Embedding Nonisolated Singularities into Isolated Singularities

Abstract

Let f : (ℂⁿ, 0) → (ℂ,0) be a complex analytic function germ with arbitrary singular locus. By adding to f a Brieskorn-Pham polynomial $x_1^{{N_1}} + \cdots + x_n^{{N_n}}$ in generic coordinates and with high enough powers N _i , one obtains a function germ g with isolated singularity. If we put coefficients in the Brieskorn-Pham polynomial, then f can be viewed as a very singular deformation of an isolated singularity g. In case dim(Sing f) = 1 one would have, for k ≫ 0, a series of functions with isolated singularities gk = f + εy ^k, sometimes called Iomdin series. The most familiar one is maybe the A _d -series: x ²₁ + ... + x ² _n−1 + εx _n ^d+1 with limit A _∞: x ²₁ + ... +x ² _n−1 . For Iomdin series, one has formulas a certain (topological) invariant of f to the same invariant of g _k , for instance the Euler characteristic of the Milnor fiber [Io], [Lê-1], the zeta function of the monodromy [Si], the spectrum [St], [Sa-1].

I thank Claude Sabbah for discussions related to this paper and the referee for some remarks which improved the presentation