Semicompleteness of homogeneous quadratic vector fields

Adolfo Guillot[1]

  • [1] Unidad Cuernavaca Instituto de Matemáticas UNAM Av. Universidad s/n, col. Lomas de Chamilpa C.P. 62210, Cuernavaca, Morelos (Mexico)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 5, page 1583-1615
  • ISSN: 0373-0956

Abstract

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We investigate the quadratic homogeneous holomorphic vector fields on  C n that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove that the foliations of  C P n - 1 induced by semicomplete quadratic vector fields are linearizable in a neighborhood of their singular points and give some new families of examples in  C 3 . Finally, we classify the semicomplete isochoric vector fields in  C 3 having an isolated singularity at the origin.

How to cite

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Guillot, Adolfo. "Semicompleteness of homogeneous quadratic vector fields." Annales de l’institut Fourier 56.5 (2006): 1583-1615. <http://eudml.org/doc/10185>.

@article{Guillot2006,
abstract = {We investigate the quadratic homogeneous holomorphic vector fields on $\mathbf\{C\}^n$ that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove that the foliations of $\mathbf\{C\}\mathbf\{P\}^\{n-1\}$ induced by semicomplete quadratic vector fields are linearizable in a neighborhood of their singular points and give some new families of examples in $\mathbf\{C\}^3$. Finally, we classify the semicomplete isochoric vector fields in $\mathbf\{C\}^3$ having an isolated singularity at the origin.},
affiliation = {Unidad Cuernavaca Instituto de Matemáticas UNAM Av. Universidad s/n, col. Lomas de Chamilpa C.P. 62210, Cuernavaca, Morelos (Mexico)},
author = {Guillot, Adolfo},
journal = {Annales de l’institut Fourier},
keywords = {Complex differential equation; semicomplete vector field; holomorphic foliation; complex differential equation; classification; equivalence classes; singularity},
language = {eng},
number = {5},
pages = {1583-1615},
publisher = {Association des Annales de l’institut Fourier},
title = {Semicompleteness of homogeneous quadratic vector fields},
url = {http://eudml.org/doc/10185},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Guillot, Adolfo
TI - Semicompleteness of homogeneous quadratic vector fields
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 5
SP - 1583
EP - 1615
AB - We investigate the quadratic homogeneous holomorphic vector fields on $\mathbf{C}^n$ that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove that the foliations of $\mathbf{C}\mathbf{P}^{n-1}$ induced by semicomplete quadratic vector fields are linearizable in a neighborhood of their singular points and give some new families of examples in $\mathbf{C}^3$. Finally, we classify the semicomplete isochoric vector fields in $\mathbf{C}^3$ having an isolated singularity at the origin.
LA - eng
KW - Complex differential equation; semicomplete vector field; holomorphic foliation; complex differential equation; classification; equivalence classes; singularity
UR - http://eudml.org/doc/10185
ER -

References

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