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

Indranil Biswas, Sujoy Chakraborty

TL;DR

This work advances Brauer-group computations for moduli of parabolic principal bundles on curves. It proves that for generic weights, the Brauer group of the parabolic $\text{PGL}(r,\mathbb{C})$ moduli stack matches the Brauer group of the smooth locus of the coarse moduli space, connecting stack-theoretic and geometric invariant-theory data. It also establishes vanishing of both analytic and algebraic Brauer groups for moduli stacks of quasi-parabolic $G$-bundles when $G$ is simple and simply connected, using spectral-curve techniques, parabolic push-forward, and stack-cohomology comparisons. Collectively, the results generalize prior work on parabolic bundles and provide robust Brauer-group invariants and vanishing properties that inform rationality questions, obstruction theories, and the geometry of parabolic moduli spaces. The methods combine fixed-point codimension estimates, the spectral-curve framework, and Leray-type cohomological analyses to bridge stacky and coarse moduli perspectives.

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.

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

TL;DR

This work advances Brauer-group computations for moduli of parabolic principal bundles on curves. It proves that for generic weights, the Brauer group of the parabolic moduli stack matches the Brauer group of the smooth locus of the coarse moduli space, connecting stack-theoretic and geometric invariant-theory data. It also establishes vanishing of both analytic and algebraic Brauer groups for moduli stacks of quasi-parabolic -bundles when is simple and simply connected, using spectral-curve techniques, parabolic push-forward, and stack-cohomology comparisons. Collectively, the results generalize prior work on parabolic bundles and provide robust Brauer-group invariants and vanishing properties that inform rationality questions, obstruction theories, and the geometry of parabolic moduli spaces. The methods combine fixed-point codimension estimates, the spectral-curve framework, and Leray-type cohomological analyses to bridge stacky and coarse moduli perspectives.

Abstract

We prove that the Brauer group of the moduli stack of parabolic stable principal -bundles on a curve , 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 -bundles. We also show that for any simple and simply connected complex linear algebraic group , the analytic and algebraic Brauer groups of the moduli stack of quasi-parabolic principal -bundles on vanish.
Paper Structure (6 sections, 13 theorems, 82 equations)

This paper contains 6 sections, 13 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on parabolic vector bundles
  3. Parabolic push-forward and pull-back
  4. Fixed points of moduli of parabolic $\text{SL}(r,\mathbb{C})$-bundles
  5. Brauer group of the moduli stack of parabolic $\text{PGL}(r,\mathbb{C})$-bundles
  6. Brauer group of moduli stack of quasi-parabolic $G$--bundles

Key Result

Theorem 1.1

Let $\boldsymbol{\alpha}$ be a generic system of weights. Let $\mathop{\mathrm{\mathcal{N}^{\boldsymbol{m,\alpha}}_X(r,\delta)}}\nolimits^{sm}$ denote the smooth locus of $\mathop{\mathrm{\mathcal{N}^{\boldsymbol{m,\alpha}}_X(r,\delta)}}\nolimits$. Then,

Theorems & Definitions (29)

  • Theorem 1.1: Theorem \ref{['thm:brauer-group-of-moduli-stack']}
  • Theorem 1.2: Proposition \ref{['prop:analytic-brauer-group-vanishing']} and Theorem \ref{['thm:brauer-group-vanishing']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 19 more