Sphericity of a real hypersurface via projective geometry

TL;DR

This paper provides a novel geometric characterization of when a real-analytic Levi-nondegenerate hypersurface in complex two-space is spherical, linking it to a classical projective geometry property called Desargues' theorem.

Contribution

It establishes a surprising equivalence between sphericity of hypersurfaces and a combinatorial property of Segre varieties, connecting complex geometry with projective geometric principles.

Findings

Sphericity characterized by Segre varieties' combinatorial property

Connection between hypersurface geometry and Desargues' theorem

New geometric criterion for sphericity in complex analysis

Abstract

In this work, we obtain an unexpected geometric characterization of sphericity of a real-analytic Levi-nondegenerate hypersurface $M\subset\mathbb C^{2}$. We prove that $M$ is spherical if and only if its Segre\,(-Webster) varieties satisfy an elementary combinatorial property, identical to a property of straight lines on the plane and known in Projective Geometry as the {\em Desargues Theorem}.