## Kodai Mathematical Journal

### 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. 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 ft at 0, as t varies, does not imply the topological triviality of the family ft 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.

#### Article information

Source
Kodai Math. J., Volume 39, Number 1 (2016), 189-201.

Dates
First available in Project Euclid: 22 March 2016

https://projecteuclid.org/euclid.kmj/1458651699

Digital Object Identifier
doi:10.2996/kmj/1458651699

Mathematical Reviews number (MathSciNet)
MR3478278

Zentralblatt MATH identifier
1341.32022

#### Citation

Eyral, Christophe; Ruas, Maria Aparecida Soares. Topological triviality of linear deformations with constant Lê numbers. Kodai Math. J. 39 (2016), no. 1, 189--201. doi:10.2996/kmj/1458651699. https://projecteuclid.org/euclid.kmj/1458651699