Higher Multiplier Ideals and Other Applications of Hodge Theory in Algebraic Geometry

这是一个纯数学的项目，更准确地说是在代数几何领域。代数几何研究由多项式方程定义的几何对象，使用代数、复分析、微分几何、拓扑学和偏微分方程的方法。代数几何也为理论物理的某些部分，如弦理论，提供了有用的例子。PI提出的大多数研究都与代数几何的一个子领域霍奇理论有关。霍奇理论使得用一种更接近代数而非分析的语言，将偏微分方程理论的结果应用于代数几何问题成为可能。PI提出的一些工作是关于进一步发展霍奇理论（并使其更容易为研究生所接受）；剩下的就是用霍奇理论来解决代数几何中的几个非常具体的问题。PI还将编写一本关于这一主题的综合书籍，这将对研究界有巨大的用处。更具体地说，本项目有几个不同的研究目标，都与Hodge理论有关：(1)与杨瑞杰合作，通过使用混合Hodge模块的附近环函数，开发有效因子的更高乘数理想理论。(2)将此理论应用于主极化阿贝尔变体上的因子，特别是Casalaina-Martin关于其奇异点多重性的猜想。(3)继续研究Hodge结构积分变分的自对偶类轨迹的局部和全局结构；自对偶类是霍奇类的一种推广，对理论物理很有兴趣。(4)进一步研究了光滑态射下Kodaira维的行为，特别是Campana-Peternell和Popa的几个相关猜想。(5)继续研究高维Hodge结构的退化复变，特别是Hodge度规的精确局部行为及其与半单李代数表示理论的关系。(6)继续“混合Hodge模块项目”（与Claude Sabbah联合），其目的是编写具有复杂系数的混合Hodge模块理论的易于理解的处理方法。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。

This is a project in pure mathematics, more precisely in the field of algebraic geometry. Algebraic geometry studies geometric objects that are defined by polynomial equations, using methods from algebra, complex analysis, differential geometry, topology, and partial differential equations. Algebraic geometry also provides useful examples for certain parts of theoretical physics such as string theory. Most of the research proposed by the PI has to do with a subfield of algebraic geometry called Hodge theory. Hodge theory makes it possible to apply results from the theory of partial differential equations to problems in algebraic geometry, in a language that is closer to algebra than to analysis. Some of the work proposed by the PI is about developing Hodge theory further (and to make it more accessible to graduate students); the rest is about using Hodge theory to solve several very specific problems in algebraic geometry. The PI will also work on a comprehensive book on this subject which will be of enormous use to the research community.More specifically, this project has several different research objectives, all related to Hodge theory:(1) To develop a theory of higher multiplier ideals for effective divisors, in joint work with Ruijie Yang, by using the nearby cycles functor for mixed Hodge modules.(2) To apply this theory to theta divisors on principally polarized abelian varieties, in particular to a conjecture by Casalaina-Martin about the multiplicities of their singular points.(3) To continue to study the local and global structure of the locus of self-dual classes for integral variations of Hodge structure; self-dual classes are a generalization of Hodge classes that are of interest in theoretical physics.(4) To further investigate the behavior of Kodaira dimension under smooth morphisms, especially several related conjectures by Campana-Peternell and by Popa.(5) To continue the study of degenerating complex variations of Hodge structure in higher dimensions, in particular the precise local behavior of the Hodge metric, and its relation with the representation theory of semisimple Lie algebras.(6) To continue the "Mixed Hodge Module Project" (joint with Claude Sabbah), whose aim is to write an accessible treatment of the theory of mixed Hodge modules with complex coefficients.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.