Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Embedded topological triviality of separable families of singularities

R. Giménez Conejero, Andreas Lind, Aurélio Menegon

TL;DR

The paper addresses when perturbations of analytic function-germs on a complex space maintain embedded topological type, focusing on separable deformations with a uniform singular set. It develops an embedded trivialization framework by leveraging Δ-regularity and a uniform critical set, and constructs an ambient trivialization away from the discriminant that is then extended across a codimension-2 set. The main result shows that such separable families have constant embedded topological type, with immediate implications for μ-constant deformations of ICIS and progress toward the μ-constant conjecture. The work integrates stratification-based control, Verdier lifting, and extension lemmas to bridge abstract topological triviality and embedded triviality.

Abstract

Understanding how singularities behave under small perturbations is a central theme in singularity theory. In this paper we establish sufficient conditions for families of analytic function-germs on a germ of a complex analytic space to admit an embedded topological trivialization. Our results extend previous work of the third author and collaborators, moving from abstract triviality to the embedded setting. As an application, we obtain new instances of topological stability, including a broad class of $μ$-constant deformations. These findings provide a new insight into the long-standing $μ$-constant conjecture, one of the major open problems in the field.

Embedded topological triviality of separable families of singularities

TL;DR

The paper addresses when perturbations of analytic function-germs on a complex space maintain embedded topological type, focusing on separable deformations with a uniform singular set. It develops an embedded trivialization framework by leveraging Δ-regularity and a uniform critical set, and constructs an ambient trivialization away from the discriminant that is then extended across a codimension-2 set. The main result shows that such separable families have constant embedded topological type, with immediate implications for μ-constant deformations of ICIS and progress toward the μ-constant conjecture. The work integrates stratification-based control, Verdier lifting, and extension lemmas to bridge abstract topological triviality and embedded triviality.

Abstract

Understanding how singularities behave under small perturbations is a central theme in singularity theory. In this paper we establish sufficient conditions for families of analytic function-germs on a germ of a complex analytic space to admit an embedded topological trivialization. Our results extend previous work of the third author and collaborators, moving from abstract triviality to the embedded setting. As an application, we obtain new instances of topological stability, including a broad class of -constant deformations. These findings provide a new insight into the long-standing -constant conjecture, one of the major open problems in the field.

Paper Structure

This paper contains 4 sections, 6 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Delta-regularity and uniform critical set
  3. Trivialization outside the discriminant
  4. Extending the partial trivialization

Key Result

Theorem 1.1

Let $f_s:(X,0)\to(\mathbb{C},0)$ be a separable family of analytic function-germs, with $s\in\mathbb{C}$. If $(f_s)$ has a uniform singular set $\Sigma=\{0\}$ with respect to some $(w)$-regular stratification $\mathcal{S}$ of $X$, then the family has constant embedded topological type. That is, for sending $(V_0,0)$ onto $(V_s,0)$.

Figures (1)

  • Figure 1: Depiction of the different objects in our proof.

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.1
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Example 2.4: cf. Menegon2023
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Lemma 4.1
  • ...and 1 more