Kawamata-Viehweg vanishing on rational surfaces in positive characteristic
Among the generalizations of the famous Kodaira vanishing theorem we certainly find the Kawamata-Viehweg vanishing theorem which is recalled in the paper in its form dealing with ample \(\mathbb{Q}\)-divisors (Theorem 1.1) and in its most general form stated for log pairs which have only KVT (i.e. Kawamata log terminal) singularities (Theorem 1.2). Then other generalizations are recalled from the Kodaira-Akizuki-Nakano vanishing theorem in positive characteristic [\textit{P. Deligne} and \textit{L. Illusie}, Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)] under the lifting condition to the ring of Witt vectors of length \(2\), to the more recent Kawamata-Viehweg vanishing theorem in positive characteristic (Theorem 1.3) under the lifting condition for certain log pairs. In this context we find the main result of the paper (Theorem 1.4) where the author proves that when \(X\) is a normal projective rational surface, \(D\) a \(\mathbb{Q}\)-Cartier Weil divisor on \(X\), and \(B\) an effective \(\mathbb{Q}\)-divisor such that \((X, B)\) is KLT and \(D - (K_X + B)\) is ample then \(H^1(X, D) = 0\). Consequences of Theorem 1.4 (Corollaries 1.5 and 1.6) give rise to vanishing results on log Del Pezzo surfaces and a weak version of the logarithmic Kollár vanishing theorem and the logarithmic semipositivity theorem (Corollary 1.9). Finally in Corollary 1.10 the author provides examples of log pairs which cannot be lifted to the ring of Witt vectors of length \(2\) showing that the lifting condition is in general very strong.
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