Decomposition of De Rham Complexes with Smooth Horizontal Coefficients for Semistable Reductions

TL;DR

This paper extends the decomposition of de Rham complexes with smooth horizontal coefficients to semistable morphisms in positive characteristic, leading to a proof of Kollár's vanishing theorem in this setting.

Contribution

It generalizes Illusie's result to a broader class of semistable morphisms liftable over z/p^2Z, and applies this to establish Kollár's vanishing theorem in positive characteristic.

Findings

Decomposition of de Rham complexes with smooth horizontal coefficients for semistable morphisms.

Proof of Kollár's vanishing theorem in positive characteristic under certain liftability conditions.

Extension of Illusie's result to more general semistable reductions.

Abstract

We generalize Illusie's result to prove the decomposition of the de Rham complex with smooth horizontal coefficients for a semistable $S$-morphism $f:X\ra Y$ which is liftable over $\Z/p^2\Z$. As an application, we prove the Koll\'ar vanishing theorem in positive characteristic for a semistable $S$-morphism $f:X\ra Y$ which is liftable over $\Z/p^2\Z$, where all concerned horizontal divisors are smooth over $Y$.