On singular real analytic Levi-flat foliations

A singular real analytic foliation 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 L which is tangent to F. In this article, we classify germs of Levi-flat foliations at (Cⁿ,0) under the hypothesis that L is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for L, from which the classification of 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 Pⁿ = PⁿC.