Vector fields and foliations on compact surfaces of class VII 0
Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1503-1545.

Il est bien connu qu’une surface compacte complexe minimale avec b 2 >0 contenant une coquille sphérique globale est dans la classe VII 0 de Kodaira. En fait, il n’y a pas d’autres exemples connus. Dans cet article nous démontrons que toute surface avec une coquille sphérique globale admet un feuilletage holomorphe singulier. L’existence d’un diviseur numériquement anticanonique est une condition nécessaire pour l’existence d’un champ de vecteurs holomorphe non trivial. Réciproquement, s’il existe un diviseur numériquement anticanonique, les surfaces admettant un champ de vecteurs se trouvent au-dessus d’un hypersurface dans la base de la déformation logarithmique verselle.

It is well-known that minimal compact complex surfaces with b 2 >0 containing global spherical shells are in the class VII 0 of Kodaira. In fact, there are no other known examples. In this paper we prove that all surfaces with global spherical shells admit a singular holomorphic foliation. The existence of a numerically anticanonical divisor is a necessary condition for the existence of a global holomorphic vector field. Conversely, given the existence of a numerically anticanonical divisor, surfaces with a global vector field lie over a hypersurface in the base of the versal logarithmic deformation.

@article{AIF_1999__49_5_1503_0,
     author = {Dloussky, Georges and Oeljeklaus, Karl},
     title = {Vector fields and foliations on compact surfaces of class ${\rm VII}_0$},
     journal = {Annales de l'Institut Fourier},
     pages = {1503--1545},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {5},
     year = {1999},
     doi = {10.5802/aif.1728},
     mrnumber = {2000k:32017},
     zbl = {0978.32021},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1728/}
}
TY  - JOUR
AU  - Dloussky, Georges
AU  - Oeljeklaus, Karl
TI  - Vector fields and foliations on compact surfaces of class ${\rm VII}_0$
JO  - Annales de l'Institut Fourier
PY  - 1999
SP  - 1503
EP  - 1545
VL  - 49
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1728/
DO  - 10.5802/aif.1728
LA  - en
ID  - AIF_1999__49_5_1503_0
ER  - 
%0 Journal Article
%A Dloussky, Georges
%A Oeljeklaus, Karl
%T Vector fields and foliations on compact surfaces of class ${\rm VII}_0$
%J Annales de l'Institut Fourier
%D 1999
%P 1503-1545
%V 49
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1728/
%R 10.5802/aif.1728
%G en
%F AIF_1999__49_5_1503_0
Dloussky, Georges; Oeljeklaus, Karl. Vector fields and foliations on compact surfaces of class ${\rm VII}_0$. Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1503-1545. doi : 10.5802/aif.1728. http://archive.numdam.org/articles/10.5802/aif.1728/

[1] P. Baum, R. Bott, On the zeroes of meromorphic vector fields, Essais en l'honneur de G. De Rham, (1970), 29-47. | MR | Zbl

[2] M. Brunella, Feuilletages holomorphes sur les surfaces complexes compactes, Annales Scient. Ec. Norm. Sup., 4e série, tome 30 (1997), 569-594. | Numdam | MR | Zbl

[3] J. Carrell, A. Howard, C. Kosniowski, Holomorphic vector fields on complex surfaces, Math. Ann., 204 (1973), 73-81. | MR | Zbl

[4] C. Camacho, P. Sad, Invariant varieties through singularities of holomorphic vector fields, Annals of Math., 115 (1982), 579-595. | MR | Zbl

[5] G. Dloussky, Structure des surfaces de Kato, Mémoire de la S.M.F 112, n° 14 (1984). | Numdam | MR | Zbl

[6] G. Dloussky, Sur la classification des germes d'applications holomorphes contractantes, Math. Ann., 280 (1988), 649-661. | MR | Zbl

[7] G. Dloussky, Une construction élémentaire des surfaces d'Inoue-Hirzebruch, Math. Ann., 280 (1988), 663-682. | MR | Zbl

[8] G. Dloussky, F. Kohler, Classification of singular germs of mappings and deformations of compact surfaces of the VII0 class, Ann. Math. Pol., LXX (1998), 49-83. | MR | Zbl

[9] G. Dloussky, K. Oeljeklaus, Surfaces de la classe VII0 et automorphismes de Hénon, C.R.A.S., 328, Série I (1999), 609-612. | MR | Zbl

[10] I. Enoki, Surfaces of class VII0 with curves, Tôhoku Math. J., 33 (1981), 453-492. | MR | Zbl

[11] C. Gellhaus, P. Heinzner, Komplexe Flächen mit holomorphen Vektorfeldern, Abh. Math. Sem. Hamburg, 60 (1990), 37-46. | MR | Zbl

[12] X. Gomez-Mont, Singularités d'équations différentielles, Astérisque 150-151, SMF (1987). | Numdam | Zbl

[13] P. Griffith, J. Harris, Principles of Algebraic Geometry, Wiley & Sons, 1978. | Zbl

[14] H. Hironaka, Introduction to the theory of infinitesimaly near singular points, Memorias de Mathematica del Instituto “Jorge Juan” 28, Madrid 1974. | MR | Zbl

[15] J. Hausen, Zur Klassifikation glatter kompakter ℂ*-Flaechen, Math. Ann., 301 (1995), 763-769. | EuDML | MR | Zbl

[16] J.H. Hubbard, R.W. Oberste-Vorth, Hénon mappings in the complex domain I, Pub. Math. IHES, 79 (1994), 5-46. | EuDML | Numdam | MR | Zbl

[17] M. Inoue, New surfaces with no meromorphic functions II, Complex Analysis and Alg. Geom., 91-106, Iwanami Shoten Pb. (1977). | MR | Zbl

[18] Ma. Kato, Compact complex manifolds containing “global spherical shells”, Proceedings of the Int. Symp. Alg. Geometry, Kyoto 1977. Kinokuniya Book Store, Tokyo, 1978. | Zbl

[19] F. Kohler, Feuilletages holomorphes singuliers sur les surfaces contenant une coquille sphérique globale, Ann. Inst. Fourier, 45-1 (1995), 161-182. Erratum Ann. Inst. Fourier, 46-2 (1996). | EuDML | Numdam | MR | Zbl

[20] K. Kodaira, On the structure of compact complex analytic surfaces I, Am. Jour. Math., 86 (1964), 651-698. | MR | Zbl

[21] K. Kodaira, On the structure of compact complex analytic surfaces II, Am. Jour. Math., 88 (1966), 682-721. | MR | Zbl

[22] B. Khanedani, T. Suwa, First variation of holomorphic forms and some applications, Hokkaido Math. Jour., Vol. XXVI, No. 2 (1997), 323-335. | MR | Zbl

[23] Y. Kawamata, On deformations of compactifiable complex manifolds, Math. Ann., 235 (1978), 247-265. | EuDML | MR | Zbl

[24] I. Nakamura, On surfaces of class VII0 with curves, Invent. Math., 78 (1984), 393-443. | EuDML | MR | Zbl

[25] I. Nakamura, On surfaces of class VII0 with curves II, Tohoku Math. J., 42 (1990), 475-516. | MR | Zbl

[26] E.H. Spanier, Algebraic Topology, Mc Graw-Hill, 1966. | MR | Zbl

[27] T. Suwa, Indices of holomorphic vector fields relative to invariant curves, Proc. AMS., 123, No.10 (1995), 2989-2997. | MR | Zbl

Cité par Sources :