Quotients by Parabolic Groups and Moduli Spaces of Unstable Objects

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.

GIT for $\hat U$-actions on algebraic $\mathbb C$-schemes

“…Given a grading 1PS λ, let Û denote the semi-direct product U ⋊ λ G m , or Û alternatively. Geometric invariant theory of linearisations of Û on projective varieties were studied in [Bér+18b], [BK17], [BDK17], [Bér+16], [HJ21], with conditions on dimensions of U -stabilisers or conditions on an extension of the U -action to a suitable reductive group.…”

“…Recently, moduli of pure sheaves of a fixed Harder-Narasimhan type of length 2 on projective varieties are constructed in [Jac21]. More applications can be found in [Bér+18b], [Ham20], [HJ21].…”