Vector fields and foliations on compact surfaces of class VII 0

Georges Dloussky; Karl Oeljeklaus

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 5, page 1503-1545
  • ISSN: 0373-0956

Abstract

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It is well-known that minimal compact complex surfaces with b 2 > 0 containing global spherical shells are in the class VII 0 of Kodaira. In fact, there are no other known examples. In this paper we prove that all surfaces with global spherical shells admit a singular holomorphic foliation. The existence of a numerically anticanonical divisor is a necessary condition for the existence of a global holomorphic vector field. Conversely, given the existence of a numerically anticanonical divisor, surfaces with a global vector field lie over a hypersurface in the base of the versal logarithmic deformation.

How to cite

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Dloussky, Georges, and Oeljeklaus, Karl. "Vector fields and foliations on compact surfaces of class ${\rm VII}_0$." Annales de l'institut Fourier 49.5 (1999): 1503-1545. <http://eudml.org/doc/75392>.

@article{Dloussky1999,
abstract = {It is well-known that minimal compact complex surfaces with $b_\{2\}&gt;0$ containing global spherical shells are in the class VII$\{\}_\{0\}$ of Kodaira. In fact, there are no other known examples. In this paper we prove that all surfaces with global spherical shells admit a singular holomorphic foliation. The existence of a numerically anticanonical divisor is a necessary condition for the existence of a global holomorphic vector field. Conversely, given the existence of a numerically anticanonical divisor, surfaces with a global vector field lie over a hypersurface in the base of the versal logarithmic deformation.},
author = {Dloussky, Georges, Oeljeklaus, Karl},
journal = {Annales de l'institut Fourier},
keywords = {compact complex surface; class VII; holomorphic vector field; singular holomorphic foliation},
language = {eng},
number = {5},
pages = {1503-1545},
publisher = {Association des Annales de l'Institut Fourier},
title = {Vector fields and foliations on compact surfaces of class $\{\rm VII\}_0$},
url = {http://eudml.org/doc/75392},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Dloussky, Georges
AU - Oeljeklaus, Karl
TI - Vector fields and foliations on compact surfaces of class ${\rm VII}_0$
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 5
SP - 1503
EP - 1545
AB - It is well-known that minimal compact complex surfaces with $b_{2}&gt;0$ containing global spherical shells are in the class VII${}_{0}$ of Kodaira. In fact, there are no other known examples. In this paper we prove that all surfaces with global spherical shells admit a singular holomorphic foliation. The existence of a numerically anticanonical divisor is a necessary condition for the existence of a global holomorphic vector field. Conversely, given the existence of a numerically anticanonical divisor, surfaces with a global vector field lie over a hypersurface in the base of the versal logarithmic deformation.
LA - eng
KW - compact complex surface; class VII; holomorphic vector field; singular holomorphic foliation
UR - http://eudml.org/doc/75392
ER -

References

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