Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic III

TL;DR

This paper develops methods to lift smooth schemes and divisors over W_2(k), proving a vanishing theorem in positive characteristic and extending previous results to the Witt ring context.

Contribution

It introduces a Kummer covering trick over W_2(k) for constructing liftable varieties and generalizes Kawamata-Viehweg vanishing to schemes over the Witt ring.

Findings

Constructed a large class of liftable smooth projective varieties.

Provided a direct proof of Kawamata-Viehweg vanishing in this setting.

Extended previous results to schemes over the Witt ring.

Abstract

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable over W_2(k), if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, we first deduce the Kummer covering trick over W_2(k), which can be used to construct a large class of smooth projective varieties liftable over W_2(k), and to give a direct proof of the Kawamata-Viehweg vanishing theorem on strongly liftable schemes. Secondly, we generalize almost all of the results in [Xie10, Xie11] to the case where everything is considered over W(k), the ring of Witt vectors of k.