A Cohomological Non Abelian Hodge Theorem in Positive Characteristic

TL;DR

This paper establishes a cohomological non-abelian Hodge correspondence in positive characteristic, revealing new periodicity phenomena and cohomology ring isomorphisms for moduli spaces of Higgs bundles and connections, using specialization and lifting techniques.

Contribution

It proves a cohomological Simpson Correspondence in positive characteristic and uncovers a new p-multiplicative periodicity, linking cohomology rings across different degrees and characteristics.

Findings

Cohomological Simpson Correspondence in characteristic p

New p-multiplicative periodicity in cohomology rings

Cohomology ring isomorphisms across degrees coprime to rank

Abstract

We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitrary characteristic, cohomology ring isomorphisms between the cohomology rings of Dolbeault moduli spaces for different degrees coprime to the rank. It is interesting that this last result is proved as follows: we prove a weaker version in positive characteristic; we lift and strengthen the weaker version to the result in characteristic zero; finally, we specialize the result to positive characteristic. The moduli spaces we work with admit certain natural morphisms (Hitchin, de Rham-Hitchin, Hodge-Hitchin), and all the cohomology ring isomorphisms we find are filtered isomorphisms for the resulting perverse Leray filtrations.