A real hypersurface H in an open set of a complex variety of dimension n is Levi-flat if its intrinsic distribution of complex tangent spaces is integrable. It thus defines a foliation on H (the Levi foliation) whose leaves are immersed complex varieties of dimension n-1. We will give an introduction to the theory of real analytic flat Levi hypersurfaces by analyzing in particular the case in which the Levi foliation extends to a holomorphic foliation in the ambient space. Our objective is to present two lines of results:
1) Local case: existence of meromorphic first integral for a holomorphic foliatiion germ which leaves invariant a real analytic Levi-flat hypersurface (Cerveau-Lins Neto Theorem).
2) Global case: Chow Levi-flat theorems: conditions for a real analytic Levi-flat hypersurface in the complex projective space to be semi-algebraic.
This mini-course will have 3 lectures of 1 hour each.