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

该项目侧重于代数和复杂几何两个方向。第一部分是一些具体的投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，在代数变种地理学中占有重要地位，具有丰富的几何和解析性质。这包括假射影平面和更一般的假厄米对称空间的分类，它们具有与复射影平面或紧化厄米对称空间相同的贝蒂数。该分类方案将导致对潜在候选示例的有限群作用的理解，并产生以前不知道的具有拓扑不变量的新代数流形。另一个课题是澄清二维复二次曲面上奇异复杂结构的存在与否问题，这是一个众所周知的问题。所提出的方法是超越和代数技术的结合。这也是本提案的第二个方向，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寻找法诺流形上有理曲线存在性的证明的方案。二是研究紧致厄密对称空间的变形刚度问题。简而言之，该提案有两个主要主题。首先是建立和分类几何模型，这些模型可以为我们从代数、几何或数论的角度理解各种数学理论和猜想提供具体的例子。二是进一步发展近年来在理解变异代数结构方面非常成功的解析技术。除了有可能丰富所涉及的不同数学领域，包括各种理论的计算方面，它也将有助于使学生具备一个合理的广泛的数学知识基础。

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.