Topological triviality of linear deformations with constant Lê numbers

Abstract

Let f(t,z) = f₀(z) + tg(z) be a holomorphic function defined in a neighbourhood of the origin in C × Cⁿ. It is well known that if the one-parameter deformation family {f_t} defined by the function f is a μ-constant family of isolated singularities, then {f_t} is topologically trivial—a result of A. Parusiński. It is also known that Parusiński's result does not extend to families of non-isolated singularities in the sense that the constancy of the Lê numbers of f_t at 0, as t varies, does not imply the topological triviality of the family f_t in general—a result of J. Fernández de Bobadilla. In this paper, we show that Parusiński's result generalizes all the same to families of non-isolated singularities if the Lê numbers of the function f itself are defined and constant along the strata of an analytic stratification of C × (f₀⁻¹(0) $\cap$ g⁻¹(0)). Actually, it suffices to consider the strata that contain a critical point of f.