CAREER: Hodge Theory and D-Modules in Algebraic Geometry

职业：代数几何中的 Hodge 理论和 D 模

• 批准号: 1551677
• 负责人: Christian Schnell
• 金额: $ 40万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1551677&HistoricalAwards=false
• 关键词: CAREER; Hodge; Theory; Modules; Algebraic

This project involves both research in pure mathematics and the mathematics training of science majors. Broadly speaking, the research component of the project is research is algebraic geometry, which is the study of study of solution sets of systems of polynomial equations. More specifically, the PI will use methods from the theory of mixed Hodge modules to solve problems in Hodge theory and algebraic geometry. The cohomology groups of a complex algebraic variety carry mixed Hodge structures, and Hodge theory studies these structures and their interaction with the geometry of the variety. The theory of mixed Hodge modules provides a modern framework for this, using the language of perverse sheaves and D-modules. The main educational component of the project is to improve the teaching of mathematics to undergraduate students pursuing a major in a non-mathematics STEM field. The PI will design and teach, in consultation with faculty from another department, a new two-semester course that meets the current mathematical needs of their students. The course will have a non-traditional format, based on an "active learning" model of instruction. In addition, the PI and co-authors will complete a book about mixed Hodge modules, that will make this powerful theory accessible to non-experts. In more technical language, this project has the following research objectives: (1) to construct compactifications for images of period mappings by using the theory of limit Hodge classes; (2) to prove several instances of a conjecture about the structure of Fourier-Mukai transforms of holonomic D-modules on complex abelian varieties; (3) to simplify the construction of singular hermitian metrics on direct images of pluricanonical bundles, due to Paun and Takayama, as well as the recent proof of Ueno's conjecture by Cao and Paun.

该项目既涉及纯数学的研究，又涉及理科专业的数学训练。从广义上讲，该项目的研究部分是代数几何研究，即多项式方程组解集的研究。更具体地说，PI将使用混合Hodge模理论的方法来解决Hodge理论和代数几何中的问题。复杂代数簇的上同调群带有混合霍奇结构，霍奇理论研究这些结构及其与簇几何的相互作用。混合 Hodge 模的理论为此提供了一个现代框架，使用反常滑轮和 D 模的语言。该项目的主要教育内容是改善对非数学 STEM 领域专业本科生的数学教学。 PI 将与另一个系的教师协商，设计和教授一门新的两学期课程，以满足学生当前的数学需求。该课程将采用非传统形式，基于“主动学习”教学模式。此外，PI 和合著者将完成一本关于混合 Hodge 模块的书，这将使非专家也能理解这一强大的理论。用更专业的语言来说，该项目有以下研究目标：（1）利用极限Hodge类理论构造周期映射图像的紧化； (2) 证明复杂阿贝尔簇上完整 D 模的 Fourier-Mukai 变换结构猜想的几个实例； (3) 简化多元束直接图像上的奇异埃尔米特度量的构造，归因于Paun和Takayama，以及Cao和Paun最近对上野猜想的证明。

项目成果

Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

将全纯形式从复杂空间的正则轨迹扩展到奇点的解决

• DOI: 10.1090/jams/962
• 发表时间: 2021
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: Kebekus, Stefan;Schnell, Christian
• 通讯作者: Schnell, Christian

Vanishing theorems for perverse sheaves on abelian varieties, revisited

重新审视阿贝尔簇上反常滑轮的消失定理

• DOI: 10.1007/s00029-017-0377-8
• 发表时间: 2018
• 期刊: Selecta Mathematica
• 作者: Bhatt, Bhargav;Schnell, Christian;Scholze, Peter
• 通讯作者: Scholze, Peter