Complex Geometric and Lie Theoretic Aspects of Hodge Theory

代数几何是与多项式方程(系统)的解有关的数学领域。这些解集被称为代数簇，人们对了解它们的代数和几何性质很感兴趣。这些结构可能相当复杂和神秘。获得一些洞察力的一种方法是考虑一下该品种的“霍奇结构”。这是一个具有线性代数性质的更简单的对象，尽管如此，它仍然经常编码关于多样性的大量信息。这个项目的主题是应用表示论(线性代数的一种复杂推广)来获得对这些变量的几何洞察。本项目涉及的主题包括：(A)Hodge结构的退化，及其在模空间及其紧化研究中的应用；(B)Hodge结构变体的特征簇(该程序与Hwang-Mok程序非常相似，通过其最小有理切线的变体来研究Fano流形)；和(C)代表Looijenga-LUNTS-Verbitsky代数的Hyperkahler上同调的结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry is a field of mathematics concerned with the solutions of (systems of) polynomial equations. These solution sets are called algebraic varieties, and one is interested in understanding their algebraic and geometric properties. These structures can be quite complicated and mysterious. One way to gain some insight is to consider the variety's "Hodge structure". This is a simpler object with a linear algebraic nature, that nonetheless often encodes a great deal of information about the variety. The theme of this project is the application of representation theory (a sophisticated generalization of linear algebra) to obtain geometric insight into these varieties.Topics addressed in this project include: (a) degenerations of Hodge structure, and their application to the study of moduli spaces and their compactifications; (b) characteristic varieties of a variation of Hodge structure (this program is driven by a close analogy with the Hwang-Mok program to study Fano manifolds via their varieties of minimal rational tangents); and (c) structure of hyperkahler cohomology as a representation of the Looijenga-Lunts-Verbitsky algebra.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.