On deformations of isolated singularity functions

Abstract

AbstractWe study multi-parameter deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic space. We prove that if such a deformation has no coalescing of singular points, then it has weak constant topological type. This extends some classical results due to Lê and Ramanujam (Am J Math 98:67–78, 1976) and Parusiński (Bull Lond Math Soc 31(6):686–692, 1999), as well as a recent result due to Jesus-Almeida and the first author (Int Math Res Notices 2023(6):4869–4886, 2023). It also provides a sufficient condition for a one-parameter family of complex isolated singularity surfaces in $${\mathbb {C}}^3$$ C 3 to have weak constant topological type. On the other hand, for complex isolated singularity families defined on an isolated determinantal singularity, we prove that $$\mu$$ μ -constancy implies weak constant topological type.