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 {ft} defined by the function f is a μ-constant family of isolated singularities, then {ft} is topologically trivial—a result of A. Parusinski. It is also known that Parusinski's result does not extend to families of non-isolated singularities in the sense that the constancy of the Le numbers of ft at 0, as t varies, does not imply the topological triviality of the family ft in general—a result of J. Fernandez de Bobadilla. In this paper, we show that Parusinski's result generalizes all the same to families of non-isolated singularities if the Le 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.