Holomorphic Foliations Tangent to Levi-Flat Subsets

• Published in: The Journal of geometric analysis 2019-04, Vol.29 (2), p.1407-1427
• Main Authors: Bretas, Jane, Fernández-Pérez, Arturo, Mol, Rogério
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Algebra; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Integrals; Mathematical analysis; Mathematics; Mathematics and Statistics
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-018-0043-1

An irreducible real analytic subvariety H of real dimension 2 n + 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 ı 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 F of dimension n in M that is tangent to H is also tangent to H ı . In this paper, we prove integration results of local and global nature for the restriction to H ı of a singular holomorphic foliation F tangent to a real analytic Levi-flat subset H . From a local viewpoint, if n = 1 and H ı has an isolated singularity, then F | H ı has a meromorphic first integral. From a global perspective, when M = P N and H is coherent and of low codimension, H ı extends to an algebraic variety. In this case, F | H ı has a rational first integral provided infinitely many leaves of F in H are algebraic.