Complex codimension one singular foliations and Godbillon-Vey sequences

Abstract

A part of version 1 will appear in Comment. Math. Helv. 81 (2006), namely the main theorem which actually did not need use of Godbillon-Vey sequences ; this was observed by E. Ghys. In this new version, we weaken assumptions for the main theorem and really use Godbillon-Vey sequences to prove it. Auxiliary results are left unchanged.International audienceLet F be a codimension one singular holomorphic foliation on a compact complex manifold M. Assume that there exists a meromorphic vector field X on M generically transversal to F. Then, we prove that F is the meromorphic pull-back of an algebraic foliation on an algebraic manifold N, or F is transversely projective outside a compact hypersurface, improving our previous work (see version 1). Such a vector field insures the existence of a global meromorphic Godbillon-Vey sequence for the foliation F. We derive sufficient conditions on this sequence insuring such alternative. For instance, if there exists a finite Godbillon-Vey sequence or if the Godbillon-Vey invariant is zero, then either F is the pull-back of a foliation on a surface, or F is transversely projective. We illustrate these results with many examples