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

TL;DR

This paper establishes the strong liftability of smooth toric varieties and extends the Kawamata-Viehweg vanishing theorem to certain positive characteristic cases, introducing new techniques for constructing liftable varieties.

Contribution

It proves that smooth toric varieties are strongly liftable and extends vanishing theorems to new classes of varieties in positive characteristic.

Findings

Smooth toric varieties are strongly liftable.

Kawamata-Viehweg vanishing holds for certain positive characteristic varieties.

Introduces cyclic cover trick over W_2(k) for constructing liftable varieties.

Abstract

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, first we prove that smooth toric varieties are strongly liftable. As a corollary, we obtain the Kawamata-Viehweg vanishing theorem for smooth projective toric varieties. Second, we prove the Kawamata-Viehweg vanishing theorem for normal projective surfaces which are birational to a strongly liftable smooth projective surface. Finally, we deduce the cyclic cover trick over W_2(k), which can be used to construct a large class of liftable smooth projective varieties.