Brauer group of moduli stacks of parabolic principal bundles over a curve

TL;DR

This paper investigates the Brauer groups of moduli stacks and coarse moduli spaces of parabolic principal bundles over a curve, establishing their equality in certain cases and vanishing results for specific groups.

Contribution

It proves the Brauer group of the moduli stack matches that of the coarse moduli space for parabolic PGL(r,C)-bundles and shows vanishing of Brauer groups for simple, simply connected groups.

Findings

Brauer group of the moduli stack equals that of the coarse moduli space for PGL(r,C).

Brauer groups of the moduli stack vanish for simple, simply connected groups.

Results hold for generic weights and arbitrary parabolic divisors.

Abstract

We prove that the Brauer group of the moduli stack of parabolic stable principal $\text{PGL}(r,\mathbb{C})$-bundles on a curve $X$, for a generic system of weights along an arbitrary parabolic divisor, coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of parabolic stable principal $\text{PGL}(r,\mathbb{C})$-bundles. We also show that for any simple and simply connected complex linear algebraic group $G$, the analytic and algebraic Brauer groups of the moduli stack of quasi-parabolic principal $G$-bundles on $X$ vanish.