A generalization of Milnor's formula

Abstract

We describe a generalization of Milnor's formula for the Milnor number of an isolated hypersurface singularity to the case of a function $f$ whose restriction $f|(X,0)$ to an arbitrarily singular reduced complex analytic space $(X,0) \subset (\mathbb C^n,0)$ has an isolated singularity in the stratified sense. The corresponding analogue of the Milnor number, $μ_f(α;X,0)$, is the number of Morse critical points in a stratum $\mathscr S_α$ of $(X,0)$ in a morsification of $f|(X,0)$. Our formula expresses $μ_f(α;X,0)$ as a homological index based on the derived geometry of the Nash modification of the closure of the stratum $\mathscr S_α$. While most of the topological aspects in this setup were already understood, our considerations provide the corresponding analytic counterpart. We also describe how to compute the numbers $μ_f(α;X,0)$ by means of our formula in the case where the closure $\overline{ \mathscr S_α} \subset X$ of the stratum in question is a hypersurface.