Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder-Narasimhan type

TL;DR

This paper develops a method to construct quotients of unstable strata in GIT and applies it to build moduli spaces of sheaves with fixed Harder-Narasimhan types and additional data.

Contribution

It introduces a new approach for quotienting unstable GIT strata and constructs moduli spaces of sheaves with specified Harder-Narasimhan types and extra rigidification data.

Findings

A method for constructing quotients of unstable GIT strata.

Construction of moduli spaces of sheaves with fixed Harder-Narasimhan type.

Inclusion of additional data ('n-rigidification') in moduli space construction.

Abstract

When a reductive group $G$ acts linearly on a complex projective scheme $X$ there is a stratification of $X$ into $G$-invariant locally closed subschemes, with an open stratum $X^{ss}$ formed by the semistable points in the sense of Mumford's geometric invariant theory which has a categorical quotient $X^{ss} \to X//G$. In this article we describe a method for constructing quotients of the unstable strata. As an application, we construct moduli spaces of sheaves of fixed Harder-Narasimhan type with some extra data (an '$n$-rigidification') on a projective base.