# Algebraic Reduction Theorem for complex codimension one singular foliations

### Frank Loray

Université Lille I, Villeneuve D'ascq, France### Dominique Cerveau

Université de Rennes I, France### Alcides Lins-Neto

Instituto de Matemática Pura e Aplicada, Rio De Janeiro, Brazil### Jorge Vitório Pereira

IMPA, Rio De Janeiro, Brazil### Frédéric Touzet

Université de Rennes I, France

## 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 $\ge2$ 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.

## Cite this article

Frank Loray, Dominique Cerveau, Alcides Lins-Neto, Jorge Vitório Pereira, Frédéric Touzet, Algebraic Reduction Theorem for complex codimension one singular foliations. Comment. Math. Helv. 81 (2006), no. 1, pp. 157–169

DOI 10.4171/CMH/47