Moduli spaces of sheaves that are semistable with respect to a K\"ahler polarisation

TL;DR

This paper constructs a proper moduli space for rank two sheaves on complex projective manifolds that are semistable with respect to a Kähler polarization, expanding the understanding of sheaf moduli.

Contribution

It introduces a new method to build moduli spaces of semistable sheaves using an existence criterion for good moduli spaces of Artin stacks.

Findings

Constructed a proper moduli space for rank two sheaves with fixed Chern classes.

Extended the moduli space construction to sheaves semistable with respect to Kähler classes.

Demonstrated the applicability of Alper-Fedorchuk-Smyth's criterion in this context.

Abstract

Using an existence criterion for good moduli spaces of Artin stacks by Alper-Fedorchuk-Smyth we construct a proper moduli space of rank two sheaves with fixed Chern classes on a given complex projective manifold that are Gieseker-Maruyama-semistable with respect to a fixed K\"ahler class.