Vector fields and foliations on compact surfaces of class <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>VII</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math>

Abstract

It is well-known that minimal compact complex surfaces with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> containing global spherical shells are in the class VII <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow/> <mml:mn>0</mml:mn> </mml:msub> </mml:math> 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.