On CR singular CR images

TL;DR

This paper investigates the conditions under which CR singularities on real-analytic submanifolds are removable, characterizing them as CR images and analyzing their stability, invariants, and perturbation behavior.

Contribution

It provides a characterization of removable CR singularities as CR images and studies their stability and invariants under perturbations.

Findings

Removability of CR singularities is equivalent to being a CR image.

The stability of CR singularities under perturbation is analyzed.

A lemma on perturbing zeros of holomorphic functions on CR submanifolds is proved.

Abstract

We say that a CR singular submanifold $M$ has a removable CR singularity if the CR structure at the CR points of $M$ extends through the singularity as an abstract CR structure on $M$. We study such real-analytic submanifolds, in which case removability is equivalent to $M$ being the image of a generic real-analytic submanifold $N$ under a holomorphic map that is a diffeomorphism of $N$ onto $M$, what we call a CR image. We study the stability of the CR singularity under perturbation, the associated quadratic invariants, and conditions for removability of a CR singularity. A lemma is also proved about perturbing away the zeros of holomorphic functions on CR submanifolds, which could be of independent interest.