Fake hermitian symmetric manifolds and analytic approach to some problems in algebraic geometry

假埃尔米特对称流形及代数几何中若干问题的解析方法

• 批准号: 0758078
• 负责人: Sai Kee Yeung
• 金额: $ 12.2万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-15 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0758078&HistoricalAwards=false
• 关键词: Fake; hermitian; symmetric; manifolds; analytic

The project focuses on two directions in algebraic and complex geometry. The first one is on the classification and construction of some concrete projective algebraic manifolds which have small topological invariants,play important roles in geography of algebraic varieties, and are equipped with rich geometric and analytic properties. This includes classification of fake projective planes and more generally fake Hermitian symmetric spaces which have the same Betti numbers as the complex projective plane or a compact Hermitian symmetric space. The scheme of classification will lead to understanding of finite group actions on potential candidates of examples,and give rise tonew algebraic manifolds with topological invariants which are not known before. Another project proposed is to clarify the problem about existence ornon-existence of exotic complex structures on the complex quadric of dimension two, a problem which is well-known and has been open for a long time. Themethod proposed is a combination of both transcendental and algebraic techniques. This is also the second direction of the proposal, namely, to develop and apply analytic techniques to study problems which are algebraicin nature. A project proposed is to search for a proof of the existenceof rational curves on a Fano manifold using completely analytic method. Another is to study the problem of deformation rigidity of compact Hermitian symmetric spaces.In short, there are two main themes in the proposal. The first one is to construct and classify geometric models which can provide concrete examples for us to understand various mathematical theories and conjectures from algebraic, geometric or number theoretical point of view. The second is to further develop analytic techniques which have been very successful in recent years in understanding algebraic structures of varieties. Apart from thepossibility of enriching different areas of mathematics involved, including computational aspects of the various theories, it will also help to equip students with a reasonable broad base of mathematical knowledge.

该项目侧重于代数和复杂几何的两个方向。第一部分是关于一些具体的射影代数流形的分类和构造，这些流形具有较小的拓扑不变量，在代数簇地理中发挥着重要作用，并具有丰富的几何和分析性质。 这包括伪射影平面的分类，更一般地，伪厄米特对称空间具有与复射影平面或紧致厄米特对称空间相同的贝蒂数。 分类方案将导致理解有限群作用于潜在的候选例子，并产生新的代数流形与拓扑不变量，这是不知道的。另一个被提出的项目是澄清二维复二次曲面上奇异复结构的存在性或不存在性问题，这是一个众所周知的问题，并且已经公开了很长时间。 所提出的方法是超越和代数技术的结合。 这也是该建议的第二个方向，即发展和应用分析技术来研究代数性质的问题。 提出了一个用完全解析方法证明Fano流形上有理曲线存在的方案。 二是研究紧致埃尔米特对称空间的形变刚性问题。一是几何模型的构造和分类，为我们从代数、几何或数论的角度理解各种数学理论和方法提供了具体的例子。第二个是进一步发展分析技术已非常成功，近年来在理解代数结构的品种。除了可以丰富不同的数学领域，包括各种理论的计算方面外，它还有助于为学生提供合理的广泛数学知识基础。