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Brauer group of moduli of stable parabolic $\text{SL}(r,\mathbb{C})$ and $\text{PGL}(r,\mathbb{C})$-connections and Higgs bundles over a curve

Pavan Adroja, Sujoy Chakraborty

Abstract

Let $X$ be a compact Riemann surface of genus at least $3$. We compute the Brauer groups of the moduli spaces of stable parabolic $\text{SL}(r,\mathbb{C})$-connections and stable strongly parabolic $\text{SL}(r,\mathbb{C})$-Higgs bundles over $X$. We also establish an equality of the Brauer group of the moduli stack of stable parabolic $\text{PGL}(r,\mathbb{C})$-connections and the smooth locus of its coarse moduli space.

Brauer group of moduli of stable parabolic $\text{SL}(r,\mathbb{C})$ and $\text{PGL}(r,\mathbb{C})$-connections and Higgs bundles over a curve

Abstract

Let be a compact Riemann surface of genus at least . We compute the Brauer groups of the moduli spaces of stable parabolic -connections and stable strongly parabolic -Higgs bundles over . We also establish an equality of the Brauer group of the moduli stack of stable parabolic -connections and the smooth locus of its coarse moduli space.
Paper Structure (12 sections, 11 theorems, 82 equations)

This paper contains 12 sections, 11 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Logarithmic connections and residues
  4. parabolic vector bundles and parabolic connections
  5. Generic systems of weights
  6. Moduli space of (twisted) stable parabolic $\text{SL}(r,\mathbb{C})-$connections
  7. Brauer group of $\mathcal{M}^{\boldsymbol{m},\boldsymbol{\alpha}}_{pc}(r,\xi)$
  8. Brauer group of the moduli of stable parabolic $\text{SL}(r,\mathbb{C})$-Higgs bundles
  9. Moduli of stable parabolic $\text{PGL}(r,\mathbb{C})$-connections
  10. Action of torsion line bundles on the moduli of parabolic connections
  11. Dimension estimate of fixed point locus
  12. Brauer group of the moduli stack of stable parabolic $\text{PGL}(r,\mathbb{C})$-connections

Key Result

Theorem 1.1

Let $\boldsymbol{\alpha}$ be a generic system of weights as in Definition def:generic-weights. Let $m_i^j$ denote the multiplicity of the flag at $x_j$. The Brauer groups of $\mathcal{M}_{pc}^{\boldsymbol{m,\alpha}}\left(r,\xi\right)$ and $\mathcal{M}^{\boldsymbol{m},\boldsymbol{\alpha}}_{Higgs}(r,\

Theorems & Definitions (31)

  • Theorem 1.1: Theorem \ref{['Br of Para Conn']} and Theorem \ref{['Br of Higgs']}
  • Theorem 1.2: Theorem \ref{['thm:brauer-group-of-moduli-stack']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: BY
  • Definition 2.6: BY96
  • Definition 3.1: IS
  • Lemma 4.1
  • ...and 21 more