Singularities of mappings on ICIS and applications to Whitney equisingularity

TL;DR

This paper extends the concept of the image Milnor number to germs of analytic maps on ICIS, characterizes Whitney equisingularity in terms of invariants, and explores their behavior under deformations.

Contribution

It introduces a generalized image Milnor number for ICIS, and provides criteria for Whitney equisingularity based on the constancy of these invariants during deformations.

Findings

The image Milnor number is conserved under deformations.

Whitney equisingularity is characterized by constant $I^*$-sequences.

The approach generalizes known smooth case results to ICIS.

Abstract

We study germs of analytic maps $f:(X,S)\rightarrow(\mathbb{C}^p,0)$, when $X$ is an ICIS of dimension $n<p$. We define an image Milnor number, generalizing Mond's definition, $\mu_I(X,f)$ and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs $f_t\colon(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instabilities in terms of the constancy of the $\mu_I^*$-sequences of $f_t$ and the projections $\pi\colon D^2(f_t)\to\mathbb{C}^n$, where $D^2(f_t)$ is the ICIS given by double point space of $f_t$ in $\mathbb{C}^n\times\mathbb{C}^n$. The $\mu_I^*$-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices.