Singular Levi-flat real analytic hypersurfaces

We initiate a systematic local study of singular Levi-flat real analytic hy persurfaces, concentrating on the simplest nontrivial case of quadratic sin gularities. We classify the possible tangent cones to such hypersurfaces an d prove the existence and convergence of a rigid normal form in the case of generic (Morse) singularities. We also characterize when such a hypersurfa ce 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 variet ies 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 su ch a hypersurface to be equivalent to an algebraic one.