强可提升概型和正特征的Kawamata-Viehweg消灭定理I.docVIP

强可提升概型和正特征的 Kawamata-Viehweg 消灭定理 I 摘要：正特征域 k 上的光滑概形 X 称为强可提升的，如果 X 和 X 上的所有素除子可以同时提 升到 W2(k) 上。本文给出了强可提升概形的很多例子和性质。作为应用，证明了如果正特征的 正规射影曲面 S 双有理等价于一个强可提升曲面，则 S 上的 Kawamata-Viehweg 消灭定理成 立。 关键词：代数几何，正特征，强可提升概形，Kawamata-Viehweg 消灭定理 中图分类号： O187.2 Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic I Xie Qi-Hong School of Mathematical Sciences, Fudan University, Shanghai 200433 Abstract: A smooth scheme X over a ?eld k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W2(k). Some concrete examples and properties of strongly liftable schemes are given in this paper. As an application, the author proves that the Kawamata-Viehweg vanishing theorem in positive characteristic holds on any normal projective surface which is birational to a strongly liftable surface. Key words: Algebraic geometry, positive characteristic, strongly liftable scheme, Kawamata-Viehweg vanishing theorem 1  Introduction As is well known, the Kawamata-Viehweg vanishing theorem plays a crucial role in bira- tional geometry of algebraic varieties, and it is of several forms, where the most general form is stated for log pairs which have only Kawamata log terminal singularities [KMM87, Theorem 1-2-5]. Theorem 1.1 (Kawamata-Viehweg vanishing). Let X be a normal projective variety over an algebraically closed ?eld k with char(k) = 0, B = biBi an e?ective Q-divisor on X, and D a Q-Cartier Weil divisor on X. Assume that (X, B) is Kawamata log terminal (KLT for short), and D ? (KX + B) is ample. Then H i(X, D) = 0 holds for any i 0. In what follows, we always work over an algebraically closed ?eld k of characteristic p 0 unless otherwise stated. The Kawamata-Viehweg vanishing theorem for smooth projective va- rieties in positive characteristic has ?rst been proved by Hara [Ha98] under the lifting condition over W2(k) of certain log pairs. Theorem 1.2 (Kawamata-Viehweg vanishing in char.