Kodaira-Saito vanishing via Higgs bundles in positive characteristic

TL;DR

This paper provides a new proof of a special case of the Kodaira-Saito vanishing theorem using Higgs bundles in positive characteristic, avoiding mixed Hodge modules and leveraging properties of Higgs bundles.

Contribution

It introduces a novel proof technique for Kodaira-Saito vanishing based on Higgs bundles in positive characteristic, bypassing traditional mixed Hodge module methods.

Findings

Proves a vanishing theorem for semistable nilpotent Higgs bundles in positive characteristic.

Reduces the Kodaira-Saito vanishing to a more general Higgs bundle vanishing result.

Provides insights into Higgs bundle behavior in positive characteristic settings.

Abstract

The goal of this paper is to give a new proof of a special case of the Kodaira-Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.