Geometry of Moduli Spaces, Arithmetic Quotients and Theta Divisors

这个项目的目的是几何模空间参数化代数和几何对象。重点将放在有几个自然的方法，特别是通过几何不变理论，模对和霍奇理论的模数问题的空间。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。 后一种方法可以允许使用算术方法，包括模块形式。 要考虑的具体问题包括模块化的解释边界轨迹的算术代数，描述日志规范模型的模空间的曲线，并调查的局部结构的紧致雅可比。阿贝尔簇和θ因子也将是特别感兴趣的。代数几何是数学领域，重点是多项式方程的解集。 由于多项式方程是无处不在的，这一主题坐落在许多不同领域的十字路口，包括复杂的几何，数论和理论物理。代数几何学家的一个激励性问题是通过不变量对解集进行分类。 例如，我们可以尝试对那些可以用环面（圆环的表面）识别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。 PI打算研究一些这样的空间。 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。

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.