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 P n , n ≥ 2. More specifically, we prove that a real analytic Levi-flat hypersurface M ⊂ P n , with singular set of real dimension at most 2 n − 4 and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in P n. As a consequence, M is a semialgebraic set. We also prove that a Levi foliation on P n — a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one — satisfying similar conditions — singular set of real dimension at most 2 n − 4 and all leaves algebraic — is defined by the level sets of a rational function. [ABSTRACT FROM AUTHOR]