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Holomorphic Foliations Tangent to Levi-Flat Subsets

  • Jane Bretas
  • Arturo Fernández-Pérez
  • Rogério Mol
Article

Abstract

An irreducible real analytic subvariety H of real dimension \(2n +1\) in a complex manifold M is a Levi-flat subset if its regular part carries a complex foliation of dimension n. Locally, a germ of real analytic Levi-flat subset is contained in a germ of irreducible complex variety \(H^{\imath }\) of dimension \(n+1\), called intrinsic complexification, which can be globalized to a neighborhood of H in M provided H is a coherent analytic subvariety. In this case, a singular holomorphic foliation \(\mathcal {F}\) of dimension n in M that is tangent to H is also tangent to \(H^{\imath }\). In this paper, we prove integration results of local and global nature for the restriction to \(H^{\imath }\) of a singular holomorphic foliation \(\mathcal {F}\) tangent to a real analytic Levi-flat subset H. From a local viewpoint, if \(n=1\) and \(H^{\imath }\) has an isolated singularity, then \(\mathcal {F}|_{H^{\imath }}\) has a meromorphic first integral. From a global perspective, when \(M = \mathbb {P}^N\) and H is coherent and of low codimension, \(H^{\imath }\) extends to an algebraic variety. In this case, \(\mathcal {F}|_{H^{\imath }}\) has a rational first integral provided infinitely many leaves of \(\mathcal {F}\) in H are algebraic.

Keywords

Holomorphic foliations CR-manifolds Levi-flat varieties 

Mathematics Subject Classification

Primary 32S65 Secondary 32V40 

Notes

Funding

Funding was provided by CNPq—Conselho Nacional de Desenvolvimento Científico e Tecnológico.

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Copyright information

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Authors and Affiliations

  1. 1.Departamento de MatemáticaCentro Federal de Educação Tecnológica de Minas GeraisBelo HorizonteBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

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