Beyond Geometric Invariant Theory

在数学的许多分支中，人们都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，根据这些对称性的作用对所有解进行分类通常非常有用，如果两个解通过对称变换相关，则它们被认为是等效的。在代数几何领域，几何不变量理论为这个分类问题提供了很好的答案。有理由怀疑几何不变量理论的方法可以扩展到更广泛的背景。在粒子物理学中，宇宙被描述为一些方程的一组解，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，包括高能物理中的一些研究问题。该项目设想了一种解决代数几何中模问题的新通用方法。主要的技术工具是一种特殊的代数堆栈分层，称为 theta 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模空间的 Harder-Narasimhan 分层的常见推广。经典地，在平滑堆栈的情况下，使用这些分层来研究半稳定物体的模堆栈的贝蒂数。该项目将对这些结果进行几种概括，以适应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。示例包括同义反复 K 理论类积分的穿墙公式，以及更一般地提取有关堆栈上相干滑轮的派生类别和半稳定轨迹的信息。

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.