JournalscmhVol. 81, No. 1pp. 157–169

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
Algebraic Reduction Theorem for complex codimension one singular foliations cover
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Abstract

Let MM be a compact complex manifold equipped with n=dim(M)n=\dim(M) meromorphic vector fields that are linearly independent at a generic point. The main theorem is the following. If MM is not bimeromorphic to an algebraic manifold, then any codimension one complex foliation F\mathcal F with a codimension 2\ge2 singular set is the meromorphic pull-back of an algebraic foliation on a lower dimensional algebraic manifold, or F\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 MM 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