Moduli spaces of sheaves via affine Grassmannians

TL;DR

This paper introduces a novel infinite-dimensional GIT approach to study moduli spaces of sheaves, constructing stratifications and good moduli spaces using higher-dimensional affine Grassmannians.

Contribution

It develops a new method for analyzing moduli of sheaves via affine Grassmannians, creating stratifications and moduli spaces for stacks of sheaves and related objects.

Findings

Constructed $ heta$-stratifications for moduli stacks.

Built higher-dimensional affine Grassmannians for these moduli problems.

Established existence of good moduli spaces for semistable points.

Abstract

We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of geometric invariant theory. We apply this to two familiar moduli problems: the stack of $\Lambda$-modules and the stack of pairs. In both examples, we construct a $\Theta$-stratification of the stack, defined in terms of a polynomial numerical invariant, and we construct good moduli spaces for the open substacks of semistable points. One of the essential ingredients is the construction of higher dimensional analogues of the affine Grassmannian for the moduli problems considered.