On Segre-degenerate Levi-flat hypervarieties
Jiří Lebl, Luka Mernik
TL;DR
The paper analyzes singular real-analytic Levi-flat hypersurfaces in $\mathbb{C}^n$, proving that Segre-degeneracy at a point $p$ is equivalent to the existence of a holomorphic support curve through $p$ and to two-sided curve-support, which in turn yields families of analytic discs attached to the hypersurface that fill a neighborhood of $p$. The main advance is the demonstration that two-sided support induces a Hartogs-type extension mechanism via Kontinuitätssatz and forces the rational hull to contain a neighborhood of $p$, implying non-convexity of Levi-flat Segre-degenerate hypersurfaces. The results are established by reducing to the two-dimensional case through complexification and finite mappings, then lifting back to $\mathbb{C}^n$, and by analyzing Segre varieties and their principal components. Overall, the work extends prior understanding of Levi-flat and Segre-degenerate structures, revealing robust local hull phenomena and holomorphic extension properties near dicritical singularities.
Abstract
We prove that a singular real-analytic Levi-flat hypersurface $H$ in $\mathbb C^n$ being Segre-degenerate at a point $p$ is equivalent to the existence of a so-called support curve, that is, a holomorphic curve that intersects $H$ at exactly one point, which in turn is equivalent to the existence of support curves on at least two sides of $H$ at $p$. The existence of such two-sided support provides families of analytic discs attached to $H$ that covers a neighborhood of $p$. The existence of such discs has two corollaries. First, any function holomorphic on a neighborhood of a Segre-degenerate $H$ extends to a fixed neighborhood of $p$. Second, the rational hull of $H$ is a neighborhood of $p$, and thus no Levi-flat Segre-degenerate hypersurface in $\mathbb C^n$ can be rationally convex.