On automorphisms of semistableG-bundles with decorations

Abstract

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