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Intrinsic complexification of real-analytic varieties

Bernhard Lamel, Jiri Lebl

TL;DR

The paper extends Segre-variety techniques to singular real-analytic subvarieties in complex spaces by introducing precomplexifications and Segre varieties $\Sigma_p^U$ relative to good neighborhoods, and by defining intrinsic complexification as the smallest complex-analytic subvariety containing the real-analytic set. It establishes dimension formulas linking the intrinsic complexification to the Segre variety via $\dim_{\mathbb{R}} X - \dim_{\mathbb{C}} \Sigma_p$ under Segre nondegeneracy, and proves local constancy results for coherent or polynomial-defining cases, including when the intrinsic complexification is generically generic. It also analyzes behavior under finite holomorphic maps, showing preservation of complexifications while noting that coherence is not generally preserved, and provides examples illustrating nontrivial dependence on singular structure and Segre degeneracy. These results advance the understanding of intrinsic complexifications and Segre geometry in singular real-analytic CR contexts with implications for genericity and CR-geometry at singular points.

Abstract

We introduce a framework for Segre varieties for singular real-analytic subvarieties of a complex space and utilize it to study the intrinsic complexifications of these subvarieties. Many examples illustrate the subtle issues arising in the singular setting.

Intrinsic complexification of real-analytic varieties

TL;DR

The paper extends Segre-variety techniques to singular real-analytic subvarieties in complex spaces by introducing precomplexifications and Segre varieties relative to good neighborhoods, and by defining intrinsic complexification as the smallest complex-analytic subvariety containing the real-analytic set. It establishes dimension formulas linking the intrinsic complexification to the Segre variety via under Segre nondegeneracy, and proves local constancy results for coherent or polynomial-defining cases, including when the intrinsic complexification is generically generic. It also analyzes behavior under finite holomorphic maps, showing preservation of complexifications while noting that coherence is not generally preserved, and provides examples illustrating nontrivial dependence on singular structure and Segre degeneracy. These results advance the understanding of intrinsic complexifications and Segre geometry in singular real-analytic CR contexts with implications for genericity and CR-geometry at singular points.

Abstract

We introduce a framework for Segre varieties for singular real-analytic subvarieties of a complex space and utilize it to study the intrinsic complexifications of these subvarieties. Many examples illustrate the subtle issues arising in the singular setting.
Paper Structure (4 sections, 13 theorems, 17 equations)

This paper contains 4 sections, 13 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Varieties and complexifications
  3. Intrinsic complexification
  4. Intrinsic complexification under finite maps

Key Result

Proposition 2.6

Let $(X,0)\subset ({\mathbb{C}}^n,0)$ be the germ of a real-analytic subvariety. Then there exists a connected neighborhood $U\subset {\mathbb{C}}^n$ of the origin and a real-analytic function $\rho \colon U \to {\mathbb{R}}^\ell$ such that: Furthermore, $U \subset {\mathbb{C}}^n$ is a neighborhood satisfying the conditions above and $W \subset U$ is another neighborhood of the origin, then there

Theorems & Definitions (41)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Example 2.9
  • ...and 31 more