Special Complex Surfaces, Moduli Spaces, and Some Analytic Approach

特殊复杂曲面、模空间和一些分析方法

• 批准号: 1101149
• 负责人: Sai Kee Yeung
• 金额: $ 16.73万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101149&HistoricalAwards=false
• 关键词: Special; Complex; Surfaces; Moduli; Spaces

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.

原理调查者建议在代数和复杂几何中的三个方向上工作。第一个是关于标准极化流形的模空间的双曲性的有关性质的研究，例如Viehweg的猜想，即这些空间是对数普通型的。我们的想法是在已知的曲线模空间原理的指导下，研究高维变种的模数，尽管必须采取全新的方法。主要使用的工具有代数几何和微分几何。第二个是关于特殊曲面的分类和理解，如一般的伪射影平面、奇异二次曲面和复杂球商。在一个由NSF先前拨款部分支持的项目中，Gopal Prasad和PI对虚假投射平面进行了代数分类。这里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的技术。第三个是关于代数或拓扑学性质的问题的分析工具的开发和应用。这包括探索奇异二次曲线的分类，以及开发分析工具以求某些变种上有理曲线的存在。主要使用的是分析的方法，比如偏微分方程中的估计和几何分析。数学的一个有趣的方面是它内在的连贯性和美感。一次又一次，来自不同数学领域的想法和工具被聚集在一起，以壮观的方式解决看似无关的问题。该提案的一个目标是通过开发和应用不同数学学科的技术来理解各种代数和复杂的几何问题，其中一些问题是长期存在的，从而在这个方向上进行更多的研究。它从研究相同拓扑类型的变种的集合的模空间的一般性质，到详细研究和分类特殊变种，如伪射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所提出的数学项目，并学会欣赏研究的过程。他还希望通过研讨会演讲和暑期班演讲将数学的兴趣传授给其他学生，无论是在美国还是在国外。