A generalization of Milnor\u2019s formula

Abstract

The Milnor number mu _f of a holomorphic function f :({mathbb {C}}^n,0) rightarrow ({mathbb {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 M_f, 3) the degree of the differential {text {d}}f at the origin, and 4) the length of an analytic algebra due to Milnor’s formula mu _f = dim _{mathbb {C}}{mathcal {O}}_n/{text {Jac}}(f). Let (X,0) subset ({mathbb {C}}^n,0) be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let f :({mathbb {C}}^n,0) rightarrow ({mathbb {C}},0) be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum {mathscr {S}}_alpha let mu _f(alpha ;X,0) be the number of critical points on {mathscr {S}}_alpha in a morsification of f|(X, 0). We show that the numbers mu _f(alpha ;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 M_{f|(X,0)} in terms of the mu _f(alpha ;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.