Vector fields, separatrices and Kato surfaces

Guillot, Adolfo

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1331-1361.

On prouve qu’un espace analytique complexe de dimension deux admettant un champ de vecteurs complet qui n’a pas de séparatrice passant par un point singulier de la surface s’obtient à partir d’une surface de Kato en effondrant un diviseur (en particulier, l’espace est compact). On prouve que, dans un espace analytique de Stein de dimension deux muni d’un champ de vecteurs complet, un point singulier de l’espace qui est un point d’équilibre isolé du champ est soit une singularité quasi-homogène, soit une singularité de Klein. On redémontre quelques résultats concernant la classification des surfaces complexes compactes admettant des champs de vecteurs holomorphes. Les preuves reposent sur des travaux récents de Rebelo et de l’auteur donnant une description combinatoire des champs de vecteurs complets.

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.

Classification : 32S65, 32C20, 34M45
Keywords: semicompleteness, separatrix, vector field, Kato surface, Stein surface.
Mot clés : semicomplétude, séparatrice, champ de vecteurs, surface de Kato, surface de Stein.

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