Vector fields, separatrices and Kato surfaces

Adolfo Guillot[1]

  • [1] Instituto de Matemáticas, Unidad Cuernavaca Universidad Nacional Autónoma de México A.P. 273-3 Admon. 3 Cuernavaca, Morelos, 62251 Mexico

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 3, page 1331-1361
  • ISSN: 0373-0956

Abstract

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We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeros of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. We give new proofs of some results concerning the classification of compact complex surfaces admitting holomorphic vector fields. Our proofs rely in a combinatorial description of the vector field on a resolution of the singular point based on previous work of Rebelo and the author.

How to cite

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Guillot, Adolfo. "Vector fields, separatrices and Kato surfaces." Annales de l’institut Fourier 64.3 (2014): 1331-1361. <http://eudml.org/doc/275572>.

@article{Guillot2014,
abstract = {We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeros of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. We give new proofs of some results concerning the classification of compact complex surfaces admitting holomorphic vector fields. Our proofs rely in a combinatorial description of the vector field on a resolution of the singular point based on previous work of Rebelo and the author.},
affiliation = {Instituto de Matemáticas, Unidad Cuernavaca Universidad Nacional Autónoma de México A.P. 273-3 Admon. 3 Cuernavaca, Morelos, 62251 Mexico},
author = {Guillot, Adolfo},
journal = {Annales de l’institut Fourier},
keywords = {semicompleteness; separatrix; vector field; Kato surface; Stein surface; separatrix, holomorphic vector fields; Kato surfaces; Stein surfaces},
language = {eng},
number = {3},
pages = {1331-1361},
publisher = {Association des Annales de l’institut Fourier},
title = {Vector fields, separatrices and Kato surfaces},
url = {http://eudml.org/doc/275572},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Guillot, Adolfo
TI - Vector fields, separatrices and Kato surfaces
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 3
SP - 1331
EP - 1361
AB - We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeros of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. We give new proofs of some results concerning the classification of compact complex surfaces admitting holomorphic vector fields. Our proofs rely in a combinatorial description of the vector field on a resolution of the singular point based on previous work of Rebelo and the author.
LA - eng
KW - semicompleteness; separatrix; vector field; Kato surface; Stein surface; separatrix, holomorphic vector fields; Kato surfaces; Stein surfaces
UR - http://eudml.org/doc/275572
ER -

References

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