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A CR singular analogue of Severi's theorem

Jiri Lebl, Alan Noell, Sivaguru Ravisankar

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.

A CR singular analogue of Severi's theorem

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 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.

Paper Structure

This paper contains 7 sections, 15 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Examples
  4. Extending CR polynomials on quadrics
  5. Extending real-analytic CR functions
  6. Flattening
  7. CR singular images of CR manifolds

Key Result

Theorem 1.1

Let $(z,w) \in {\mathbb{C}}^n \times {\mathbb{C}}$, $n \geq 2$, be the coordinates and, near the origin, let $M \subset {\mathbb{C}}^{n+1}$ be a codimension-2 submanifold given by where $\rho$ is real-analytic, $A,B,C$ are complex $n \times n$ matrices, $B$ and $C$ are symmetric, and $E$ is $O(\lVert {z} \rVert^3)$. Assume Suppose $f(z,\bar{z})$ is a real-analytic function defined near the origi

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Proposition 4.1
  • proof
  • ...and 22 more