A Nonabelian Hodge Correspondence for Principal Bundles in Positive Characteristic

TL;DR

This paper establishes a nonabelian Hodge correspondence for principal bundles in positive characteristic, extending known results to higher dimensions and more complex structures like logahoric torsors and root stacks.

Contribution

It generalizes the Ogus-Vologodsky correspondence to principal bundles and extends the framework to logahoric torsors and higher-dimensional root stacks.

Findings

Proves a nonabelian Hodge correspondence for principal bundles in positive characteristic.

Extends the correspondence to logahoric torsors over log pairs.

Establishes a link between principal bundles on root stacks and parahoric torsors.

Abstract

In this paper, we prove a nonabelian Hodge correspondence for principal bundles on a smooth variety $X$ in positive characteristic, which generalizes the Ogus-Vologodsky correspondence for vector bundles. Then we extend the correspondence to logahoric torsors over a log pair $(X,D)$, where $D$ a reduced normal crossing divisor in $X$. As an intermediate step, we prove a correspondence between principal bundles on root stacks $\mathscr{X}$ and parahoric torsors on $(X,D)$, which generalizes the correspondence on curves given by Balaji--Seshadri to higher dimensional case.