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On CR singular CR images

Jiří Lebl, Alan Noell, Sivaguru Ravisankar

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.

On CR singular CR images

Abstract

We say that a CR singular submanifold has a removable CR singularity if the CR structure at the CR points of extends through the singularity as an abstract CR structure on . We study such real-analytic submanifolds, in which case removability is equivalent to being the image of a generic real-analytic submanifold under a holomorphic map that is a diffeomorphism of onto , 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.

Paper Structure

This paper contains 7 sections, 19 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Perturbing away zeros of holomorphic functions on CR submanifolds
  3. CR images
  4. Perturbing away singularities in equidimensional case
  5. Perturbing away singularities in general
  6. Quadratic terms for a codimension-2 CR image
  7. Codimension-2 CR images in C3

Key Result

Proposition 1.2

A point $q$ of a real-analytic submanifold $M \subset {\mathbb{C}}^m$ is a removable CR singularity if and only if there exist a real-analytic generic submanifold $N \subset {\mathbb{C}}^n$, $n \leq m$, a neighborhood $U$ of $N$ in ${\mathbb{C}}^n$, a neighborhood $W$ of $q$ in ${\mathbb{C}}^m$, and

Theorems & Definitions (51)

  • Definition 1.1
  • Proposition 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Lemma \ref{['lemma:main']}
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • ...and 41 more