A Simpson correspondence in positive characteristic

TL;DR

This paper develops a positive characteristic analogue of the Simpson correspondence by constructing a Frobenius map on differential operators, leading to an equivalence between Higgs modules and differential modules of level m.

Contribution

It introduces a new Frobenius map on differential operators of level m and establishes a Simpson-type correspondence in positive characteristic.

Findings

Defined the p^m-curvature map dual to divided power maps.

Constructed a Frobenius map leading to an Azumaya splitting.

Established an equivalence between Higgs modules and differential modules.

Abstract

We define the $p^m$-curvature map on the sheaf of differential operators of level $m$ on a scheme of positive characteristic $p$ as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of $p^m$-curvature on differential modules of level $m$. We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level $m$. Then, using a lifting modulo $p^2$ of Frobenius, we are able to define a Frobenius map on differential operators of level $m$ as dual to some divided Frobenius on infinitesimal neighborhhods. We use this map to build a true Azumaya splitting of the completed sheaf of differential operators of level $m$ (up to an automorphism of the center). From this, we derive the fact that Frobenius pull back gives, when restricted to quasi-nilpotent objects, an equivalence between Higgs-modules and differential modules of level $m$. We end by explaining the relation with related work of Ogus-Vologodski and van der Put in level zero as well as Berthelot's Frobenius descent.