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Algebraic Reduction Theorem for complex codimension one singular foliations

2006; European Mathematical Society; Volume: 81; Issue: 1 Linguagem: Inglês

10.4171/cmh/47

1420-8946

Dominique Cerveau, Alcides Lins-Neto, Frank Loray, Jorge Vitório Pereira, Frédéric Touzet,

Geometry and complex manifolds

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.