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Global Euler obstruction, global Brasselet numbers and critical points

Part of: Theory of singularities and catastrophe theory Analytic spaces Local theory

Published online by Cambridge University Press:  14 May 2019

Nicolas Dutertre  and
Nivaldo G. Grulha Jr.
Nicolas Dutertre
Affiliation:
Laboratoire angevin de recherche en mathématiques, LAREMA, UMR6093, CNRS, UNIV. Angers, SFR MathStic, 2 Bd Lavoisier 49045 Angers Cedex 01, France ([email protected])
Nivaldo G. Grulha Jr.
Affiliation:
Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação - USP Av. Trabalhador São-carlense, 400 - Centro, Caixa Postal: 668 - CEP: 13560-970 - São Carlos - SP - Brazil ([email protected])
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Abstract

Let X ⊂ ℂn be an equidimensional complex algebraic set and let f: X → ℂ be a polynomial function. For each c ∈ ℂ, we define the global Brasselet number of f at c, a global counterpart of the Brasselet number defined by the authors in a previous work, and the Brasselet number at infinity of f at c. Then we establish several formulas relating these numbers to the topology of X and the critical points of f.

Keywords

Euler obstructionglobal Euler obstructionconstructible functionsMorsefications

MSC classification

Primary: 14B05: Singularities
Secondary: 32C18: Topology of analytic spaces 58K45: Singularities of vector fields, topological aspects
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2503 - 2534
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Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Artal, E., Luengo, I. and Melle, A.. On the topology of a generic fibre of a polynomial function. Comm. Algebr. 28 (2000), 1767–1787.Google Scholar
2Brasselet, J. P. and Schwartz, M. H.. Sur les classes de Chern d'un ensemble analytique complexe. Astérisque 82–83 (1981), 93–147.Google Scholar
3Brasselet, J. P., Lê, D. T. and Seade, J.. Euler obstruction and indices of vector fields. Topology 6 (2000), 1193–1208.CrossRefGoogle Scholar
4Brasselet, J. P., Massey, D., Parameswaran, A. and Seade, J.. Euler obstruction and defects of functions on singular varieties. J. London Math. Soc. (2) 70 (2004), 59–76.CrossRefGoogle Scholar
5Broughton, S. A.. Milnor number and the topology of polynomial hypersurfaces. Invent. Math. 92 (1988), 217–241.CrossRefGoogle Scholar
6Brylinski, J., Dubson, A. and Kashiwara, M.. Formule de l'indice pour modules holonomes et obstruction d'Euler locale. C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 573–576.Google Scholar
7Dias, L. R. G.. On regularity conditions at infinity. J. Singul. 10 (2014), 54–66.Google Scholar
8Dias, L. R. G., Ruas, M. A. S. and Tibăr, M.. Regularity at infinity of real mappings and a Morse-Sard theorem. J. Topol. 5 (2012), 323–340.CrossRefGoogle Scholar
9Dubson, A.. Classes caractéristiques des variétés singulières. C. R. Acad. Sci. Paris Sér. A-B 287 (1978), 237–240.Google Scholar
10Dubson, A.. Calcul des invariants numériques des singularités et applications, Sonderforschungsbereich 40 Theoretische Mathematik. Universitaet Bonn (1981).Google Scholar
11Dutertre, N.. Euler obstruction and Lipschitz-Killing curvatures. Israel J. Math. 213 (2016), 109–137.Google Scholar
12Dutertre, N. and Grulha, N. G. Jr.Lê-Greuel type formula for the Euler obstruction and applications. Adv. Math. 251 (2014), 127–146.Google Scholar
13Dutertre, N. and Grulha, N. G., Jr. Some notes on the Euler obstruction of a function. J. Singul. 10 (2014), 82–91.Google Scholar
14Goresky, M. and Mac-Pherson, R.. Stratified Morse theory (Berlin: Springer-Verlag, 1988).CrossRefGoogle Scholar
15Grulha, N.G. Jr., The Euler obstruction and Bruce-Roberts Milnor number. Q. J. Math. 60 (2009), 291–302.CrossRefGoogle Scholar
16Hà, H. V. and Lê, D. T.. Sur la topologie des polynômes complexes. Acta Math. Vietnam. 9 (1984), 21–32.Google Scholar
