Chow's theorem for real analytic Levi-flat hypersurfaces

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space Pⁿ, n≥2. More specifically, we prove that a real analytic Levi-flat hypersurface M⊂Pⁿ, with singular set of real dimension at most 2n−4 and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in Pⁿ. As a consequence, M is a semialgebraic set. We also prove that a Levi foliation on Pⁿ — a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one — satisfying similar conditions — singular set of real dimension at most 2n−4 and all leaves algebraic — is defined by the level sets of a rational function.