Degeneration of K3 surfaces
type: Item
instance of: scholarly article
title: Degeneration of K3 surfaces
zbMATH Open document ID: 0707.14035
author: Kenji Nishiguchi
published in: Journal of Mathematics of Kyoto University
publication date: 1988-01-01T00:00:00Z
review text: 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 degen…
reviewed by: Sorin Popescu
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J30
zbMATH DE Number: 4160936
zbMATH Keywords: semistable degeneration of K3-surfaces
zbMATH Keywords: Hopf surface
zbMATH Keywords: (CB)-surface
zbMATH Keywords: surface of class VII
zbMATH Keywords: semi-stable modification
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520482
OpenAlex ID: W1522507129
DOI: 10.1215/KJM/1250520482
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description: scientific article; zbMATH DE number 4160936