Degeneration of K3 surfaces
Let \(\pi\) : \(X\to \Delta\) be a semistable degeneration of K3-surfaces (i.e. a proper surjective holomorphic map with connected fibers of a 3- dimensional complex manifold X to a disk \(\Delta =\{t\in {\mathbb{C}}| \quad | t| <\epsilon \}\) such that \(\pi\) is smooth at each point of \(\pi^{-1}(\Delta \setminus \{0\})\), the general fiber \(X_ t=\pi^{-1}(t)\), \(t\neq 0\), is a K3-surface and the singular fiber \(X_ 0\) is a reduced divisor with simple normal crossings). \textit{V. S. Kulikov} [Math. USSR, Izv. 11, 957-989 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 1008-1042 (1977; Zbl 0367.14014)] and \textit{U. Persson} and \textit{H. Pinkham} [Ann. Math., II. Ser. 113, 45-66 (1981; Zbl 0426.14015)] studied such degenerations under the assumption that either \(\pi\) is projective or every component of \(X_ 0\) is algebraic, showing that there exists a semi-stable modification \(\pi ': X'\to \Delta\) of \(\pi\) : \(X\to \Delta\) such that the canonical bundle of \(X'\) is trivial. In {\S} 1 of this paper the author generalizes the previous result under no assumption of Kählerity proving that for a semistable degeneration \(\pi\) : \(X\to \Delta\) of K3-surfaces either (i) there exists a semistable modification with trivial canonical bundle as above; or (ii) there exists in \(X_ 0\) a component which is a Hopf surface or a blowing-up of a Hopf surface; or (iii) there exists in \(X_ 0\) a component which is a (CB)-surface or a blowing up of a (CB)-surface (a (CB)-surface being a surface of class VII with no exceptional curves of the first kind, having an effective reduced divisor consisting of a rational cycle and some trees of nonsingular rational curves sprouting from the cycle). In {\S} 2 the author gives a classification of semistable degenerations of K3-surfaces with trivial canonical bundle and in {\S} 3 are discussed which surfaces can be a component of the singular fiber \(X_ 0\) in this case. In {\S} 4 the author gives examples of degenerations of K3-surfaces for which no semi-stable modification has trivial canonical bundle and which belong to the \(case\quad (ii)\quad or\quad (iii)\) of the mentioned result. In the last part of the paper, there are constructed examples of semi- stable degenerations which contain some series of (CB)-surfaces in their singular fibers and also it is given a sufficient condition that such a surface becomes a component in the singular fibre of a semi-stable degeneration.
scientific article; zbMATH DE number 4160936