Moduli Theory of Sheaves Over Low-Dimensional Varieties

这个项目是在代数几何领域，最古老的，但目前最活跃的数学领域之一。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希腊古代，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。对于理论物理学，代数几何中最相关的领域是模理论，它涉及相同类型的重要几何对象的分类和变形特性。目前的项目将研究层的模量理论，这些层对应于物理（规范）场的概念，如电磁场。与层的模空间相关的数学不变量计算粒子物理学中高能过程的振幅。因此，研究这些不变量的结构关系到几何和高能物理中的基本问题;这个项目的完成将提高我们对数学和理论物理的理解。这一奋进符合当前重要的研究目标，即为当代理论物理学奠定必要的数学基础。本项目将重点研究低维簇上层的模空间几何，包括允许簇模变化时的重要设置。特别地，在相对条件下，利用K3曲面上Grothendieck Quot格式的虚交理论，研究了拟极化K3曲面模空间的重言式Chow环.特殊的Chow类，如模空间上Verlinde层的Chern类，也将被研究。更一般地说，虚交理论的高阶Grothendieck报价计划在任意表面上将被用来推导结果的结构上的重要不变量的基本空间，如希尔伯特计划的点。高阶的报价方案几何也承担有趣的联系，在表示论的问题。

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.