Local topological algebraicity of analytic function germs
Authors:
Marcin Bilski, Adam Parusiński and Guillaume Rond
Journal:
J. Algebraic Geom. 26 (2017), 177-197
DOI:
https://doi.org/10.1090/jag/667
Published electronically:
June 20, 2016
MathSciNet review:
3570585
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Abstract |
References |
Additional Information
Abstract: Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.
References
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- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
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- S. Łojasiewicz, Ensembles semi-analytiques, preprint IHES (1965). Available from https://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf.
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- Tadeusz Mostowski, Topological equivalence between analytic and algebraic sets, Bull. Polish Acad. Sci. Math. 32 (1984), no. 7-8, 393–400 (English, with Russian summary). MR 782754
- A. Płoski, Note on a theorem of M. Artin, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 1107–1109 (English, with Russian summary). MR 367257
- Dorin Popescu, General Néron desingularization, Nagoya Math. J. 100 (1985), 97–126. MR 818160, DOI https://doi.org/10.1017/S0027763000000246
- Ronan Quarez, The Artin conjecture for ${\bf Q}$-algebras, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 229–263. MR 1605646
- Mark Spivakovsky, A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms, J. Amer. Math. Soc. 12 (1999), no. 2, 381–444. MR 1647069, DOI https://doi.org/10.1090/S0894-0347-99-00294-5
- Richard G. Swan, Néron-Popescu desingularization, Algebra and geometry (Taipei, 1995) Lect. Algebra Geom., vol. 2, Int. Press, Cambridge, MA, 1998, pp. 135–192. MR 1697953
- Bernard Teissier, Résultats récents sur l’approximation des morphismes en algèbre commutative (d’après André, Artin, Popescu et Spivakovsky), Astérisque 227 (1995), Exp. No. 784, 4, 259–282 (French, with French summary). Séminaire Bourbaki, Vol. 1993/94. MR 1321650
- A. N. Varchenko, Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings, Math. USSR Izvestiya 6 (1972), 949–1008.
- A. N. Varchenko, The relation between topological and algebro-geometric equisingularities according to Zariski, Funkcional. Anal. Appl. 7 (1973), 87–90.
- A. N. Varchenko, Algebro-geometrical equisingularity and local topological classification of smooth mappings, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 427–431. MR 0424805
- 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
- Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509 (2000a:14067)
- Jacek Bochnak and Wojciech Kucharz, Local algebraicity of analytic sets, J. Reine Angew. Math. 352 (1984), 1–14. MR 758691 (86e:32010)
- Javier Fernández de Bobadilla, On algebraic representatives of topological types of analytic hypersurface germs, J. Algebraic Geom. 21 (2012), no. 4, 789–797. MR 2957697, DOI https://doi.org/10.1090/S1056-3911-2011-00575-4
- 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 (55 \#9151)
- Heisuke Hironaka, Stratification and flatness, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 199–265. MR 0499286 (58 \#17187)
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 0173675 (30 \#3885)
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181 (33 \#7330)
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 0238860
- K. Kurdyka and G. Raby, Densité des ensembles sous-analytiques, Ann. Inst. Fourier (Grenoble) 39 (1989), no. 3, 753–771 (French, with English summary). MR 1030848 (90k:32026)
- S. Łojasiewicz, Ensembles semi-analytiques, preprint IHES (1965). Available from https://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf.
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344 (82i:13003)
- Tadeusz Mostowski, Topological equivalence between analytic and algebraic sets, Bull. Polish Acad. Sci. Math. 32 (1984), no. 7-8, 393–400 (English, with Russian summary). MR 782754 (86h:32010)
- A. Płoski, Note on a theorem of M. Artin, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 1107–1109 (English, with Russian summary). MR 0367257 (51 \#3499)
- Dorin Popescu, General Néron desingularization, Nagoya Math. J. 100 (1985), 97–126. MR 818160 (87f:13019)
- Ronan Quarez, The Artin conjecture for $\textbf {Q}$-algebras, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 229–263. MR 1605646 (99f:13010)
- Mark Spivakovsky, A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms, J. Amer. Math. Soc. 12 (1999), no. 2, 381–444. MR 1647069 (99j:13008), DOI https://doi.org/10.1090/S0894-0347-99-00294-5
- Richard G. Swan, Néron-Popescu desingularization, Algebra and geometry (Taipei, 1995) Lect. Algebra Geom., vol. 2, Int. Press, Cambridge, MA, 1998, pp. 135–192. MR 1697953 (2000h:13006)
- Bernard Teissier, Résultats récents sur l’approximation des morphismes en algèbre commutative (d’après André, Artin, Popescu et Spivakovsky), Astérisque 227 (1995), Exp. No. 784, 4, 259–282 (French, with French summary). Séminaire Bourbaki, Vol. 1993/94. MR 1321650 (96c:13023)
- A. N. Varchenko, Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings, Math. USSR Izvestiya 6 (1972), 949–1008.
- A. N. Varchenko, The relation between topological and algebro-geometric equisingularities according to Zariski, Funkcional. Anal. Appl. 7 (1973), 87–90.
- A. N. Varchenko, Algebro-geometrical equisingularity and local topological classification of smooth mappings, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 427–431. MR 0424805 (54 \#12763)
- 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 (32 \#5924)
Additional Information
Marcin Bilski
Affiliation:
Department of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
Email:
marcin.bilski@im.uj.edu.pl
Adam Parusiński
Affiliation:
Université Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06108 Nice, France
Email:
adam.parusinski@unice.fr
Guillaume Rond
Affiliation:
Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France
MR Author ID:
759916
Email:
guillaume.rond@univ-amu.fr
Received by editor(s):
February 24, 2014
Received by editor(s) in revised form:
April 11, 2014, May 20, 2014, and August 15, 2014
Published electronically:
June 20, 2016
Additional Notes:
The authors were partially supported by ANR project STAAVF (ANR-2011 BS01 009). The first author is partially supported by the NCN grant 2014/13/B/ST1/00543. The third author is partially supported by the ANR project SUSI (ANR-12-JS01-0002-01)
Article copyright:
© Copyright 2016
University Press, Inc.
