  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.
