Semicompleteness of homogeneous quadratic vector fields
Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 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: 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
Guillot, Adolfo 1

1 Unidad Cuernavaca Instituto de Matemáticas UNAM Av. Universidad s/n, col. Lomas de Chamilpa C.P. 62210, Cuernavaca, Morelos (Mexico)
@article{AIF_2006__56_5_1583_0,
     author = {Guillot, Adolfo},
     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 = {http://archive.numdam.org/articles/10.5802/aif.2221/}
}
TY  - JOUR
AU  - Guillot, Adolfo
TI  - Semicompleteness of homogeneous quadratic vector fields
JO  - Annales de l'Institut Fourier
PY  - 2006
SP  - 1583
EP  - 1615
VL  - 56
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2221/
DO  - 10.5802/aif.2221
LA  - en
ID  - AIF_2006__56_5_1583_0
ER  - 
%0 Journal Article
%A Guillot, Adolfo
%T Semicompleteness of homogeneous quadratic vector fields
%J Annales de l'Institut Fourier
%D 2006
%P 1583-1615
%V 56
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2221/
%R 10.5802/aif.2221
%G en
%F 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://archive.numdam.org/articles/10.5802/aif.2221/

[1] Baum, Paul; Bott, Raoul Singularities of holomorphic foliations, J. Differential Geometry, Volume 7 (1972), pp. 279-342 | MR | Zbl

[2] Cairns, Grant; Ghys, Étienne The local linearization problem for smooth SL(n)-actions, Enseign. Math. (2), Volume 43 (1997) no. 1-2, pp. 133-171 | MR | Zbl

[3] Camacho, César; Sad, Paulo Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2), Volume 115 (1982) no. 3, pp. 579-595 | DOI | MR | Zbl

[4] Cerveau, D.; Lins Neto, A. Irreducible components of the space of holomorphic foliations of degree two in CP(n), n3, Ann. of Math. (2), Volume 143 (1996) no. 3, pp. 577-612 | DOI | MR | Zbl

[5] Erdös, P.; Graham, R. L. Old and new problems and results in combinatorial number theory, L’Enseignement Mathématique, Université de Genève, Geneva, 1980, pp. 128 | MR | Zbl

[6] Fulton, William; Harris, Joe Representation theory, Springer-Verlag, New York, 1991 (A first course, Readings in Mathematics) | MR | Zbl

[7] Ghys, É.; Rebelo, J.-C. Singularités des flots holomorphes II, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 4, pp. 1117-1174 | DOI | EuDML | Numdam | MR | Zbl

[8] Guillot, Adolfo Champs quadratiques uniformisables, ÉNS-Lyon (2001) (Ph. D. Thesis)

[9] Guillot, Adolfo Sur les exemples de Lins Neto de feuilletages algébriques, C. R. Math. Acad. Sci. Paris, Volume 334 (2002) no. 9, pp. 747-750 | MR | Zbl

[10] Guillot, Adolfo Un théorème de point fixe pour les endomorphismes de l’espace projectif avec des applications aux feuilletages algébriques, Bull. Braz. Math. Soc. (N.S.), Volume 35 (2004) no. 3, pp. 345-362 | DOI | MR | Zbl

[11] Guillot, Adolfo The Painlevé property for quasihomogenous systems and a many-body problem in the plane, Comm. Math. Phys., Volume 256 (2005) no. 1, pp. 181-194 | DOI | MR | Zbl

[12] Guillot, Adolfo Sur les équations d’Halphen et les actions de SL 2 (C) (2006) (in preparation)

[13] Gunning, R. C. Lectures on Riemann surfaces, Princeton University Press, Princeton, N.J., 1966 (Princeton Mathematical Notes) | MR | Zbl

[14] Halphen, G.-H. Sur certains systèmes d’équations différentielles, Comptes Rendus Hebdomadaires de l’Académie des Sciences, Volume XCII (1881) no. 24, pp. 1404-1406

[15] Hille, Einar Ordinary differential equations in the complex domain, Dover Publications Inc., Mineola, NY, 1997 (Reprint of the 1976 original) | MR | Zbl

[16] Landau, L.; Lifchitz, E. Physique théorique. Tome I. Mécanique, Éditions Mir, Moscou, 1982 (Quatrième édition revue et complétée)

[17] Lins Neto, Alcides Some examples for the Poincaré and Painlevé problems, Ann. Sci. École Norm. Sup. (4), Volume 35 (2002) no. 2, pp. 231-266 | Numdam | MR | Zbl

[18] Palais, Richard S. A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. No., Volume 22 (1957), pp. iii+123 | MR | Zbl

[19] Rebelo, Julio C. Singularités des flots holomorphes, Ann. Inst. Fourier (Grenoble), Volume 46 (1996) no. 2, pp. 411-428 | DOI | Numdam | MR | Zbl

[20] Thurston, William P. Three-dimensional geometry and topology. Vol. 1, Princeton University Press, Princeton, NJ, 1997 (Edited by Silvio Levy) | MR | Zbl

Cited by Sources: