On deformations of isolated singularity functions

TL;DR

This paper investigates how multi-parameter deformations of isolated singularity functions affect their topological type, establishing conditions under which the topological structure remains unchanged, extending classical results and providing new criteria.

Contribution

It extends classical results on singularity deformations by proving topological invariance under certain conditions and introduces criteria for constant topological type in complex singularity families.

Findings

Deformations without singular point coalescence preserve topological type.

Provides a sufficient condition for constant topological type in one-parameter complex singularities.

Shows that for determinantal singularities, μ-constancy implies topological invariance.

Abstract

We study multi-parameters deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic spaces. We prove that if such a deformation has no coalescing of singular points, then it has constant topological type. This extends some classical results due to L\^e \& Ramanujam (1976) and Parusi\'nski (1999), as well as a recent result due to Jesus-Almeida and the first author. It also provides a sufficient condition for a one-parameter family of complex isolated singularity surfaces in $\C^3$ to have constant topological type. On the other hand, for complex isolated singularity families defined on an isolated determinantal singularity, we prove that $\mu$-constancy implies constant topological type.