TOPOLOGICAL TRIVIALITY OF LINEAR DEFORMATIONS WITH CONSTANT LE NUMBERS

Author(s)

Eyral, Christophe ^[1] ; Ruas, Maria Aparecida Soares ^[2]

Affiliation

^[1] Polskiej Akad Nauk, Inst Matemat, Ul Sniadeckich 8, PL-00656 Warsaw - Poland

^[2] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Ave Trabalhador Sao Carlense 400 Ctr, BR-13566590 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: KODAI MATHEMATICAL JOURNAL; v. 39, n. 1, p. 189-201, MAR 2016.
• Web of Science Citations: 1

Abstract

Let f (t, z) = f(0) (z) + tg(z) be a holomorphic function defined in a neighbourhood of the origin in C x C-n. It is well known that if the one-parameter deformation family [f(t)] defined by the function f is a mu-constant family of isolated singularities, then [f(t)] 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 f(t) at 0, as t varies, does not imply the topological triviality of the family [f(t)] 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 x (f(0)(-1)(0) boolean AND g(-1)(0)). Actually, it suffices to consider the strata that contain a critical point of f. (AU)

• FAPESP's process: 14/00304-2 - Singularities of differentiable mappings: theory and applications
• Grantee: Maria Aparecida Soares Ruas
• Support Opportunities: Research Projects - Thematic Grants