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On some topological equivalences for moduli spaces of $G$-bundles

Sumit Roy

Abstract

Let $X$ be a smooth projective curve of genus $g \geq 3$, and let $G$ be a nontrivial connected reductive affine algebraic group over $\mathbb{C}$. Examining the moduli spaces of regularly stable $G$-Higgs bundles and holomorphic $G$-connections with a fixed topological type $d\in π_1(G)$ over $X$, we establish that the $k$-th homotopy groups of these two moduli spaces are isomorphic for $k \leq 2g-4$. We also prove that the mixed Hodge structures on the rational cohomology groups of these two moduli spaces are pure and isomorphic. Lastly, we explicitly describe the homotopy groups of the moduli space of $\mathrm{SL}(n,\mathbb{C})$-connections over $X$.

On some topological equivalences for moduli spaces of $G$-bundles

Abstract

Let be a smooth projective curve of genus , and let be a nontrivial connected reductive affine algebraic group over . Examining the moduli spaces of regularly stable -Higgs bundles and holomorphic -connections with a fixed topological type over , we establish that the -th homotopy groups of these two moduli spaces are isomorphic for . We also prove that the mixed Hodge structures on the rational cohomology groups of these two moduli spaces are pure and isomorphic. Lastly, we explicitly describe the homotopy groups of the moduli space of -connections over .
Paper Structure (8 sections, 7 theorems, 72 equations)

This paper contains 8 sections, 7 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Principal $G$-bundles
  4. Principal $G$-Higgs bundles
  5. Holomorphic $G$-conections
  6. Mixed Hodge structures
  7. Topological Invariances
  8. $\mathrm{SL}_n$ case

Key Result

Proposition 2.10

Let $Y$ be a complex algebraic variety. Then a mixed Hodge structure exists on the cohomology $H^i(Y,\mathbb{C})$. Furthermore, the weight filtration is given by and the Hodge filtration is given by

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10: D72, D75
  • ...and 13 more