Let be a compact complex manifold equipped with meromorphic vector fields that are linearly independent at a generic point. The main theorem is the following. If is not bimeromorphic to an algebraic manifold, then any codimension one complex foliation with a codimension singular set is the meromorphic pull-back of an algebraic foliation on a lower dimensional algebraic manifold, or is transversely projective outside a proper analytic subset. The two ingredients of the proof are the Algebraic Reduction Theorem for the complex manifold and an algebraic version of Lie's first theorem which is due to J. Tits.
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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. 157–169DOI 10.4171/CMH/47