Pull-back of singular Levi-flat hypersurfaces

TL;DR

This paper investigates conditions under which singular Levi-flat hypersurfaces in complex projective spaces can be represented as pull-backs of simpler geometric objects, enhancing understanding of their structure and classification.

Contribution

It provides new sufficient conditions for Levi-flat subsets to be pull-backs of semianalytic hypersurfaces or algebraic curves, improving previous results in the field.

Findings

Criteria for Levi-flat subsets to be pull-backs of hypersurfaces or curves

Applications to singular Levi-flat hypersurfaces in projective spaces

Enhanced classification of Levi-flat hypersurfaces

Abstract

We study singular real analytic Levi-flat subsets invariant by singular holomorphic foliations in complex projective spaces. We give sufficient conditions for a real analytic Levi-flat subset to be the pull-back of a semianalytic Levi-flat hypersurface in a complex projective surface under a rational map or to be the pull-back of a real algebraic curve under a meromorphic function. In particular, we give an application to the case of a singular real analytic Levi-flat hypersurface. Our results improve previous ones due to Lebl and Bretas -- Fern\'andez-P\'erez -- Mol.