Uniform stable radius, Lê numbers and topological triviality for line singularities

Abstract

Let $\{f_t\}$ be a family of complex polynomial functions with line singularities. We show that if $\{f_t\}$ has a uniform stable radius (for the corresponding Milnor fibrations), then the Lê numbers of the functions $f_t$ are independent of $t$ for all small $t$. In the case of isolated singularities --- a case for which the only non-zero Lê number coincides with the Milnor number --- a similar assertion was proved by M. Oka and D. O'Shea. By combining our result with a theorem of J. Fernández de Bobadilla --- which says that families of line singularities in ${\mathbb{C}}^n$, $n\ge 5$, with constant Lê numbers are topologically trivial --- it follows that a family of line singularities in ${\mathbb{C}}^n$, $n\ge 5$, is topologically trivial if it has a uniform stable radius. As an important example, we show that families of weighted homogeneous line singularities have a uniform stable radius if the nearby fibres $f_t^{-1}(\eta )$, $\eta \ne 0$, are "uniformly" non-singular with respect to the deformation parameter $t$.