Moduli and Surfaces in Complex Geometry

在数学或物理中，为了理解一个物体的性质，有时把所有性质相似的物体放在一起，然后把结果集合作为一个整体来研究性质，会更有启发性。这在代数几何和复几何中称为模。这个概念也用于量子物理和弦理论。项目的第一部分是了解一些自然存在的几何物体的模的性质。另一方面，有时具有特殊性质的几何物体，如存在额外的对称性或特殊类型的奇点，为理解科学中的各种主题提供了很好的模型。该项目的第二部分旨在寻找特殊的复杂表面或高维几何物体，这些物体将捕捉数学家一直在寻找的一些特性。由于将使用各种各样的工具，包括计算机实施，因此拟议活动的很大一部分可用于培训学生或与年轻学者合作。更具体地说，首席研究员提议对与复杂几何和代数几何有关的几个问题进行研究。提案的第一部分详细介绍了杜永强和研究者最近开发的一些分析工具的可能应用和扩展。这些新工具使他们能够以一种新的、有效的方法来处理高维极化复流形的模空间问题。本课题旨在通过对比曲线模空间的解析、几何和算术性质，探索和理解一般极化流形的模。该项目的第二部分旨在应用Gopal Prasad和首席研究员的假投影平面分类作为构建块来研究各种代数几何或复杂几何开放问题。包括一些小正则度的未知实例的构造，小不变量的奇异辛流形的构造，伪射影平面的更多几何实现和算术伪紧厄米对称空间的分类。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

To understand properties of an object in mathematics or physics, sometimes it is more enlightening to put all objects of similar nature together and study properties on the resulted collection as a whole. This is termed as moduli in algebraic and complex geometry. The concept is used in quantum physics and string theory as well. The first part of the project is to understand properties of moduli of some naturally existing geometric objects. On the other hand, sometimes a geometric object with special properties such as existence of extra symmetries or special type of singularities provide good models for understanding of various subjects in science. The second part of the project is aimed at finding special complex surfaces or higher dimensional geometric objects that would capture some properties that mathematicians have been seeking after. Since a wide range of tools are to be used, including computer implementation, a significant portion of the proposed activities can be used to train students or collaborate with young scholars.In more concrete terms, the principal investigator proposed to conduct research in several problems related to complex and algebraic geometry. The first part of the proposal details possible applications and extensions of some analytic tool developed recently by Wing-Keung To and the investigator. The new tools allow them to approach problems related to moduli space of higher dimensional polarized complex manifolds in a new and efficient way. The project aims at exploring and understanding moduli of general polarized manifolds in comparison with analytic, geometric and arithmetic properties of moduli space of curves. The second part of the project aims at applying the classification of fake projective planes by Gopal Prasad and the principal investigator as building blocks to investigate various algebraic geometric or complex geometric open problems. It includes construction of some unknown examples with small canonical degree, construction of exotic symplectic manifolds of small invariants, more geometrical realization of fake projective planes and classification of arithmetic fake compact Hermitian symmetric spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.