17Lê, D. T.. Vanishing cycles on complex analytic sets. Proc. Sympos., Res. Inst. Math. Sci., Kyoto, Univ. Kyoto, 1975. Sûrikaisekikenkyûsho Kókyûroku, vol. 266, pp. 299–318 (1976).Google Scholar
18Lê, D. T.. Complex analytic functions with isolated singularities. J. Algebr. Geom. 1 (1992), 83–99.Google Scholar
19Lê, D. T. and Teissier, B.. Variétés polaires locales et classes de Chern des variétés singulières. Ann. Math. 114 (1981), 457–491.Google Scholar
20Lê, D. T. and Weber, C.. Polynômes à fibres rationnelles et conjecture jacobienne à 2 variables. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 581–584.Google Scholar
21Loeser, F.. Formules intégrales pour certains invariants locaux des espaces analytiques complexes. Comment. Math. Helv. 59 (1984), 204–225.CrossRefGoogle Scholar
22Mac-Pherson, R. D.. Chern classes for singular algebraic varieties. Ann. Math. 100 (1974), 423–432.CrossRefGoogle Scholar
23Massey, D. B.. Hypercohomology of Milnor fibres. Topology 35 (1996), 969–1003.CrossRefGoogle Scholar
24McCrory, C. and Parusinski, A.. Algebraically constructible functions. Ann. Sci. École Norm. Sup. (4) 30 (1997), 527–552.CrossRefGoogle Scholar
25Parusiński, A.. On the bifurcation set of complex polynomial with isolated singularities at infinity. Compositio Math. 97 (1995), 369–384.Google Scholar
26Schürmann, J.. Topology of singular spaces and constructible sheaves. Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series),vol. 63, (Basel: Birkhauser Verlag, 2003).CrossRefGoogle Scholar
27Schürmann, J. and Tibăr, M.. Index formula for MacPherson cycles of affine algebraic varieties. Tohoku Math. J. (2) 62 (2010), 29–44.CrossRefGoogle Scholar
28Seade, J., Tibăr, M. and Verjovsky, A.. Global Euler obstruction and polar invariants. Math. Ann. 333 (2005), 393–403.Google Scholar
29Seade, J., Tibăr, M. and Verjovsky, A.. Milnor Numbers and Euler obstruction. Bull. Braz. Math. Soc. (N.S) 36 (2005), 275–283.CrossRefGoogle Scholar
30Siersma, D.. A bouquet theorem for the Milnor fibre. J. Algebr. Geom. 4 (1995), 51–66.Google Scholar
31Siersma, D. and Tibăr, M.. Singularities at infinity and their vanishing cycles. Duke Math. J. 80 (1995), 771–783.Google Scholar
32Siersma, D. and Tibăr, M.. On the vanishing cycles of a meromorphic function on the complement of its poles. Real and complex singularities. Contemp. Math., vol. 354, pp. 277–289 (Providence, RI: Amer. Math. Soc., 2004).Google Scholar
33Siersma, D. and Tibăr, M.. Gauss-Bonnet defect of complex affine hypersurfaces. Bull. Sci. Math. 130 (2006), 110–122.CrossRefGoogle Scholar
34Suzuki, M.. Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace ℂ2. J. Math. Soc. Japan 26 (1974), 241–257.Google Scholar
35Tibăr, M.. Bouquet decomposition of the Milnor fibre. Topology 35 (1996), 227–241.CrossRefGoogle Scholar
36Tibăr, M.. Asymptotic equisingularity and topology of complex hypersurfaces. Internat. Math. Res. Notices 18 (1998), 979–990.CrossRefGoogle Scholar
37Tibăr, M.. Regularity at infinity of real and complex polynomial functions, singularity theory (Liverpool, 1996). London Math. Soc. Lecture Note Ser., vol. 263, pp. 249–264 (Cambridge: Cambridge Univ. Press, 1999).Google Scholar
38Tibăr, M.. Singularities and topology of meromorphic functions, trends in singularities, pp. 223–246 (Basel: Trends Math., Birkhäuser, 2002).Google Scholar
39Tibăr, M.. Vanishing cycles of pencils of hypersurfaces. Topology 43 (2004), 619–633.CrossRefGoogle Scholar
40Tibăr, M.. Duality of Euler data for affine varieties, singularities in geometry and topology 2004, Adv. Stud. Pure Math., vol. 46, pp. 251–257 (Tokyo: Math. Soc. Japan, 2007).CrossRefGoogle Scholar
41Tibăr, M.. Polynomials and vanishing cycles, Cambridge Tracts in Mathematics, vol. 170 (Cambridge: Cambridge University Press, 2007).CrossRefGoogle Scholar
42Viro, O.. Some integral calculus based on Euler characteristic, topology and geometry, Rohlin Seminar. Lecture Notes in Math., vol. 1346, pp. 127–138 (Berlin: Springer, 1988).CrossRefGoogle Scholar