Some observations on cohomologically \(p\)-ample bundles (Q1312943)

The author obtains a vanishing theorem for the cohomology of a \(p\)-ample bundle. A key step is the use of a result of \textit{P. Deligne} and \textit{L. Illusie} which gives an algebraic proof of the Kodaira-Akizuki-Nakano vanishing theorem [Invent. Math. 89, 247-270 (1987; Zbl 0632.14017)]. It is proved that no projective scheme of dimension greater than 1, liftable over \(W_ 2(k)\), has cohomologically \(p\)-ample cotangent bundle. There are discussed the cohomologically \(p\)-ample bundles on \(\mathbb{P}^ d_{F_ p}\). There are obtained sufficient conditions for a Frobenius morphism not to be extended to any lifting of a projective \(X\) to \(W_ 2 (k)\).