Strongly liftable schemes and the Kawamata–Viehweg vanishing in positive characteristic III

A smooth scheme X over a field k   of positive characteristic is said to be strongly liftable over W2(k)W2(k), if X and all prime divisors on X   can be lifted simultaneously over W2(k)W2(k). In this paper, we first deduce the Kummer covering trick over W2(k)W2(k), which can be used to construct a large class of smooth projective varieties liftable over W2(k)W2(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 [18] and [19] to the case where everything is considered over W(k)W(k), the ring of Witt vectors of k.