A generalization of Milnor’s formula

Abstract

The Milnor number μf of a holomorphic function f: (Cn, 0) → (C, 0) with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber Mf, 3) the degree of the differential d f at the origin, and 4) the length of an analytic algebra due to Milnor’s formula μf= dim COn/ Jac (f). Let (X, 0) ⊂ (Cn, 0) be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let f: (Cn, 0) → (C, 0) be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum Sα let μf(α; X, 0) be the number of critical points on Sα in a morsification of f|(X, 0). We show that the numbers μf(α; X, 0) generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber Mf|(X,) in terms of the μf(α; X, 0) and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.