Moduli Theory of Sheaves Over Low-Dimensional Varieties

这个项目属于代数几何领域，这是数学中最古老但目前最活跃的领域之一。代数几何的核心是研究由多项式方程组的解集切割出来的几何空间。一方面，这门学科可以追溯到古希腊，而它的现代发展为当前理解早期宇宙物理学的努力提供了数学基础。在理论物理中，与代数几何最相关的领域是模理论，模理论研究相同类型的重要几何对象的分类和变形特性。目前的项目将研究与物理（规范）场（如电磁场）概念相对应的物体的模理论。与模空间相关的数学不变量计算粒子物理中高能过程的振幅。因此，研究这些不变量的结构涉及几何和高能物理的基本问题；这个项目的完成将提高我们对数学和理论物理的理解。这一努力符合为当代理论物理奠定所需数学基础的重要研究目标。本项目将重点研究低维品种上的轮轴模空间几何，包括品种允许模变化时的重要设置。特别是，在相对环境下，将利用K3曲面上Grothendieck Quot格式的虚相交理论，研究准极化K3曲面模空间的重言Chow环。特殊的Chow类，如模空间上Verlinde轴的chen类，也将被研究。更一般地说，任意曲面上的高阶Grothendieck Quot格式的虚相交理论将用于推导基本空间（如点的Hilbert格式）上重要不变量结构的结果。高阶图式几何也与表征理论中的问题有着有趣的联系。

This project is in the field of algebraic geometry, one of the oldest yet currently most active areas of mathematics. At its core, algebraic geometry is the study of geometric spaces cut out by solution sets of systems of polynomial equations. The subject goes back to Greek antiquity on the one hand, while its modern development provides the mathematical foundation for current efforts to understand the physics of the early universe. For theoretical physics, the most relevant area of algebraic geometry is moduli theory, which deals with the classificaiton and deformation properties of important geometric objects of the same type. The current project will investigate moduli theory of sheaves, objects which correspond to the notion of physical (gauge) fields, such as the electromagnetic field. Mathematical invariants associated with moduli spaces of sheaves calculate amplitudes of high-energy processes in particle physics. Studying the structure of these invariants therefore bears on essential questions in both geometry and high-energy physics; the completion of this project will thus improve our understanding of both mathematics and theoretical physics. The endeavor fits in the important current research goal of setting needed mathematical foundations to contemporary theoretical physics.This project will focus on the study of the geometry of moduli spaces of sheaves on low-dimensional varieties, including the important setting when the variety is allowed to vary in moduli. In particular, the virtual intersection theory of Grothendieck Quot schemes over K3 surfaces, in a relative setting, will be used to study the tautological Chow ring of the moduli space of quasipolarized K3 surfaces. Special Chow classes, such as the Chern classes of Verlinde sheaves over the moduli space, will also be investigated. More generally, the virtual intersection theory of higher-rank Grothendieck Quot schemes over an arbitrary surface will be used to deduce results on the structure of important invariants on fundamental spaces such as Hilbert schemes of points. The higher-rank Quot scheme geometry also bears interesting connections to questions in representation theory.