Semicompleteness of homogeneous quadratic vector fields
Annales de l'Institut Fourier, Volume 56 (2006) no. 5, p. 1583-1615

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 CP 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.

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.

DOI : https://doi.org/10.5802/aif.2221
Classification:  32S65,  34M05,  34M15,  34M35,  37F75
Keywords: Complex differential equation, semicomplete vector field, holomorphic foliation
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     author = {Guillot, Adolfo},
     title = {Semicompleteness of homogeneous quadratic vector fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {5},
     year = {2006},
     pages = {1583-1615},
     doi = {10.5802/aif.2221},
     mrnumber = {2273865},
     zbl = {1110.37040},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_5_1583_0}
}
Guillot, Adolfo. Semicompleteness of homogeneous quadratic vector fields. Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1583-1615. doi : 10.5802/aif.2221. http://www.numdam.org/item/AIF_2006__56_5_1583_0/

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