Chow's theorem for real analytic Levi-flat hypersurfaces

Abstract

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space ${\mathbb{P}}^n$, $n\ge 2$. More specifically, we prove that a real analytic Levi-flat hypersurface $M\subset {\mathbb{P}}^n$, 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 ${\mathbb{P}}^n$. As a consequence, $M$ is a semialgebraic set. We also prove that a Levi foliation on ${\mathbb{P}}^n$ - a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one - satisfying similar conditions - singular set of real dimension at most $2n-4$ and all leaves algebraic - is defined by the level sets of a rational function.

18 pages