| Let M be a compact complex manifold equipped with n=dim(M) meromorphic vector fields that are independant 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 F with a codimension 2 singular set satisfies the following alternative: either F is the meromorphic pull-back of an algebraic foliation on a lower dimensional algebraic manifold, or F is transversely projective outside a compact hypersurface. The ingredients are essentially the Algebraic Reduction Theorem for M, Lie's classification of geometries on the line and algebraic manipulations with the (meromorphic) Godbillon-Vey sequences associated to the foliation. We also derive from our study (even in the case M algebraic) several sufficient conditions on the Godbillon-Vey 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 our results with many examples. |
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