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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)].

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 14D15 Formal methods and deformations in algebraic geometry 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
##### Keywords:
lifting to Witt ring; degenerations; $$K3$$ surfaces