  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.
