Geometry of Moduli Spaces, Arithmetic Quotients and Theta Divisors

模空间几何、算术商和 Theta 除数

• 批准号: 1101333
• 负责人: Sebastian Casalaina-Martin
• 金额: $ 13万
• 依托单位: University of Colorado at Boulder
• 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: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101333&HistoricalAwards=false
• 关键词: Geometry; Moduli; Spaces; Arithmetic; Quotients

This project addresses the geometry of moduli spaces that parameterize algebraic and geometric objects. The focus will be on spaces where there are several natural approaches to the moduli problem, particularly via geometric invariant theory, moduli of pairs, and Hodge theory. The former two approaches provide a method where the geometry of the objects parameterized plays a central role. The latter approach can allow for the use of arithmetic methods, including modular forms. Specific problems to be considered include giving modular interpretations to boundary loci of arithmetic quotients, describing log-canonical models of the moduli space of curves, and investigating the local structure of compactified Jacobians. Abelian varieties and theta divisors will also be of special interest.Algebraic geometry is the field of mathematics that focuses on the solution sets of polynomial equations. Inasmuch as polynomial equations are ubiquitous, the subject sits at the crossroads of many different fields including complex geometry, number theory and theoretical physics. A motivating question for algebraic geometers has been to classify solution sets by their invariants. For instance, one could try to classify those complex valued solution sets that can be identified with a torus (the surface of a donut); this would correspond to fixing the invariants known as the (complex) dimension and genus equal to the number one. Often the collection of solution sets with fixed invariants can itself be viewed naturally as a solution set of polynomial equations. These are known as moduli spaces, and their algebraic and geometric properties yield a tremendous amount of information about the original solution sets of interest. The PI intends to study a number of such spaces. The proposed research will have a significant impact on a field that plays a central role in mathematics and interacts with numerous other fields.

这个项目解决了模空间的几何学，它将代数对象和几何对象参数化。重点将放在那些有几种自然的方法来解决模问题的空间上，特别是通过几何不变理论、对的模和霍奇理论。前两种方法提供了一种方法，其中参数化对象的几何形状起着核心作用。后一种方法允许使用算术方法，包括模块化形式。要考虑的具体问题包括对算术商的边界轨迹进行模解，描述曲线的模空间的对数规范模型，以及研究紧致雅可比的局部结构。Abelian簇和theta因子也将是特别有趣的。代数几何是专注于多项式方程的解集的数学领域。由于多项式方程无处不在，这门学科位于许多不同领域的十字路口，包括复杂几何、数论和理论物理。代数几何学家的一个激励问题是根据它们的不变量对解集进行分类。例如，人们可以尝试对那些可以用环面(甜甜圈的表面)标识的复值解集进行分类；这将对应于将称为(复)维和亏格的不变量固定为数字1。通常，具有固定不变量的解集的集合本身可以自然地被视为多项式方程的解集。这些被称为模空间，它们的代数和几何性质产生了关于感兴趣的原始解集的大量信息。国际和平研究所打算研究一些这样的空间。这项拟议的研究将对一个在数学中发挥核心作用并与许多其他领域相互作用的领域产生重大影响。