Levi-flats in $\mathbb{CP}^n$: a survey for nonexperts
Rasul Shafikov
TL;DR
The paper surveys the problem of real analytic Levi-flat hypersurfaces in complex projective spaces, focusing on $\mathbb{CP}^n$ with $n>2$ where such hypersurfaces are shown to not exist. It collects three self-contained proofs (two by Lins Neto and one by Siu) and outlines their distinct strategies: extending Levi foliations to holomorphic foliations across Stein domains, leveraging simple-connectedness and Haefliger’s theorem, and employing curvature/extension methods to contradict plurisubharmonicity. The work also connects Levi-flat hypersurfaces to minimal sets and demonstrates how the Levi foliation extends locally to holomorphic foliations, leading to global obstructions. Beyond $\mathbb{CP}^n$, it reviews broader generalizations to Kähler and other complex manifolds, highlighting the role of normal-bundle positivity and related rigidity phenomena. The results collectively constrain the existence of Levi-flat structures in high-dimensional projective and Kähler settings and inform the interplay between foliation theory, several complex variables, and complex geometry.
Abstract
This survey paper, aimed at nonexperts in the field, explores various proofs of nonexistence of real analytic Levi-flat hypersurfaces in $\mathbb CP^n$, $n>2$. Some generalizations and other related results are also discussed.