CAREER: Hodge Theory and D-Modules in Algebraic Geometry

点击翻译按钮获取中文摘要

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.