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Complex codimension one singular foliations and Godbillon-Vey sequences

Dominique Cerveau·Alcides Lins Neto·Frank Loray·Jorge Vitorio Pereira·Frederic Touzet
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Abstract

Let 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.

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Dominique Cerveau, Alcides Lins Neto, Frank Loray, Jorge Vitorio Pereira, Frederic Touzet. 2005-12-09. Complex codimension one singular foliations and Godbillon-Vey sequences. https://arxiv.org/abs/math/0406293

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