to his earlier research directions.In mathematics, people are interested in models which on one hand are elegant and relatively simple to describe and on the other hand display rich mathematicalstructures. Locally Hermitian symmetric spaces are such nice geometric models forwhich different disciplines of mathematics meet. The main purpose of this proposal is to understand several geometric and arithmetic aspects of such modelsand explain their interrelationships. Understanding of the various propertiesand problems proposed will enhance the knowledge of the subject and other disciplines of mathematics involved.