Chow's theorem for real analytic Levi-flat hypersurfaces

TL;DR

This paper extends Chow's theorem to real analytic Levi-flat hypersurfaces in complex projective space, showing they are algebraically characterized by rational functions under certain conditions.

Contribution

It establishes a version of Chow's theorem for Levi-flat hypersurfaces with singularities and algebraic leaves, linking geometric and algebraic properties.

Findings

Levi-flat hypersurfaces are tangent to rational function levels.

Such hypersurfaces are semialgebraic sets.

Levi foliations with algebraic leaves are defined by rational functions.

Abstract

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space $\mathbb{P}^{n}$, $n \geq 2$. 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.