Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

TL;DR

This paper studies Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian, revealing their geometric structure, algebraic properties, and limitations of algebraic characterization.

Contribution

It proves Levi-flatness and describes the structure of singular sets, establishing conditions for algebraicity and extending Levi-foliations, with explicit constructions showing non-algebraic examples.

Findings

Levi-flat hypersurfaces are unions of complex hyperplanes.

Hypersurfaces with certain singularities are algebraic and admit meromorphic first integrals.

Counterexamples show Levi-flat hypersurfaces can be non-algebraic and defy Chow's theorem.

Abstract

Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming $H$ has a global defining function, we prove $H$ is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension $2n-2$ or dimension $2n-4$. If the singular set is of dimension $2n-4$, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ${\mathbb P}^n$ with a meromorphic (rational of degree 1) first integral. In this case, $H$ is in some sense simply a complex cone over an algebraic curve in ${\mathbb P}^1$. Similarly if $H$ has a degenerate singularity, then $H$ is also algebraic. If the dimension of the singular set is $2n-2$ and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ${\mathbb P}^2$ of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ${\mathbb P}^2$. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.