On the topological triviality along moduli of deformations of $J_{k,0}$ singularities

• Autorzy: Piotr Jaworski
• Języki publikacji: EN

Abstrakty

EN

It is well known that versal deformations of nonsimple singularities depend on moduli. However they can be topologically trivial along some or all of them. The first step in the investigation of this phenomenon is to determine the versal discriminant (unstable locus), which roughly speaking is the obstacle to analytic triviality. The next one is to construct continuous liftable vector fields smooth far from the versal discriminant and to integrate them. In this paper we extend the results of J. Damon and A. Galligo, concerning the case of the Pham singularity ($J_{3,0}$ in Arnold's classification) (see [2, 3, 4]), and deal with deformations of general $J_{k,0}$ singularities.

Słowa kluczowe

EN

$J_{k,0}$ singularities; topological trivialization; moduli of singularities

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 75
• Numer: 3
• Strony: 193-212

Daty

wydano

2000

otrzymano

1997-10-27

poprawiono

2000-09-18

Twórcy

autor

Piotr Jaworski

• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

• [1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1, Birkhäuser, 1985.
• [2] J. Damon, On the Pham example and the universal topological stratification of singularities, in: Singularities, Banach Center Publ. 20, PWN-Polish Scientific Publ., Warszawa, 1988, 161-167.
• [3] J. Damon, A-equivalence and the equivalence of sections of images and discriminants, in: Singularity Theory and its Applications, Part 1 (Coventry 1988/1989), Lecture Notes in Math. 1492, Springer, Berlin, 1991, 93-121.
• [4] J. Damon and A. Galligo, Universal topological stratification for the Pham example, Bull. Soc. Math. France 121 (1993), 153-181.
• [5] A. du Plessis and C. T. C. Wall, Topological stability, in: Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser. 201, Cambridge Univ. Press, Cambridge, 1994, 351-362.
• [6] A. du Plessis and C. T. C. Wall, The Geometry of Topological Stability, London Math. Soc. Monogr. (N.S.) 9, Oxford University Press, New York, 1995.
• [7] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982.
• [8] P. Jaworski, Decompositions of hypersurface singularities of type $J_{k,0}$, Ann. Polon. Math. 59 (1994), 117-131.
• [9] P. Jaworski, On the versal discriminant of the $J_{k,0}$ singularities, ibid. 63 (1996), 89-99.
• [10] P. Jaworski, On the uniqueness of the quasihomogeneity, in: Geometry and Topology of Caustics - Caustics '98, Banach Center Publ. 50, Inst. Math., Polish Acad. Sci., Warszawa, 1999, 163-167.
• [11] E. Looijenga, Semi-universal deformation of a simple elliptic hypersurface singularity, I: Unimodularity, Topology 16 (1977), 257-262.
• [12] K. Wirthmüller, Universell topologische triviale Deformationen, Ph.D. thesis, Univ. of Regensburg, 1979.