CAREER: Hodge Theory and D-Modules in Algebraic Geometry

该项目既涉及纯数学研究，也涉及理工科学生的数学培养。广义上讲，该项目的研究内容是研究代数几何，即对多项式方程组解集的研究。更具体地说，PI将使用混合Hodge模块理论中的方法来解决Hodge理论和代数几何中的问题。复杂代数变体的上同调群带有混合Hodge结构，Hodge理论研究这些结构及其与变体几何的相互作用。混合Hodge模块的理论为这个问题提供了一个现代的框架，使用了反常捆和d模块的语言。该项目的主要教育组成部分是改善对攻读非数学STEM专业的本科生的数学教学。PI将与其他系的教员协商，设计并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的形式，以“主动学习”的教学模式为基础。此外，PI和合著者将完成一本关于混合霍奇模块的书，这将使这个强大的理论对非专家来说也很容易理解。用更专业的语言来说，本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)证明了复阿贝尔变集上完整d模的Fourier-Mukai变换结构猜想的几个实例；(3)简化了复数形束直接像上奇异厄米度量的构造，这是由paan和Takayama，以及Cao和paan最近对上野猜想的证明。

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.