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

强可提升概型和正特征的 Kawamata-Viehweg 消灭定理 II 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。首先，本文证明了光滑 toric 代数簇是强可提升概形，从而光滑射影 toric 代数 簇上的 Kawamata-Viehweg 消灭定理成立。其次，本文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的光滑射影曲面，则 S 上的 Kawamata-Viehweg 消灭定理成立。 最后，本文得到了 W2(k) 上的循环覆盖技巧，利用它可构造出可提升到 W2(k) 上的一大类光 滑射影代数簇。 关键词：代数几何，正特征，强可提升概形，toric 代数簇，Kawamata-Viehweg 消灭定理，循 环覆盖 中图分类号： O187.2 Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic II 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). In this paper, ?rst the author proves that smooth toric varieties are strongly liftable, hence the Kawamata-Viehweg vanishing theorem holds for smooth projective toric varieties. Second, the author proves the Kawamata-Viehweg vanishing theorem for normal projective surfaces which are birational to a strongly liftable smooth projective surface. Finally, the author deduces the cyclic cover trick over W2(k), which can be used to construct a large class of liftable smooth projective varieties. Key words: Algebraic geometry, positive characteristic, strongly liftable scheme, toric variety, Kawamata-Viehweg vanishing theorem, cyclic cover 1  Introduction Throughout this paper, we always work over an algebraically closed ?eld k of character- istic p 0 unless otherwise stated. A smooth scheme X is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W2(k). This notion was ?rst introduced in [Xie10c] to study the Kawamata-Viehweg vanishing theorem in positive charac- teristic, furthermore, some examples and properties of strongly liftable schemes were also given in [Xie10c]. In this paper, we shall continue to study strongly liftable schemes. First of all, we ?nd an important class of strongly liftable schemes with simp