Tame Parahoric Nonabelian Hodge Correspondence in Positive Characteristic over Algebraic Curves

TL;DR

This paper establishes a nonabelian Hodge correspondence for tame G-local systems and logarithmic G-Higgs bundles over algebraic curves in positive characteristic, utilizing parahoric group schemes for noncompact cases.

Contribution

It introduces a novel framework using parahoric group schemes to extend nonabelian Hodge correspondence to noncompact curves in positive characteristic.

Findings

Proves a version of nonabelian Hodge correspondence in positive characteristic.

Uses parahoric group schemes to handle noncompact cases.

Provides a full description of the correspondence for tame G-local systems.

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.