On singular real analytic Levi-flat foliations

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

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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