Mathematical Research Letters

Volume 18 (2011)

Number 2

Strongly liftable schemes and the Kawamata-Viehweg Vanishing in positive characteristic II

Pages: 315 – 328

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n2.a9

Author

Qihong Xie (School of Mathematical Sciences, Fudan University, Shanghai 200433, China)

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.

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