Topological triviality of families of functions on analytic varieties

Abstract

Abstract. We present in this paper sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety V . The main result is an infinitesimal criterion based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu in [8]. When V is a weighted homogeneous variety, we obtain as a corollary, the topological triviality of deformations by terms of non negative weights of a weighted homogeneous germ consistent with V . Application of the results to deformations of Newton non-degenerate germs with respect to a given variety is also given. §1. Introduction Let V, 0 be the germ of an analytic subvariety of k n , k = R, or C and let R V (respectively C 0 -R V ) be the group of germs of diffeomorphisms (respectively homeomorphisms) preserving V, 0, acting on germs h 0 : k n , 0 → k, 0. The aim of this paper is to study topologically trivial deformations of R Vfinitely determined germs h 0 . The main result is Theorem 3.4 in which we introduce a sufficient condition for the C 0 -R V -triviality of families of map germs h : k n × k, 0 → k, 0, h(x, 0) = h 0 (x), based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu in [8]. A non weighted version of this result first appeared in [13]. There, the sufficient condition for topological triviality is formulated in terms of the integral closure of the tangent space to the R V -orbit of h t .As an application of the results, when V is a weighted homogeneous analytic variety, we prove that any deformation by non negative weights of an R V -finitely determined weighted homogeneous germ (consistent with V ) is topologically trivial. This result was previously proved by J. Damon

Non-Negative Deformations of Weighted Homogeneous Singularities

“…We see that the implication (3) ⇒ (2) is also true even if f is weighted homogeneous with different weights of (X, 0) (that is, it is not "consistent with (X, 0)" in the terminology of [6,21]). However, the converse is not true in general in that case, as the following example shows.…”

“…We recall that a deformation f t of a function germ f : ‫ރ(‬ n , 0) → ‫,ރ(‬ 0) is called C 0 -R X -trivial if there is a homeomorphism : ‫ރ(‬ × ‫ރ‬ n ) → ‫ރ(‬ × ‫ރ‬ n ) which is an unfolding of the identity map in ‫ރ‬ n such that f t • ψ t = f and ψ t preserves (X, 0) (i.e., ψ t (X) = X for all t ∈ ‫. )ރ‬ THEOREM 3.2 [6,21]. Let (X, 0) be a weighted homogeneous variety with an isolated singularity and let f be a weighted homogeneous R X -finitely determined function.…”

“…REMARK 3.3. In [6,21], it is used the term "consistent with (X, 0)" to refer to the fact that f is weighted homogenous with the same weights of (X, 0), but here this is not necessary since we fix the weights from the beginning.…”

Invariants of topological relative right equivalences

“…Moreover, F is a deformation by nonnegative weights of the R V -finitely determined weighted homogeneous germ f 0 = y + ax 2 . Then F is C 0 -R V -trivial (see [6] and [15]). One can also verify from direct computations that μ B R (V, f t ) is constant.…”

Equivalências de contato topológica e bi-Lipschitz de germes de aplicações diferenciáveis

“…A proposição a seguir é uma aplicação do Teorema 5.12 para germes de classe C r . argumento aparece em muitos trabalhos [52], [55], [17], entre outros. O conceito de fecho integral de ideais ou módulos é usado por vários autores para a resolução de problemas de trivialidade de famílias.…”