Quotients by Parabolic Groups and Moduli Spaces of Unstable Objects

TL;DR

This paper develops new methods in non-reductive GIT to construct quotients by parabolic groups, enabling the creation of moduli spaces for unstable objects with specific stratifications, expanding the scope of moduli theory.

Contribution

It introduces a staged approach to parabolic quotients in non-reductive GIT, allowing for the construction of moduli spaces of unstable objects under verifiable stabiliser conditions.

Findings

Constructed moduli spaces for certain sheaves with fixed Harder-Narasimhan types.

Developed a staged method for parabolic quotients in non-reductive GIT.

Verified stabiliser assumptions in specific cases.

Abstract

Motivated by constructing moduli spaces of unstable objects, we use new ideas in non-reductive GIT to construct quotients by parabolic group actions. For moduli problems with semistable moduli spaces constructed by reductive GIT, we consider associated instability (or HKKN) stratifications, which are often closed related to Harder-Narasimhan stratifications, and construct quotients of the unstable strata under various stabiliser assumptions by further developing ideas of non-reductive GIT. Our approach is to construct parabolic quotients in stages, in order for the required stabiliser assumptions to be more readily verified. To illustrate these ideas, we construct moduli spaces for certain sheaves of fixed Harder-Narasimhan type on a projective scheme in cases where our stabiliser assumptions can be verified.cases where our stabiliser assumptions can be verified.