Special Complex Surfaces, Moduli Spaces, and Some Analytic Approach

首席研究员建议在代数和复杂几何的三个方向上工作。第一部分研究了正则极化流形模空间的双曲性，如Viehweg关于模空间为对数一般型的一个猜想。这个想法是在已知的曲线模空间原理的指导下研究高维变量的模，尽管必须采用全新的方法。主要使用的工具是代数几何和微分几何。第二部分是关于特殊曲面的分类和理解，如假投影平面、奇异二次曲面和一般的复球商。假投影平面已经由Gopal Prasad和PI在一个项目中进行了代数分类，该项目部分得到了NSF的资助。我们的目标是从几何上理解它们。该项目涉及代数几何、数论、Bruhat-Tits理论和计算机辅助计算等技术。第三部分是分析工具在代数或拓扑问题上的发展和应用。这包括对奇异二次曲线的分类的探索，以及对某些异形有理曲线存在性的分析工具的发展。使用的主要技术本质上是解析的，例如偏微分方程的估计和几何分析。数学的一个有趣的方面是它内在的连贯性和美。一次又一次，来自不同数学领域的思想和工具被汇集在一起，以惊人的方式解决看似无关的问题。该提案的目标是通过开发和应用各种数学学科的技术来理解各种代数和复杂的几何问题，在这个方向上进行更多的研究，其中一些是长期存在的。它从模空间的一般性质的研究（模空间是相同拓扑类型的变种集合）到特殊变种的详细研究和分类，如假投影平面和奇异二次曲面（它们是具有小拓扑不变量的代数曲面）。PI希望他的研究生积极参与到他提出的数学项目中，并学会欣赏研究的过程。他还希望通过在美国和国外的研讨会讲座和暑期学校演讲，将对数学的兴趣传递给其他学生。

The Principle Investigator proposes to work on three directions in algebraic and complex geometry. The first is on investigation of properties related to hyperbolicity of moduli spaces of canonically polarized manifolds, such as a conjecture of Viehweg that the spaces are of log-general type. The idea is to study moduli of higher dimensional varieties guided by principles known for the moduli spaces of curves, though completely new approaches have to be taken. The main tools to be used are algebraic geometric and differential geometric in nature. The second is on the classification and understanding of special surfaces, such as fake projective planes, exotic quadrics, and complex ball quotients in general. Fake projective planes have been classified algebraically by Gopal Prasad and the PI in a project partially supported by a prior NSF grant. The goal here is to understand them geometrically. The project involves techniques from algebraic geometry, number theory, Bruhat-Tits theory and computer aided computations. The third is on development and application of analytic tools to problems which are algebraic or topological in nature. This includes a quest for classification of exotic quadrics and development of analytic tools for existence of rational curves on some varieties. The main techniques to be used are analytic in nature, such as estimates in partial differential equations and geometric analysis.An interesting aspect of mathematics is its intrinsic coherence and beauty. Time and again, ideas and tools from different areas of mathematics are brought together to solve seemingly unrelated problems spectacularly. A goal of the proposal is to study more in this direction by developing and applying techniques from various mathematical disciplines to understand various algebraic and complex geometric problems, some of which are long standing. It varies from investigation of general properties of moduli spaces which are collections of varieties of the same topological type, to detailed study and classification of special varieties such as fake projective planes and exotic quadrics which are algebraic surfaces with small topological invariants. The PI expects his graduate students to be actively involved in the mathematical projects proposed and to learn to appreciate the process of research. He also hopes that the interests in mathematics may be passed to other students through seminar talks and summer school presentations, both in US and in foreign countries.