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

本文针对DMS-0104089，重点研究了非紧型局部厄米特对称空间，它被认为是特殊的非正截面曲率的复流形。复双曲性可以被认为是负曲率的弱概念。从解析的角度出发，研究局部厄米对称空间向量丛的模空间中切丛的刚性；从同一化的角度，研究复两球算术格的刻划.他建议研究Fijita猜想的相似之处，这些猜想与代数几何有关，涉及这种流形上的多正则线丛的非常广度。他建议理解和估计与流形自然相关的覆盖塔上Betti数的增长，将拓扑学与几何联系起来。在数论的指导下，他提出研究代数数域上两个球商模型上有理点的有限性问题。他还建议研究复双曲性和Kaehler几何领域的其他几个问题，这与他早期的研究方向有关。在数学中，人们对模型感兴趣，这些模型一方面优雅，相对简单，描述起来相对简单，另一方面显示出丰富的数学结构。局部厄米特对称空间是不同数学学科相遇的很好的几何模型。这项建议的主要目的是了解这种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增强对所涉及的学科和其他数学学科的知识。

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.