Vector fields and foliations on compact surfaces of class ${\mathrm{VII}}_0$

Dloussky, Georges; Oeljeklaus, Karl

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.

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