Smooth deformations of singular contractions of class VII surfaces
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Mathematische Zeitschrift
Smooth deformations of singular contractions of class VII surfaces Georges Dloussky1 · Andrei Teleman1 Received: 10 April 2018 / Accepted: 15 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider normal compact surfaces Y obtained from a minimal class VII surface X by contraction of a cycle C of r rational curves with C 2 < 0. Our main result states that, if the obtained cusp is smoothable, then Y is globally smoothable. The proof is based on a vanishing theorem for H 2 (Y ). If r < b2 (X ) any smooth small deformation of Y is rational, and if r = b2 (X ) (i.e. when X is a half-Inoue surface) any smooth small deformation of Y is an Enriques surface. The condition “the cusp is smoothable” in our main theorem can be checked in terms of the intersection numbers of the cycle, using the Looijenga conjecture (which has recently become a theorem). Therefore this is a “decidable” condition. We prove that this condition is always satisfied if r < b2 (X ) 11. Therefore the singular surface Y obtained by contracting a cycle C of r rational curves in a minimal class VII surface X with r < b2 (X ) 11 is always smoothable by rational surfaces. The statement holds even for unknown class VII surfaces.
1 Introduction 1.1 The results Class VII surfaces are not classified yet. The global spherical shell (GSS) conjecture, which, if true, would complete the classification of this class, can be stated as follows: The GSS conjecture Any minimal class VII surface with b2 > 0 is a Kato surface. Recall that a Kato surface is a minimal class VII surface [1] with positive b2 which admits a GSS. Kato surfaces are considered to be the known surfaces in the class VII; they can be
The authors thank M. Manetti for a fruitful exchange of mails about deformation theory. We also thank very much the unknown referee for the careful reading of the article, and for many valuable suggestions and remarks.
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Andrei Teleman [email protected] Georges Dloussky [email protected]
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Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France
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G. Dloussky, A. Teleman
obtained by a simple two-step construction: iterated blown up of the standard ball B ⊂ C2 over the origin, followed by a glueing procedure (see [4,7] for details). This important conjecture has been stated by Nakamura in [29] and has been proved for surfaces with b2 = 1 in [34]. Any Kato surface X has b2 (X ) rational curves, and some of these curves form a cycle of rational curves (see Sect. 1.2). Therefore any Kato surface contains a cycle of r rational curves with 0 < r b2 (X ). The standard approach for proving the GSS conjecture has two steps corresponding to the following two conjectures considered by experts to be more accessible: Conjecture 1 Any minimal class VII surface with b2 > 0 has a cycle of rational curves. Conjecture 2 Any minimal class VII surface with b2 > 0 containing a cycle of rational curves is a Kato surface. Conjecture 1 is itself very imp
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