Vector fields, separatrices and Kato surfaces
Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1331-1361.

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.

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.

DOI: 10.5802/aif.2882
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.
Guillot, Adolfo 1

1 Instituto de Matemáticas, Unidad Cuernavaca Universidad Nacional Autónoma de México A.P. 273-3 Admon. 3 Cuernavaca, Morelos, 62251 Mexico
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Guillot, Adolfo. Vector fields, separatrices and Kato surfaces. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1331-1361. doi : 10.5802/aif.2882. http://archive.numdam.org/articles/10.5802/aif.2882/

[1] Barth, W.; Peters, C.; Van de Ven, A. Compact complex surfaces, 4, Springer-Verlag, Berlin, 1984, pp. x+304 | MR | Zbl

[2] Bondil, Romain; Lê, Dũng Tráng Résolution des singularités de surfaces par éclatements normalisés (multiplicité, multiplicité polaire, et singularités minimales), Trends in singularities (Trends Math.), Birkhäuser, Basel, 2002, pp. 31-81

[3] Briot; Bouquet Recherches sur les propriétés des fonctions définies par des équations différentielles, Comptes rendus hebdomadaires des scéances de l’Académie des Sciences, Volume 39 (1854), pp. 368-371

[4] Brunella, Marco Birational geometry of foliations, Publicações Matemáticas do IMPA, IMPA, Rio de Janeiro, 2004, pp. iv+138 | MR | Zbl

[5] Brunella, Marco Nonuniformisable foliations on compact complex surfaces, Mosc. Math. J., Volume 9 (2009) no. 4, p. 729-748, 934 | MR | Zbl

[6] Camacho, C.; Movasati, H.; Scárdua, B. The moduli of quasi-homogeneous Stein surface singularities, J. Geom. Anal., Volume 19 (2009) no. 2, pp. 244-260 | DOI | MR | Zbl

[7] Camacho, César Quadratic forms and holomorphic foliations on singular surfaces, Math. Ann., Volume 282 (1988) no. 2, pp. 177-184 | DOI | MR | Zbl

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

[9] Camacho, César; Sad, Paulo Pontos singulares de equações diferenciais analí ticas, 16 o Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1987, pp. iv+132

[10] Dloussky, G.; Oeljeklaus, K. Vector fields and foliations on compact surfaces of class VII 0 , Ann. Inst. Fourier (Grenoble), Volume 49 (1999) no. 5, pp. 1503-1545 | Numdam | MR | Zbl

[11] Dloussky, Georges Structure des surfaces de Kato, Mém. Soc. Math. France (1984) no. 14, pp. ii+120 | Numdam | MR | Zbl

[12] Dloussky, Georges; Oeljeklaus, Karl; Toma, Matei Surfaces de la classe VII 0 admettant un champ de vecteurs, Comment. Math. Helv., Volume 75 (2000) no. 2, pp. 255-270 | DOI | MR | Zbl

[13] Dloussky, Georges; Oeljeklaus, Karl; Toma, Matei Surfaces de la classe VII 0 admettant un champ de vecteurs. II, Comment. Math. Helv., Volume 76 (2001) no. 4, pp. 640-664 | DOI | MR | Zbl

[14] Favre, Charles Classification of 2-dimensional contracting rigid germs and Kato surfaces. I, J. Math. Pures Appl., Volume 79 (2000) no. 5, pp. 475-514 | DOI | MR | Zbl

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

[16] Ghys, Étienne À propos d’un théorème de J.-P. Jouanolou concernant les feuilles fermées des feuilletages holomorphes, Rend. Circ. Mat. Palermo (2), Volume 49 (2000) no. 1, pp. 175-180 | MR | Zbl

[17] Guillot, Adolfo; Rebelo, Julio Semicomplete meromorphic vector fields on complex surfaces, J. Reine Angew. Math., Volume 667 (2012), pp. 27-65 | DOI | MR | Zbl

[18] Kato, Masahide Compact complex manifolds containing “global” spherical shells, Proc. Japan Acad., Volume 53 (1977) no. 1, pp. 15-16 | MR | Zbl

[19] Orlik, Peter; Wagreich, Philip Isolated singularities of algebraic surfaces with C * action, Ann. of Math., Volume 93 (1971), pp. 205-228 | MR | Zbl

[20] 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

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

[22] Rebelo, Julio C. Champs complets avec singularités non isolées sur les surfaces complexes, Bol. Soc. Mat. Mexicana (3), Volume 5 (1999) no. 2, pp. 359-395 | MR | Zbl

[23] Rebelo, Julio C. Réalisation de germes de feuilletages holomorphes par des champs semi-complets en dimension 2, Ann. Fac. Sci. Toulouse Math., Volume 9 (2000) no. 4, pp. 735-763 | Numdam | MR | Zbl

[24] Rossi, Hugo Vector Fields on Analytic Spaces, Ann. of Math. (2), Volume 78 (1963) no. 3, pp. 455-467 | MR | Zbl

[25] Sánchez-Bringas, Federico Normal forms of invariant vector fields under a finite group action, Publ. Mat., Volume 37 (1993) no. 1, pp. 75-82 | MR | Zbl

[26] Seidenberg, A. Derivations and integral closure, Pacific J. Math., Volume 16 (1966), pp. 167-173 | MR | Zbl

[27] Zariski, Oscar The reduction of the singularities of an algebraic surface, Ann. of Math. (2), Volume 40 (1939), pp. 639-689 | MR | Zbl

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