Algebraic Levi-flat hypervarieties in complex projective space

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.

21 pages, latex amsart, amsrefs; A couple of very minor typos fixed, added journal reference

References in corpus (2)

• On the complexity of proper holomorphic mappings between balls
• Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

Cited by in corpus (4)

• Singular set of a Levi-flat hypersurface is Levi-flat
• Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian
• Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations
• On real-analytic Levi-flat hypersurfaces and holomorphic webs