A note on the filtered decomposition theorem

TL;DR

This paper extends the filtered decomposition theorem of Deligne-Illusie to a more general form, enabling new proofs and analogues in positive characteristic for key theorems related to algebraic geometry and Hodge structures.

Contribution

It introduces a filtered version of the decomposition theorem, providing new proofs and analogues in positive characteristic for classical results.

Findings

Provides a mod p proof for a vanishing theorem in characteristic zero

Establishes a positive characteristic analogue of Deligne's theorem on mixed Hodge structures

Generalizes the logarithmic decomposition theorem to a filtered setting

Abstract

We generalize the logarithmic decomposition theorem of Deligne-Illusie to a filtered version. There are two applications. The easier one provides a mod $p$ proof for a vanishing theorem in characteristic zero. The deeper one gives rise to a positive characteristic analogue of a theorem of Deligne on the mixed Hodge structure attached to complex algebraic varieties.