Asian Journal of Mathematics

Volume 24 (2020)

Number 6

On singular real analytic Levi-flat foliations

Pages: 1007 – 1028

DOI: https://dx.doi.org/10.4310/AJM.2020.v24.n6.a4

Authors

Arturo Fernández-Pérez (Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)

Rogério Mol (Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)

Rudy Rosas (Pontificia Universidad Católica del Perú, Lima, Peru)

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 of 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 it 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 $\mathbb{P}^n = \mathbb{P}^n_C$.

Keywords

holomorphic foliation, CR-manifold, Levi-flat variety

2010 Mathematics Subject Classification

32S65, 32V40, 37F75

The first-named author was supported by CNPq-Universal, Pronex/FAPERJ and by a CNPq grant PQ2019-302790/2019-5.

The second-named author was supported by CNPq-Universal and Pronex/FAPERJ.

The third-named author was supported by Vicerrectorado de investigación de la Pontificia Universidad Católica del Perú.

Received 29 January 2020

Accepted 17 February 2020

Published 3 September 2021