On automorphisms of semistable G-bundles with decorations

TL;DR

This paper establishes a rigidity result for automorphisms of stacks of semistable G-bundles with decorations, with applications to moduli space descriptions and smoothness in characteristic zero.

Contribution

It proves a general rigidity theorem for automorphisms of stacks with adequate moduli spaces, extending to various semistable G-bundle moduli stacks and their applications.

Findings

Automorphisms of certain stacks are rigid under specified conditions.

Stacks of semistable decorated G-bundles can be expressed as GIT quotients in characteristic 0.

The stack of semistable meromorphic G-Higgs bundles is smooth over any base in characteristic 0.

Abstract

We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of $G$-bundles on a smooth projective curve for a reductive algebraic group $G$. For example, our result applies to the stack of semistable $G$-bundles, stacks of semistable Hitchin pairs, and stacks of semistable parabolic $G$-bundles. Similar arguments apply to Gieseker semistable $G$-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic $0$ every stack of semistable decorated $G$-bundles admitting a quasiprojective good moduli space can be written naturally as a $G$-linearized global quotient $Y/G$, so the moduli problem can be interpreted as a GIT problem. Secondly, we give a proof that the stack of semistable meromorphic $G$-Higgs bundles on a family of curves is smooth over any base in characteristic $0$.