Semistable modules over Lie algebroids in positive characteristic

TL;DR

This paper investigates semistable modules over Lie algebroids in positive characteristic, establishing a Langton's type theorem, relating it to Simpson's construction, and proving a conjecture on the strong semistability of Hodge sheaves.

Contribution

It introduces a Langton's type theorem for moduli of Lie algebroid modules in positive characteristic and proves a conjecture on semistability of Hodge sheaves.

Findings

Established a Langton's type theorem for these moduli spaces.

Connected Langton's construction with Simpson's filtration.

Proved the conjecture that semistable Hodge sheaves are strongly semistable.

Abstract

We study Lie algebroids in positive characteristic and moduli spaces of their modules. In particular, we show a Langton's type theorem for the corresponding moduli spaces. We relate Langton's construction to Simpson's construction of gr-semistable Griffiths transverse filtration. We use it to prove a recent conjecture of Lan-Sheng-Zuo that semistable systems of Hodge sheaves on liftable varieties in positive characteristic are strongly semistable.