Vector fields and foliations on compact surfaces of class \(\text{VII}_0\)
type: Item
instance of: scholarly article
title: Vector fields and foliations on compact surfaces of class \(\text{VII}_0\)
zbMATH Open document ID: 0978.32021
author: Georges Dloussky
author: Karl Oeljeklaus
publication date: 1999-11-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75392
review text: Let GSS be the class of compact complex surfaces \(S\) containing global spherical shells. If \(S\) is GSS then it contains exactly \(n=b_2(S)\) rational curves \(D_0,\ldots,D_{n-1}\) each of them being regular or with a double point. Let \(\sigma(S):=-\sum_{i=0}^{n-1}D_i^2+2\text{Card}\{\text{double points}\}\). Then \(2n\leq\sigma(S)\leq 3n\). The main result proved by the authors is the following: let \(S\) be GSS and minimal; then there is always a global singular holomorphic foliation on \(S\). Furthermore if \(b_2(S)\geq 1\) then \(S\) admits at most two foliations and there are two foliations iff \(S\) is an Inoue-Hirzebruch surface. If \(2n<\sigma(S)<3n\) and there exists a numerically anticanonical divisor, there exists a logarithmic deformation of \(S\) into a surface admitting a global non-trivial vector field.
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 1355990
zbMATH Keywords: compact complex surface
zbMATH Keywords: class VII\(_0\)
zbMATH Keywords: holomorphic vector field
zbMATH Keywords: singular holomorphic foliation
reviewed by: Giuseppe Tomassini
MaRDI profile type: Publication
OpenAlex ID: W2325600645
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DOI: 10.5802/AIF.1728
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1355990