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The Fukui numerical set is known as a blow-analytic invariant for real analytic function germs. Taking into account the similarity between real blow-analytic properties and complex topological ones, we may ask if the Fukui numerical set is a topological invariant for complex analytic function germs. In this note we discuss the problem and give some related questions.
Wydawca
Czasopismo
Rocznik
Tom
Strony
285--302
Opis fizyczny
Bibliogr. 36 poz.
Twórcy
autor
autor
- Department of Mathematics Hyogo University of Teacher Education 942-1 Shimokume, Kato Hyogo 673-1494, Japan, koike@hyogo-u.ac.jp
Bibliografia
- [1] V. I. Arnol’d, Normal forms of functions in a neighbourhood of a degenerate critical point, Uspekhi Mat. Nauk 29-2 (1974), 11–49, = Russian Math. Surveys 29-2 (1974), 10–50.
- [2] V. I. Arnol’d, S. M. Guzein-Zard, A. N. Varchenko, Singularities of Differentiable Maps, Volume II, Birkhäuser, Boston, MA, 1988.
- [3] E. Artal Bartolo, Forme de Seifert des singularités de surface, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 689–692.
- [4] E. Artal Bartolo, P. Cassou-Nogues, I. Luengo, A. Melle Hernádez, The Denef-Loeser zeta function is not a topological invariant, J. London Math. Soc. (2) 65 (2002), 45–64.
- [5] J. Brian輟n, Ph. Maisonobe, M. Merle, Localisations de systèmes differentiels, tratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531–550.
- [6] P. Du Bois, F. Michel, The integral Seifert form does not determine the topological type of plane curve germs, J. Algebraic Geom. 3 (1994), 1–38.
- [7] J. Briançn, J. P. Speder, La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci. Paris Sér. I Math. 280 (1975), 365–367.
- [8] J. Damon, T. Gaffney, Topological triviality of deformations of functions and Newton filtrations, Invent. Math. 72 (1983), 335-358.
- [9] A. H. Durfee, Fibered knots and algebraic singularities, Topology 13 (1974), 47–59.
- [10] T. Fukui, E. Yoshinaga, The modified analytic trivialization of family of real analytic functions, Invent. Math. 82 (1985), 467–477.
- [11] T. Fukui, Seeking invariants for blow-analytic equivalence, Compositio Math. 105 (1997), 95–107.
- [12] T. Fukui, S. Koike, T.-C. Kuo, Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities (T. Fukuda, T. Fukui, S. Izumiya and S. Koike, ed), Pitman Res. Notes Math. Ser. 381 (1998), 8–29.
- [13] T. Fukui, L. Paunescu, Modified analytic trivialization for weighted homogeneous function-germs, J. Math. Soc. Japan 52 (2000), 433–446.
- [14] T. Fukui, L. Paunescu, On blow-analytic equivalence, in “Arc-Spaces and Additive Invariants in Real Algebraic Geometry”, Proceedings of Winter School “Real Algebraic and Analytic Geometry and Motivic Integration”, Aussoie 2003, M. Coste, K. Kurdyka and A. Parusiński eds, Panoramas et Syntheses 24 (2008), SMF, pp. 87–125.
- [15] A. M. Gabrielov, Intersection matrices for certain singularities, Funktsional. Anal. i Prilozhen 7 (1973), 18–32.
- [16] S. Izumi, S. Koike, T.-C. Kuo, Computations and stability of the Fukui invariant, Compositio Math. 130 (2002), 49–73.
- [17] M. Kato, A classification of simple spinnable structures on a 1-connected Alexander manifold, J. Math. Soc. Japan 26 (1974), 454-463.
- [18] H. King, Topological type of isolated critical points, Ann. Math. 107 (1978), 385–397.
- [19] H. King, Topological types in families of germs, Invent. Math. 62 (1980), 1–13.
- [20] S. Koike, A. Parusiński, Motivic-type invariants of blow-analytic equivalence, Ann. Inst. Fourier 53 (2003), 2061–2104.
- [21] S. Koike, A. Parusiński, Blow-analytic equivalence of two variable real analytic function germs, to appear in Journal of Algebraic Geometry.
- [22] S. Koike, A. Parusiński, Equivalence relations for two variable real analytic function germs, arXiv:0801.2650.
- [23] T.-C. Kuo, Y. C. Lu, On analytic function germs of complex variables, Topology 16 (1977), 299–310.
- [24] T.-C. Kuo, The modified analytic trivialization of singularities, J. Math. Soc. Japan 32 (1980), 605–614.
- [25] T.-C. Kuo, On classification of real singularities, Invent. Math. 82 (1985), 257–262.
- [26] K. Kurdyka, L. Paunescu, Arc-analytic roots of analytic functions are Lipschitz, Proc. Amer. Math. Soc. 132 (2004), 1693–1702.
- [27] M. Oka, On the weak simultaneous resolution of a negligible truncation of the Newton boundary, Contemp. Math. 90 (1989), 199–210.
- [28] A. Parusiński, Topological triviality of μ-constant deformations of type f(x) + tg(x), Bull. London Math. Soc. 31 (1999), 686–692.
- [29] A. Parusiński, Limits of tangent spaces to fibres and the wf condition, Duke Math. J. 72 (1993), 99–108.
- [30] A. Parusiński, A criterion for the topological equivalence of two variable complex analytic function germs, Proceedings of the Japan Academy, Series A, Math. Sci. 84 no. 8 (2008), 147–150.
- [31] K. Sakamoto, The Seifert matrices of Milnor fiberings defined by holomorphic functions, J. Math. Soc. Japan 26 (1974), 714–721.
- [32] B. Teissier, Cycles évanescents, sections planes et conditions de Whitney, Singularitér a Cargese, Asterisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, 285–362.
- [33] B. Teissier, Resolution simultanée I, II, Séinaire sur les singularités des surfaces (M. Demazure, H. C. Pinkham, B. Teissier ed), Lecture Notes in Mathematics 777 (1980), Springer-Verlag Berlin, pp. 71 –146.
- [34] O. Zariski, On the topology of algebroid singularities, Amer. J. Math. 54 (1932), 453–465.
- [35] O. Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491.
- [36] O. Zariski, Contribution to the problem of equisingularity, Collected papers, Vol. IV, MIT Press, Cambridge, London, 1978, 159–237.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA5-0027-0043