Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-15T02:41:06.765Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariants of topological relative right equivalences

Published online by Cambridge University Press:  12 June 2013

IMRAN AHMED ,
MARIA APARECIDA SOARES RUAS  and
JOÃO NIVALDO TOMAZELLA
IMRAN AHMED
Affiliation:
Department of Mathematics, COMSATS Institute of Information Technology, M.A. Jinnah Campus, Defence Road, off Raiwind Road Lahore, Pakistan. e-mail: drimranahmed@ciitlahore.edu.pk
MARIA APARECIDA SOARES RUAS
Affiliation:
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador Sãocarlense 400, São Carlos-S.P., Brazil. e-mail: maasruas@icmc.usp.br
JOÃO NIVALDO TOMAZELLA
Affiliation:
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador Sãocarlense 400, São Carlos-S.P., Brazil. e-mail: maasruas@icmc.usp.br
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (V,0) be the germ of an analytic variety in $\mathbb{C}^n$ and f an analytic function germ defined on V. For functions with isolated singularity on V, Bruce and Roberts introduced a generalization of the Milnor number of f, which we call Bruce–Roberts number, μBR(V,f). Like the Milnor number of f, this number shows some properties of f and V. In this paper we investigate algebraic and geometric characterizations of the constancy of the Bruce–Roberts number for families of functions with isolated singularities on V. We also discuss the topological invariance of the Bruce–Roberts number for families of quasihomogeneous functions defined on quasihomogeneous varieties. As application of the results, we prove a relative version of the Zariski multiplicity conjecture for quasihomogeneous varieties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 307 - 315
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ausina, C. Bivià and Nuño–Ballesteros, J. J.The deformation multiplicity of a map germ with respect to Boardman symbol. Proc. Roy. Soc. Edinburgh. 131A (2001), 1003–1022.CrossRefGoogle Scholar
[2]Bruce, J. W., Kirk, N. P. and du Plessis, A. A.Complete transversals and the classification of singularities. Nonlinearity 10 (1997), 253–275.CrossRefGoogle Scholar
[3]Briançon, J., Galligo, A. and Granger, M.Déformations équisinguliéres des germes de courbes gauches réduites. Mém. Soc. Math. France (N.S.) 69 no. 1 (1980/81).Google Scholar
[4]Bruce, J. W. and Roberts, M.Critical points of functions on analytic varieties. Topology 27 (1988), no. 1, 57–90.CrossRefGoogle Scholar
[5]Brasselet, J. P., Tráng, Lê Dũng and Seade, J. Euler obstruction and indices of vector fields. Topology (2000).CrossRefGoogle Scholar
[6]Damon, J.Topological triviality and versality for subgroups of $\cal{A}$ and $\cal{K}$: II. Sufficient conditions and applications. Nonlinearity 5 (1992), 373–412.CrossRefGoogle Scholar
[7]Gaffney, T.Integral closure of modules. Invent. Math. 107 (1992), 301–322.CrossRefGoogle Scholar
[8]Greuel, G. M.Constant Milnor Number implies Constant Multiplicity for Quasihomogeneous Singularities. Manuscripta Math. 56 (1986), 159–166.CrossRefGoogle Scholar
[9]Tráng, Lê DũngLe concept de singularité isolée de fonction analytique. Adv. Stud. Pure Math. 8 (1986), 215–227.Google Scholar
[10]Milnor, J.Singular points of complex hypersurfaces. Ann. of Math. Stud. 61 (Princeton University Press, 1968).Google Scholar
[11]Grulha, N. G. JrThe Euler obstruction and Bruce–Roberts' Milnor number. Quat. J. Math. 60 (3), (2009), 291–302.CrossRefGoogle Scholar
[12]O'Shea, D. B.Topologically trivial deformations of isolated quasihomogeneous hypersurface singularities are equimultiple. Proc. Amer. Math. Soc. 101, no. 2, (1987), 260–262.CrossRefGoogle Scholar
[13]Nuño–Ballesteros, J. J., Oréfice, B. and Tomazella, J. N.The Bruce–Roberts number of a function on a weighted homogeneous hypersurface. Quat. J. Math. 64 (1), (2013), 269–280.CrossRefGoogle Scholar
[14]Ruas, M. A. S. and Tomazella, J. N.An infinitesimal criterion for topological triviality of families of sections of analytic varieties. Adv. Stud. Pure Math. 43 (2006), 421–436.CrossRefGoogle Scholar
[15]Ruas, M. A. S. and Tomazella, J. N.Topological triviality of families of functions on analytic varieties. Nagoya Math. J. 175 (2004), 39–50.CrossRefGoogle Scholar
[16]Teissier, B.Cycles évanescents: sections planes et conditions de Whitney. Astérisque 7–8 (1973), 285–362.Google Scholar
[17]Tomazella, J. N. Seções de Variedades Analíticas. Ph.D. thesis. ICMC-USP (1999).Google Scholar