Levi-flat world: a survey of local theory

This expository paper concerns local properties of Levi-flat real analytic manifolds with singularities. Levi-flat manifolds arise naturally in Complex Geometry and Foliation Theory. In many cases (global) compact Levi-flat manifolds without singularities do not exist. These global obstructions make natural the study of Levi-flat objects with singularities because they always exist. The present expository paper deals with some recent results on local geometry of Levi-flat singularities. One of the main questions concerns an extension of the Levi foliation as a holomorphic foliation to a full neighborhood of singularity. It turns out that in general such extension does not exist. Nevertheless, the Levi foliation always extends as a holomorphic web (a foliation with branching) near a non-dicritical singularity. We also present an efficient criterion characterizing these singularities.