Topological equisingularity for isolated complete intersection singularities

Abstract

Let X, x be a germ of an analytic variety (over the complex numbers) which is a complete intersection isolated singularity. The author associates to X, x a sequence µ^∗= (µ₀, µ₁, . . . ) of numerical invariants by taking µⁱ to be the minimal value of the Milnor numbers µ(X_i, x_i) for all deformations (X_i, x_i) → (S_i, s_i) of X, x with dim S_i = i. One has µ₀ = µ(X, x) and µ_i = 0 if i is bigger than the embedding codimension of X, x. On the other hand the author defines the topological type of X, x as the class of homeomorphism of any sequence of germs (X, x) = (X₀, x) ⊂ (X₁, x) ⊂ ... ⊂ (X_k, x) where k is the embedding codimension of X, x, for each i, µ(X_i, x) = µ_i(X, x) and then the homeomorphism class does not depend on the X_i. The main result in the paper says that a µ^∗-constant family of isolated complete intersection singularities of dimension different from two is topologically equisingular. A sufficient condition for the members of the family to have isomorphic monodromy fibrations is also given.