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Topological triviality of families of functions on analytic varieties

Published online by Cambridge University Press:  22 January 2016

Maria Aparecida Soares Ruas  and
João Nivaldo Tomazella
Maria Aparecida Soares Ruas
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Departamento de Matemática, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil, [email protected]
João Nivaldo Tomazella
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, 13560-905, São Carlos, SP, Brazil, [email protected]
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Abstract

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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.

Keywords

32S1558K4058K15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 175 , 2004 , pp. 39 - 50
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[1] Bruce, J. W. and Roberts, M., Critical points of functions on analytic varieties, Topology, 27 (1988), no. 1, 57–90.Google Scholar
[2] Bruce, J. W., Kirk, N. P. and Plessis, A. A. du, Complete transversals and the classification of singularities, Nonlinearity, 10 (1997), 253–275.Google Scholar
[3] Damon, J., The unfolding and determinacy theorems for subgroups of A and K, Memoirs Am. Math. Soc., 306 (1984).Google Scholar
[4] Damon, J., Topological triviality and versality for subgroups of A and K, Memoirs Am. Math. Soc., 389 (1988).Google Scholar
[5] Damon, J., Deformations of sections of singularities and Gorenstein surface singularities, Am. Journal of Mathematics, 109 (1987), 695–722.Google Scholar
[6] Damon, J., Topological triviality and versality for subgroups of A and K: II. Sufficient conditions and applications, Nonlinearity, 5 (1992), 373–412.Google Scholar
[7] Damon, J., On the freeness of equisingular deformations of plane curve singularities, Topology and its Application, 118 (2002), 31–43.Google Scholar
[8] Fukui, T. and Paunescu, L., Stratification theory from the weighted point of view, Canad. J. Math., 53 (2001), no. 1, 73–97.CrossRefGoogle Scholar
[9] Kouchnirenko, A. G., Polyèdres de Newton et nombres de Milnor, Invent. Math., 32 (1976), 1–31.Google Scholar
[10] Mond, D., On the classification of germs of maps from ℝ2 to ℝ3 , Proc. of London Math. Soc. 3, 50 (1985), 333–369.Google Scholar
[11] Ruas, M. A. S. and Saia, M. J., Cl-determinacy of weighted homogeneous germs, Hokkaido Math. Journal, 26 (1997), 89–99.Google Scholar
[12] Saia, M., The integral closure of ideals and the Newton filtration, J. Algebraic Geometry, 5 (1996), 1–11.Google Scholar
[13] Tomazella, J. N., Seções de Variedades Analíticas, Ph.D. Thesis, ICMSC-USP.Google Scholar