On Levi-flat hypersurfaces tangent to holomorphic webs

TL;DR

This paper studies Levi-flat hypersurfaces tangent to holomorphic webs, introducing first integrals for webs, and proves that webs with finitely many invariant subvarieties tangent to such hypersurfaces have holomorphic first integrals.

Contribution

It introduces the concept of first integrals for local webs and proves the existence of holomorphic first integrals under certain tangency conditions.

Findings

Webs with finitely many invariant subvarieties tangent to Levi-flat hypersurfaces have holomorphic first integrals.

Introduces the notion of first integrals for local holomorphic webs.

Provides conditions under which Levi-flat hypersurfaces admit holomorphic first integrals.

Abstract

We investigate germs of real analytic Levi-flat hypersurfaces tangent to germs of codimension one holomorphic webs. We introduce the notion of first integrals for local webs. In particular, we prove that a $k$-web with finitely many invariant analytic subvarieties through the origin tangent to a Levi-flat hypersurface has a holomorphic first integral.