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——将方程的任何解转化为另一个解的变换。在这种情况下，根据这些对称的作用对所有解进行分类通常是非常有用的，如果两个解通过对称变换相关联，则认为它们是等价的。在代数几何领域中，被称为几何不变量理论的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量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一组解，这些方程是由一组非常大的对称性所决定的。本项目旨在扩展几何不变量理论的方法和结果，并将其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分类问题。这个项目设想了一种新的通用方法来解决代数几何中的模问题。主要的技术工具是一种特殊的代数堆栈上的分层，称为θ分层。Theta分层是几何不变理论中研究的Kempf-Ness分层和曲线上向量束模空间的Harder-Narasimhan分层的一般推广。在光滑叠的情况下，经典地使用这些分层来研究半稳定对象的模叠的贝蒂数。该项目将对这些结果进行几种概括，以解决堆栈不一定平滑的情况，并提取有关堆栈的更微妙的拓扑信息。例子包括重言k理论类积分的壁交公式，以及更一般地提取关于堆栈上相干束和半稳定轨迹的派生类别的信息。

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.