Complex Geometric and Lie Theoretic Aspects of Hodge Theory

霍奇理论的复杂几何和李理论方面

• 批准号: 1906352
• 负责人: Colleen Robles
• 金额: $ 22.4万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906352&HistoricalAwards=false
• 关键词: Complex; Geometric; Lie; Theoretic; Aspects

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.

代数几何是研究多项式方程（方程组）解的数学领域。这些解集被称为代数变种，人们对理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。一种方法是考虑该品种的“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，它经常编码大量关于变化的信息。这个项目的主题是应用表征理论（线性代数的一个复杂的推广）来获得对这些变化的几何洞察力。本课题涉及的主题包括：(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结构变体的特征变体（该程序与通过最小有理切线的变体来研究Fano流形的Hwang-Mok程序进行了密切的类比）；(c)表示Looijenga-Lunts-Verbitsky代数的超kahler上同调结构。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。

项目成果

Hodge Representations

霍奇表示

• DOI: 10.1017/exp.2020.55
• 发表时间: 2020
• 期刊: Experimental Results
• 作者: Han, Xiayimei;Robles, Colleen
• 通讯作者: Robles, Colleen