Sphericity of a real hypersurface via projective geometry

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}.