Sur une $q$-d\'eformation locale de la th\'eorie de Hodge non-ab\'elienne en caract\'eristique positive

TL;DR

This paper constructs a local $q$-deformation of the non-abelian Hodge theory in characteristic $p$, extending the Simpson correspondence using $q$-twisted differential operators and exploring connections with recent work by Bhatt and Scholze.

Contribution

It introduces a new local $q$-deformation of the non-abelian Hodge correspondence in characteristic $p$, based on Morita-equivalence of $q$-twisted differential operators.

Findings

Construction of the $q$-deformation using Morita-equivalence.

Explanation of relations with Bhatt and Scholze's recent work.

Focus on the case of dimension 1 for clarity.

Abstract

For $p$ a prime number and $q$ a non trivial $p$th root of 1, we present the main steps of the construction of a local $q$-deformation of the "Simpson correspondence in characteristic $p$" found by Ogus and Vologodsky in 2005. The construction is based on the Morita-equivalence between a ring of $q$-twisted differential operators and its center. We also explain the expected relations between this construction and those recently done by Bhatt and Scholze. For the sake of readability, we limit ourselves to the case of dimension 1.