Algebraic Levi-flat hypervarieties in complex projective space

TL;DR

This paper investigates the properties of algebraic Levi-flat hypersurfaces in complex projective space, focusing on their singularities, rank, and conditions for being pullbacks of curves, including examples of nonalgebraic hypersurfaces.

Contribution

It introduces the concept of rank for algebraic Levi-flat hypersurfaces, explores the relationship between rank, degree, and singularities, and provides criteria for hypersurfaces to be pullbacks of curves.

Findings

Defined the rank of algebraic Levi-flat hypersurfaces.

Established conditions linking rank, degree, and singularity types.

Constructed examples of nonalgebraic Levi-flat hypersurfaces with specific properties.

Abstract

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in $\mathbb{C}$ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.