Holomorphic Foliations Tangent to Levi-Flat Subsets

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.