Hodge Theory and Classifying Spaces

霍奇理论和分类空间

• 批准号: 1701237
• 负责人: Burt Totaro
• 金额: $ 35.5万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701237&HistoricalAwards=false
• 关键词: Hodge; Theory; Classifying; Spaces

Mathematics has become central to the modern economy, based on information and computation. In particular, algebraic geometry studies the geometric spaces defined by algebraic formulas. The grand challenges of algebraic geometry, such as the Hodge conjecture, seek to describe all the subspaces of a given space in computable terms. This project will engineer new connections between geometric spaces and the corresponding formulas. Looking at familiar spaces through the eyes of algebraic geometry yields new results in topology and group representation theory. The project will bring undergraduates, graduate students, and postdocs into mathematical research.In more detail, the theory of algebraic cycles aims to describe all the subvarieties of a given algebraic variety. This subject has recently led to dramatic progress on the old problem of which varieties have a rational parametrization. One main theme of the project is to study varieties in positive characteristic, where there are direct connections between algebraic cycles and more computable theories such as Hodge cohomology. The PI will use these connections to make new calculations in representation theory and the theory of algebraic groups. Finally, the PI will prove new cases of the integral Hodge conjecture for complex varieties.

数学已经成为现代经济的核心，基于信息和计算。特别是，代数几何研究由代数公式定义的几何空间。代数几何的巨大挑战，如霍奇猜想，试图用可计算的术语描述给定空间的所有子空间。这个项目将设计几何空间和相应公式之间的新连接。通过代数几何的眼睛来看待熟悉的空间，在拓扑学和群表示论中产生了新的结果。该项目将吸引本科生、研究生和博士后进入数学研究。更详细地说，代数圈理论旨在描述给定代数簇的所有子簇。这一主题最近导致了戏剧性的进展的老问题，品种有一个合理的参数化。该项目的一个主题是研究正特性的品种，其中代数循环和霍奇上同调等更可计算的理论之间有直接的联系。PI将使用这些连接在表示论和代数群理论中进行新的计算。最后，PI将证明复杂簇的积分霍奇猜想的新情况。

项目成果

Cohomological Invariants in Positive Characteristic

正特征中的上同调不变量

• DOI: 10.1093/imrn/rnaa321
• 发表时间: 2021
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Totaro, Burt

Bott vanishing for algebraic surfaces

• DOI: 10.1090/tran/8045
• 发表时间: 2018-12
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: B. Totaro

Hodge theory of classifying stacks

堆栈分类的 Hodge 理论

• DOI: 10.1215/00127094-2018-0003
• 发表时间: 2018
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Totaro, Burt

The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen-Macaulay

• DOI: 10.1090/jag/724
• 发表时间: 2017-10
• 期刊: Journal of Algebraic Geometry
• 影响因子: 1.8
• 作者: B. Totaro

The integral cohomology of the Hilbert scheme of points on a surface

曲面上点的希尔伯特格式的积分上同调

• DOI: 10.1017/fms.2020.35
• 发表时间: 2020
• 期刊: Sigma
• 作者: Totaro, Burt