Singular Levi-flat real analytic hypersurfaces

Abstract

We initiate a systematic local study of singular Levi-flat real analytic hypersurfaces, concentrating on the simplest nontrivial case of quadratic singularities. We classify the possible tangent cones to such hypersurfaces and prove the existence and convergence of a rigid normal form in the case of generic (Morse) singularities. We also characterize when such a hypersurface is defined by the vanishing of the real part of a holomorphic function. The main technique is to control the behavior of the homorphic Segre varieties contained in such a hypersurface. Finally, we show that not every such singular hypersurface can be defined by the vanishing of the real part of a holomorphic or meromorphic function, and give a necessary condition for such a hypersurface to be equivalent to an algebraic one.