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Tame Parahoric Nonabelian Hodge Correspondence in Positive Characteristic over Algebraic Curves

Mao Li, Hao Sun

Abstract

Let $G$ be a reductive group, and let $X$ be an algebraic curve over an algebraically closed field $k$ with positive characteristic. We prove a version of nonabelian Hodge correspondence for tame $G$-local systems over $X$ and logarithmic $G$-Higgs bundles over the Frobenius twist $X'$. To obtain a full description of the correspondence for the noncompact case, we introduce the language of parahoric group schemes to establish the correspondence.

Tame Parahoric Nonabelian Hodge Correspondence in Positive Characteristic over Algebraic Curves

Abstract

Let be a reductive group, and let be an algebraic curve over an algebraically closed field with positive characteristic. We prove a version of nonabelian Hodge correspondence for tame -local systems over and logarithmic -Higgs bundles over the Frobenius twist . To obtain a full description of the correspondence for the noncompact case, we introduce the language of parahoric group schemes to establish the correspondence.

Paper Structure

This paper contains 23 sections, 46 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Local Case
  5. Global Case
  6. Structure of the Paper
  7. Preliminaries
  8. Correspondence: Parahoric vs. Equivariant
  9. Torsors
  10. Logarithmic Higgs Bundles
  11. Tame Local Systems
  12. Local Tame Parahoric Nonabelian Hodge Correspondence
  13. Standard Form of Tame Connections
  14. Irrational Case
  15. Rational Case
  16. ...and 8 more sections

Key Result

Theorem 1.1

We have an equivalence of categories Moreover, the $p$-curvature of the parahoric $\mathcal{P}_\theta(\mathcal{O})$-connection is zero if and only if the corresponding parahoric $G'_{\theta+\tau}(\mathcal{O})$-Higgs bundle has zero Higgs field.

Theorems & Definitions (89)

  • Theorem 1.1: Theorem \ref{['classification for parahoric connection']}
  • Theorem 1.2: Theorem \ref{['first structure theorem for G']}
  • Proposition 1.3: Proposition \ref{['structure of X']}
  • Theorem 1.4: Theorem \ref{['parahoric non hod corr']} and Proposition \ref{['structure of parahoric X']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • ...and 79 more