Beyond Geometric Invariant Theory

在数学的许多分支中，人们会遇到具有对称性的方程组-将方程的任何解转换为另一个解。在这种情况下，将所有的解分类到这些对称的作用下通常是非常有用的，如果两个解通过对称变换相关，则认为它们是等价的。在代数几何领域，几何不变理论为这个分类问题提供了一个很好的答案。有理由怀疑几何不变理论的方法可以扩展到更广泛的范围。在粒子物理学中，宇宙被描述为一些方程的解的集合，直到一个非常大的对称性集合的作用。本计画的目的是扩大几何不变理论的方法和结果，并将它们应用于代数几何中的大量分类问题，包括高能物理中的一些研究。主要的技术工具是一种特殊的分层代数堆栈称为θ分层。Theta分层是几何不变理论中研究的Kempf-Ness分层和曲线上向量丛的模空间的Harder-Narasimhan分层的常见推广。经典的这些分层的情况下，顺利栈，研究贝蒂数的模栈的半稳定对象。该项目将追求这些结果的几个概括的情况下，堆栈不一定是光滑的，并提取更微妙的拓扑信息堆栈。例子包括重言式K-理论类积分的跨壁公式，以及更一般地提取关于栈上的相干层和半稳定轨迹的导出类别的信息。

In many branches of mathematics one encounters sets of equations with symmetries -- transformations that take any solution of the equations to another solution. In this situation, it is often very useful to classify all solutions up to the action of these symmetries, two solutions considered equivalent if they are related by a symmetry transformation. Within the field of algebraic geometry, the theory known as geometric invariant theory provides a very good answer to this classification question. There is reason to suspect that the methods of geometric invariant theory extend to a much broader context. In particle physics, the universe is described as a set of solutions of some equations up to the action of a very large set of symmetries. This project aims to broaden the methods and results of geometric invariant theory and bring them to bear on a large set of classification problems in algebraic geometry, including some of those studied in high energy physics.This project envisions a new general approach to moduli problems in algebraic geometry. The main technical tool is a special kind of stratification on an algebraic stack called a theta stratification. Theta stratifications are a common generalization of the Kempf-Ness stratification studied in geometric invariant theory and the Harder-Narasimhan stratification of the moduli space of vector bundles on a curve. Classically these stratifications were used, in the case of smooth stacks, to study the Betti numbers of the moduli stack of semistable objects. The project will pursue several generalizations of these results to situations where the stack is not necessarily smooth, and to extracting more subtle topological information about the stack. Examples include wall-crossing formulas for integrals of tautological K-theory classes, and more generally extracting information about the derived category of coherent sheaves on the stack and the semistable locus.