A CR singular analogue of Severi's theorem

TL;DR

This paper characterizes when CR functions on certain singular submanifolds can be extended holomorphically, providing conditions for extension, flattening, and classification of CR singular images in complex spaces.

Contribution

It introduces a simple condition that fully characterizes extension properties for quadric CR singular manifolds of codimension 2, advancing understanding of CR singularities.

Findings

Identifies a condition for extension of CR functions on CR singular manifolds.

Provides a classification of CR singular images up to second order.

Offers criteria for flattening of CR singular submanifolds.

Abstract

Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that completely characterizes those quadric CR singular manifolds of codimension 2 in ${\mathbb C}^{n+1}$ for which an extension result holds. Consequently, we obtain an extension result for general real-analytic CR singular submanifolds of codimension 2. As applications we give a condition for the flattening of such submanifolds, and we classify CR singular images of CR submanifolds up to second order.