Semicompleteness of homogeneous quadratic vector fields

Guillot, Adolfo

doi:10.5802/aif.2221

Semicompleteness of homogeneous quadratic vector fields
[Semicomplétude des champs quadratiques homogènes]

Guillot, Adolfo 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, Tome 56 (2006) no. 5, pp. 1583-1615.

Résumé
Abstract

On étudie les champs de vecteurs holomorphes quadratiques et homogènes de $C^{n}$ qui sont semicomplets : ceux dont les solutions sont uniformes dans leurs domaines maximaux de définition. À un champ générique on associe de façon rationnelle quelques nombres complexes qui s’avèrent entiers dans le cas semicomplet. Ceci montre que, dans l’espace des classes d’équivalence linéaire de champs de vecteurs, les semicomplets sont contenus dans une sorte de réseau. On prouve que les feuilletages de ${CP}^{n - 1}$ induits par des champs quadratiques semicomplets sont linéarisables au voisinage de leurs points singuliers et on donne quelques familles nouvelles d’exemples dans $C^{3}$ . Finalement, on classifie les champs semicomplets de $C^{3}$ qui sont isochores et à singularité isolée.

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.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.2221

Classification : 32S65, 34M05, 34M15, 34M35, 37F75
Keywords: Complex differential equation, semicomplete vector field, holomorphic foliation
Mot clés : équation différentielle complexe, champ semicomplet, feuilletage holomorphe

Affiliations des auteurs :

Guillot, Adolfo ¹

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

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     title = {Semicompleteness of homogeneous quadratic vector fields},
     journal = {Annales de l'Institut Fourier},
     pages = {1583--1615},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {5},
     year = {2006},
     doi = {10.5802/aif.2221},
     zbl = {1110.37040},
     mrnumber = {2273865},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2221/}
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Guillot, Adolfo. Semicompleteness of homogeneous quadratic vector fields. Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1583-1615. doi : 10.5802/aif.2221. https://www.numdam.org/articles/10.5802/aif.2221/

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