zbMATH — the first resource for mathematics

Liftings of simple normal crossing log \(K3\) and log Enriques surfaces in mixed characteristics. (English) Zbl 0972.14029
According to Kulikov, the degenerations of the minimal semi-stable families of \(K3\) surfaces over a disk are classified into three types: Type I (= smooth type), Type II and Type III. In the smooth type case, Deligne and Ogus have shown that a smooth \(K3\) surface over an algebraically closed field of positive characteristic has a lifting over a complete discrete valuation ring of mixed characteristics. In this paper, the author shows the existence of liftings of the projective simple normal crossing \(\log K3\) surfaces of type II and type III over an algebraically closed field of positive characteristic to the Witt ring of the field. [See also D.-Q. Zhang, J. Math. Kyoto Univ. 31, No. 2 419-466 (1991; Zbl 0759.14029) for the definition of log Enriques surfaces).] The result of the current paper is a log version of that of Deligne and Ogus and a mixed characteristics algebraic version of that of R. Friedman [Ann. Math., II. Ser. 118, 75-114 (1983; Zbl 0569.14002)] and V. Kawamata and Y. Namikawa [Invent. Math. 118, 395-409 (1994; Zbl 0848.14004)].

14J28 \(K3\) surfaces and Enriques surfaces
14D15 Formal methods and deformations in algebraic geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)