Some Lê-Greuel type formulae on stratified spaces

Abstract

We extend the circle of ideas from a previous paper on hypersurfaces to functions $f:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^k,0)$ 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 $\hat{I}¼(\hat{I}\pm ;f)$ for each stratum $V^{\hat{I}}\pm$ 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.