The de Rham stack and the variety of very good splittings of a curve

TL;DR

This paper explores the structure of very good splittings of Azumaya algebras on curves, demonstrating their role in preserving semistability in Non-Abelian Hodge Theory and establishing a moduli space with significant geometric properties.

Contribution

It introduces the concept of very good splittings, studies their properties, and shows their importance in the preservation of semistability and in the geometric structure of moduli spaces in positive characteristic.

Findings

Very good splittings form a quasi-projective tame moduli space.

Non-Abelian Hodge isomorphism preserves semistability with very good splittings.

Derived pushforwards of intersection complexes are isomorphic under Hitchin and de Rham-Hitchin morphisms.

Abstract

The stack of relative splittings of a special Azumaya algebra plays a key role in the Non-Abelian Hodge Theory for curves in positive characteristics. In this paper, we define and study an open substack consisting of the so-called very good splittings. We show that, when using very good splittings, the Non-Abelian Hodge isomorphism preserves the semistable loci on the Dolbeault and the de Rham sides. We also show that the stack of very good splittings admits a quasi-projective tame moduli space. As a consequence, we show that the derived pushforwards of the intersection complexes by the Hitchin and the de Rham-Hitchin morphisms are isomorphic and they have isomorphic perverse cohomology sheaves.