On singular real analytic Levi-flat foliations

TL;DR

This paper classifies singular Levi-flat foliations on complex manifolds, showing they are either defined by a meromorphic first integral or a closed rational 1-form, with applications to algebraic foliations on projective space.

Contribution

It provides a classification of germs of Levi-flat foliations with holomorphic tangent foliation, identifying two main types based on their defining forms.

Findings

Levi-flat foliations are either meromorphic integrals or defined by closed rational 1-forms.

Classification applies to germs at the origin in complex space.

Results extend to real algebraic Levi-flat foliations on projective space.

Abstract

A singular real analytic foliation $\mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $\mathcal{L}$ which is tangent to $\mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(\mathbb{C}^{n},0)$ under the hypothesis that $\mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $\mathcal{L}$, from which the classification of $\mathcal{F}$ derives: either it has a meromorphic first integral or is defined by a closed rational $1-$form. Our local results also allow us to classify real algebraic Levi-flat foliations on the complex projective space $\mathbb{P}^{n} = \mathbb{P}^{n}_{\mathbb{C}}$.