A Simpson correspondence for abelian varieties in characteristic p > 0

TL;DR

This paper establishes a local equivalence between the moduli stack of Higgs bundles on the Frobenius twist of an abelian variety and the moduli stack of local systems on the original variety in characteristic p > 0, using Azumaya properties and Morita equivalence.

Contribution

It introduces a Simpson correspondence for abelian varieties in characteristic p > 0, linking Higgs bundles and local systems via Azumaya algebra techniques.

Findings

Local equivalence of moduli stacks established

Uses Azumaya property and Morita equivalence

Extends Simpson correspondence to positive characteristic

Abstract

Let X/k be an abelian variety over an algebraically closed field k of characteristic p > 0. In this paper, using the Azumaya property of the sheaf of crystalline differential operators and the Morita equivalence, we show that etale locally over the Hitchin base, the moduli stack of Higgs bundles on the Frobenius twist X' is equivalent to that of local systems on X. We follow the approach of [Gro16].