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Segre nondegenerate totally real subvarieties

Bernhard Lamel, Jiri Lebl

Abstract

We study an irreducible real-analytic germ of an $n$-dimensional variety in $n$ dimensional complex space. Assuming that the variety is Segre nondegenerate we define an averaging operator that generalizes the Moser--Webster involution. This operator can be thought of as being the CR structure of the singularity, and using this operator we study the set of functions that are restrictions of holomorphic functions. We give a condition on the flattening of the singularity, that is realizing the singularity as a codimention one subvariety of a nonsingular Levi-flat hypersurface.

Segre nondegenerate totally real subvarieties

Abstract

We study an irreducible real-analytic germ of an -dimensional variety in dimensional complex space. Assuming that the variety is Segre nondegenerate we define an averaging operator that generalizes the Moser--Webster involution. This operator can be thought of as being the CR structure of the singularity, and using this operator we study the set of functions that are restrictions of holomorphic functions. We give a condition on the flattening of the singularity, that is realizing the singularity as a codimention one subvariety of a nonsingular Levi-flat hypersurface.

Paper Structure

This paper contains 10 sections, 17 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric functions and standard defining equations
  4. The Segre varieties as multifunctions
  5. Real-analytic and formal functions on $(X,0)$
  6. Remark: Formal varieties
  7. Obstructions to holomorphicity
  8. The averaging operator
  9. Flattening and other applications of averaging
  10. Examples of flattening

Key Result

Proposition 2.1

Let $(X,0) \subset ({\mathbb{C}}^n_z,0)$ be a germ of a Segre nondegenerate irreducible $n$-dimensional subvariety at the origin. Then there exist polydiscs $\Delta_z \subset {\mathbb{C}}^n$ and $\Delta_\xi \subset {\mathbb{C}}^n$, centered at the origin, and an $n$-dimensional closed complex subvar

Theorems & Definitions (40)

  • Proposition 2.1
  • Definition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Example 2.6
  • ...and 30 more