Algebraic Reduction Theorem for complex codimension one singular foliations

Abstract

Let $M$ be a compact complex manifold equipped with $n=\dim (M)$ meromorphic vector fields that are linearly independent at a generic point. The main theorem is the following. If $M$ is not bimeromorphic to an algebraic manifold, then any codimension one complex foliation $\mathcal{F}$ with a codimension $\ge 2$ singular set is the meromorphic pull-back of an algebraic foliation on a lower dimensional algebraic manifold, or $\mathcal{F}$ is transversely projective outside a proper analytic subset. The two ingredients of the proof are the Algebraic Reduction Theorem for the complex manifold $M$ and an algebraic version of Lie's first theorem which is due to J. Tits.