On algebraic representatives of topological types of analytic hypersurface germs
Author:
Javier Fernández de Bobadilla
Journal:
J. Algebraic Geom. 21 (2012), 789-797
DOI:
https://doi.org/10.1090/S1056-3911-2011-00575-4
Published electronically:
November 28, 2011
MathSciNet review:
2957697
Full-text PDF
Abstract |
References |
Additional Information
Abstract: A question of B. Teissier, inspired by a previous problem of R. Thom, asks whether for any germ of complex analytic hypersurface there exists a germ of complex algebraic hypersurface with the same topological type. Up to now, only the case of germs with an isolated singularity is known. In this article we answer in the affirmative, the case of the $1$-dimensional singular set and also for embedded topological types.
References
- M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 268188
- Christopher G. Gibson, Klaus Wirthmüller, Andrew A. du Plessis, and Eduard J. N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Mathematics, Vol. 552, Springer-Verlag, Berlin-New York, 1976. MR 0436203
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI https://doi.org/10.2307/1970547
- B. Teissier, Introduction to equisingularity problems, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 593–632. MR 0422256
- R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240–284 (French). MR 239613, DOI https://doi.org/10.1090/S0002-9904-1969-12138-5
- R. Thom. In: Problems concerning manifolds. Manifolds, Amsterdam 1970. Springer Lecture Notes in Mathematics 197.
- Hassler Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, pp. 205–244. MR 0188486
References
- M. Artin. Algebraic approximation of structures over complete local rings. Publ. Math. I.H.E.S. 36 (1969), 23-58. MR 0268188 (42:3087)
- C.G. Gibson, E. Looijenga, A. du Plessis, K. Wirthmüller. Topological stability of smooth mappings. Lecture Notes in Mathematics 552, Springer (1976). MR 0436203 (55:9151)
- H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109-203, 205-326. MR 0199184 (33:7333)
- B. Teissier. Introduction to equisingularity problems. Algebraic geometry, Humboldt State Univ., Arcata, Calif., 1974. Proc. Sympos. Pure Math., 29 (1975), 593-632. MR 0422256 (54:10247)
- R. Thom. Ensembles et morphismes stratifiés. Bull. Amer. Math. Soc. 75 (2) (1969), 240-284. MR 0239613 (39:970)
- R. Thom. In: Problems concerning manifolds. Manifolds, Amsterdam 1970. Springer Lecture Notes in Mathematics 197.
- H. Whitney. Local properties of analytic varieties. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse). Princeton Univ. Press, Princeton, N.J. (1965), 205-244. MR 0188486 (32:5924)
Additional Information
Javier Fernández de Bobadilla
Affiliation:
ICMAT (CSIC-Complutense-Autónoma- Carlos III), Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense, Plaza de Ciencias 3, 28040 Madrid, Spain
Email:
javier@mat.csic.es
Received by editor(s):
March 2, 2010
Received by editor(s) in revised form:
July 18, 2010
Published electronically:
November 28, 2011
Additional Notes:
Research partially supported by the ERC Starting Grant project TGASS and by Spanish Contract MTM2007-67908-C02-02. The author thanks the Faculty de Ciencias Matemáticas of the Universidad Complutense de Madrid for excellent working conditions.
