Beyond Geometric Invariant Theory

在数学的许多分支中，人们遇到具有对称性的一组方程--将方程的任何解转换为另一解的变换。在这种情况下，将所有的解决方案归类到这些对称的作用通常是非常有用的，如果两个解决方案通过对称变换相关，则被认为是等价的。在代数几何领域中，被称为几何不变量理论的理论很好地回答了这个分类问题。有理由怀疑几何不变量理论的方法延伸到了更广泛的背景下。在粒子物理学中，宇宙被描述为一组方程的解，直到一组非常大的对称性的作用。该项目旨在拓宽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，包括一些在高能物理中研究的问题。该项目设想了一种新的一般方法来解决代数几何中的模问题。主要的技术工具是代数堆栈上的一种特殊的分层，称为theta分层。Theta分层是几何不变量理论中研究的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。经典地，在光滑堆栈的情况下，这些分层被用来研究半稳定对象的模堆叠的Betti数。该项目将把这些结果推广到堆栈不一定平滑的情况，并提取关于堆栈的更微妙的拓扑信息。例子包括重言式K-理论类的积分的跨墙公式，以及更一般地提取关于堆栈和半稳定轨迹上的相干片层的派生范畴的信息。

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.