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

Christophe Eyral

Vol. 291 (2017), No. 2, 359–367

DOI: 10.2140/pjm.2017.291.359

Abstract

Let $\left\{f_t\right\}$ be a family of complex polynomial functions with line singularities. We show that if $\left\{f_t\right\}$ 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 $t$. A similar assertion was proved by M. Oka and D. B. O’Shea in the case of isolated singularities — a case for which the only nonzero Lê number coincides with the Milnor number.

By combining our result with a theorem of J. Fernández de Bobadilla, we conclude that a family of line singularities in ${ℂ}^n$, $n\ge 5$, 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 $f_t^{-1}\left(\eta \right)$, $\eta \ne 0$, are “uniformly” nonsingular with respect to the deformation parameter $t$.

Keywords

line singularities, uniform stable radius, Lê numbers, equisingularity

Mathematical Subject Classification 2010

Primary: 14B05, 14B07, 14J17, 14J70

Secondary: 32S05, 32S25

Milestones

Received: 27 April 2017

Accepted: 11 May 2017

Published: 14 September 2017