A note on Higgs-de Rham flows of level zero

TL;DR

This paper studies level-zero Higgs-de Rham flows, refining their definition and establishing a correspondence with certain fundamental group representations in positive characteristic, extending classical Katz results.

Contribution

It improves the definition of level-zero Higgs-de Rham flows and establishes a Hitchin-Simpson-type correspondence with fundamental group representations in positive characteristic.

Findings

Established a correspondence between level-zero Higgs-de Rham flows and fundamental group representations.

Compared deformation theories of both sides of the correspondence.

Translated Galois actions into the Higgs side.

Abstract

The notion of Higgs-de Rham flows was introduced by Lan-Sheng-Zuo, as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory. In this short note we investigate a small part of this theory, and study those Higgs-de Rham flows which are of level zero. We improve the original definition of level-zero Higgs-de Rham flows (which works for general levels), and establish a Hitchin-Simpson-type correspondence between such objects and certain representations of fundamental groups in positive characteristic, which generalizes the classical results of Katz. We compare the deformation theories of two sides in the correspondence, and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.