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Semiprojectivity of the moduli of principal $G$-bundles with $λ$-connections

Sumit Roy, Anoop Singh

Abstract

Let $X$ be a compact connected Riemann surface of genus $g \geq 2$ and $G$ a connected reductive affine algebraic group over $\mathbb{C}$. We prove the semiprojectivity of the moduli spaces of semistable $G$-Higgs bundles and $G$-bundles with $λ$-connections of fixed topological type $d\in π_1(G)$. As an application, in the smooth case we describe the resulting Bialynicki - Birula decomposition and derive cohomological and motivic consequences.

Semiprojectivity of the moduli of principal $G$-bundles with $λ$-connections

Abstract

Let be a compact connected Riemann surface of genus and a connected reductive affine algebraic group over . We prove the semiprojectivity of the moduli spaces of semistable -Higgs bundles and -bundles with -connections of fixed topological type . As an application, in the smooth case we describe the resulting Bialynicki - Birula decomposition and derive cohomological and motivic consequences.
Paper Structure (8 sections, 9 theorems, 55 equations)

This paper contains 8 sections, 9 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G$-Higgs bundles
  4. Holomorphic $G$-connections
  5. $\lambda$-connections
  6. Semiprojectivity of the moduli space of $G$-Higgs bundles
  7. Semiprojectivity of principal Hodge moduli space
  8. Cohomological and motivic consequences in the smooth case

Key Result

Lemma 2.1

The Hitchin map $h : \mathcal{M}^d_{\mathrm{Higgs}}(G) \to \mathcal{H}$ is $\mathbb{C}^*$-equivariant.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 15 more