Some L\^e-Greuel type formulae on stratified spaces

TL;DR

This paper generalizes Lê-Greuel type formulas to stratified spaces, providing a topological approach to compute Milnor numbers for functions with isolated singularities on complex analytic germs.

Contribution

It extends previous hypersurface results to stratified spaces, introducing a topological definition of Milnor numbers and proving formulas to compute them via homological indices.

Findings

Generalizes Lê-Greuel formulas to stratified spaces

Defines Milnor numbers topologically for stratified singularities

Provides formulas relating Milnor numbers to homological indices

Abstract

We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$. An extension of Tib{\u a}r's Bouquet Theorem to this setup allows for a topological definition of Milnor numbers $\mu(\alpha; f)$ for each stratum $V^\alpha$ of $X$ and we prove several formulas which compute these numbers as (alternating) sums of certain ``homological indices''. The main technical result at work in the background is a local Riemann-Roch type theorem, relating a topological obstruction to holomorphic Euler characteristics.