Moduli of flat connections in positive characteristic

TL;DR

This paper establishes a local equivalence between moduli stacks of flat connections and Higgs bundles in positive characteristic, explores stability, and extends the Langlands correspondence for D-modules to integral spectral curves.

Contribution

It demonstrates an étale local equivalence of moduli stacks in positive characteristic and extends the Langlands correspondence to a broader class of spectral curves.

Findings

Moduli stack of flat connections is étale locally equivalent to Higgs bundles.

Generalization of the properness of the Hitchin map.

Extension of the Langlands correspondence to integral spectral curves.

Abstract

Exploiting the description of rings of differential operators as Azumaya algebras on cotangent bundles, we show that the moduli stack of flat connections on a curve (allowed to acquire orbifold points) defined over an algebraically closed field of positive characteristic is \'etale locally equivalent to a moduli stack of Higgs bundles over the Hitchin base. We then study the interplay with stability and generalize a result of Laszlo-Pauly, concerning properness of the Hitchin map. Using Arinkin's autoduality of compactified Jacobians we extend the main result of Bezrukavnikov-Braverman, the Langlands correspondence for D-modules in positive characteristic for smooth spectral curves, to the locus of integral spectral curves.