CAREER: Hodge Theory and Moduli

代数变量是多项式方程的解的空间，代数几何试图用几何方法描述这些解。代数变量在数学中无处不在，是复杂几何、数论、拓扑学和表示理论研究的重要对象。理解它们的模通常是特别重要的，也就是说，当定义它们的多项式方程的系数变化时，它们是如何变化的。霍奇理论为这个问题提供了一种观点：代数变量的模可以用微分形式的积分如何变化来理解。本研究项目将使用源自模型理论、表示理论和复杂几何的新方法来理解这种联系。该项目特别旨在促进与数学密切相关领域的互动，并鼓励学生和年轻研究人员的参与。该项目将解决两类主要的研究问题。霍奇理论是代数几何中的一个强大工具，但它本质上是先验的。PI和合作者最近的工作已经证明，模型理论和复杂几何的技术可以用来系统地弥合这一鸿沟，该项目的第一个目标是更深入地开发这些技术的应用。其次，该项目将更详细地了解与Hodge理论特别密切相关的某些品种的模量，包括阿贝尔品种和超kahler品种。在先前使用微分几何和复几何技术的基础上，PI将进一步研究这些变体的模空间的几何和算术。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。

Algebraic varieties are the spaces of solutions to polynomial equations, and algebraic geometry seeks to describe those solutions geometrically. Ubiquitous in mathematics, algebraic varieties are critical objects of study in complex geometry, number theory, topology, and representation theory. It is often particularly important to understand their moduli, that is, how they vary as the coefficients of the polynomial equations defining them are varied. Hodge theory offers one perspective on this problem: the moduli of algebraic varieties can be understood in terms of how integrals of differential forms vary. This research project will use new methods stemming from model theory, representation theory, and complex geometry to understand this connection. The project especially aims to promote interactions with closely-related areas of mathematics and to encourage the participation of students and young researchers. The project will address two main types of research problems. Hodge theory is a powerful tool in algebraic geometry but it is fundamentally transcendental in nature. Recent work of the PI and coauthors has demonstrated that techniques from model theory and complex geometry can be used to bridge this divide systematically, and the first goal of the project is to develop the application of these techniques more deeply. Secondly, the project will develop a more detailed understanding of the moduli of certain varieties which are particularly closely related to Hodge theory, including abelian varieties and hyperkahler varieties. Building on previous work using techniques from differential and complex geometry, the PI will further investigate the geometry and arithmetic of the moduli spaces of these varieties.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.