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

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.