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Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 905–930 | Cite as

The local Euler obstruction and topology of the stabilization of associated determinantal varieties

  • Terence Gaffney
  • Nivaldo G. GrulhaJr.Email author
  • Maria A. S. Ruas
Article
  • 69 Downloads

Abstract

This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern–Schwartz–MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney’s \(m_{d}\) multiplicity.

Keywords

Local Euler obstruction Chern–Schwartz–MacPherson class Generic determinantal varieties Essentially isolated determinantal singularity Stabilization 

Mathematics Subject Classification

Primary 14C17 32S15 55S35 Secondary 14J17 58K05 32S60 

Notes

Acknowledgements

The authors are grateful to Jonathan Mboyo Esole, Thiago de Melo, Otoniel Silva, Jawad Snoussi and Xiping Zhang for their careful reading of the first draft of this paper and for their suggestions. We also thank the referee for his/her careful reading and for his/her suggestions and corrections to the final version of this paper. The first author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brazil, Grant PVE 401565/2014-9. The second author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP, Brazil, Grants 2015/16746-7 and 2017/09620-2 and Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brazil, Grants 474289/2013-3, 303641/2013-4 and 303046/2016-3 . The third author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP, Brazil, Grant 2014/00304-2 and Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brazil, Grant 305651/2011-0. This paper was written while the second author was visiting Northeastern University, Boston, USA. During this period the first author had also visited the Universidade de São Paulo at São Carlos, Brazil, and we would like to thank these institutions for their hospitality.

References

  1. 1.
    Arbarello, E., Cornualba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, Vol. I, volume 267 of Fundamental Principles of Mathematical Sciences. Springer, New York (1985)Google Scholar
  2. 2.
    Brasselet, J.-P.: Local Euler obstruction, old and new, XI Brazilian Topology Meeting (Rio Claro, 1998), pp. 140–147. World Sci. Publishing, River Edge, NJ (2000)Google Scholar
  3. 3.
    Brasselet, J.-P., Lê, D.T., Seade, J.: Euler obstruction and indices of vector fields. Topology 39(6), 1193–1208 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brasselet, J.-P., Schwartz, M.-H.: Sur les classes de Chern d’un ensemble analytique complexe. Astérisque 82–83, 93–147 (1981)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bruns, W., Vetter, U.: Determinantal Rings. Springer, New York (1998)zbMATHGoogle Scholar
  6. 6.
    Buchsbaum, D.A., Rim, D.S.: A generalized Koszul complex. II. Depth and multiplicity. Trans. AMS 111, 197–224 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chong, Chuan Chen, Khee Meng, Koh: Principles and Techniques in Combinatorics. World Scientific, River Edge (1992)CrossRefzbMATHGoogle Scholar
  8. 8.
    Ebeling, W., Gusein-Zade, S.M.: On indices of \(1\)-forms on determinantal singularities. Proc. Steklov Inst. Math. 267(1), 113–124 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ebeling, W., Gusein-Zade, S.M.: Radial index and Euler obstruction of a 1-form on a singular variety. Geom. Dedicata 113, 231–241 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Forstneric̆, F.: Holomorphic flexibility properties of complex manifolds. Am. J. Math. 128(1), 239–270 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gaffney, T.: Polar methods, invariants of pairs of modules and equisingularity. In: Gaffney, T., Ruas, M. (eds.) Real and Complex Singularities (São Carlos, 2002), pp. 113–136. Contemp. Math., 354, Amer. Math. Soc., Providence, RI, June (2004)Google Scholar
  12. 12.
    Gaffney, T.: The Multiplicity polar theorem. arXiv:math/0703650v1 [math.CV]
  13. 13.
    Gaffney, T.: Polar multiplicities and equisingularity of map germs. Topology 32, 185–223 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gaffney, T.: Integral closure of modules and Whitney equisingularity. Inventiones 107, 301–22 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gaffney, T., Rangachev, A.: Pairs of modules and determinatal isolated singularities. arXiv:1501.00201 [math.CV]
  16. 16.
    Gaffney, T., Ruas, M. A. S.: Equisingularity and EIDS. arXiv:1602.00362 [math.CV]
  17. 17.
    Goresky, M., MacPherson, R.: Stratified Morse theory. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gonzalez-Sprinberg, G.: Calcul de l’invariant local d’Euler pour les singulariteś quotient de surfaces, C. R. Acad. Sci. Paris Ser. A-B 288, A989–A992 (1979)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gonzalez-Sprinberg, G.: L’ obstruction locale d’Euler et le théorème de MacPherson, Séminaire de géométrie analytique de l’E.N.S. 1978–1979Google Scholar
  20. 20.
    Jorge Pérez, V.H., Saia, M.J.: Euler obstruction, polar multiplicities and equisingularityof map germs in \({\cal{O}}(n,p), n<p.\). Int. J. Math 17(8), 887–903 (2006)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kaliman, S.H., Zaidenberg, M.: A transversality theorem for holomorphic mappings and stability of Eisenman–Kobayashi measures. Trans. Am. Math. Soc. 348(2), 661–672 (1996)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kleiman, S.: The transversality of general translate. Compos. Math. 28, 287–297 (1974)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kleiman, S., Thorup, A.: A geometric theory of the Buchsbaum–Rim multiplicity. J. Algebra 167, 168–231 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lê, D.T.: Vanishing cycles on complex analytic sets, Proc. Sympos., Res. Inst. Math. Sci., Kyoto, Univ. Kyoto, 1975. Sûrikaisekikenkyûsho Kókyûroku 266, 299–318 (1976)Google Scholar
  25. 25.
    Lê, D.T.: Complex analytic functions with isolated singularities. J. Algebraic Geom. 1(1), 83–99 (1992)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Lê, D.T., Teissier, B.: Variétés polaires Locales et classes de Chern des variétés singulières. Ann. Math. 114, 457–491 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lê, D.T., Teissier, B.: Limites d’espaces tangents en géométrie analytique. Comm. Math. Helv. 63(4), 540–578 (1988)zbMATHGoogle Scholar
  28. 28.
    MacPherson, R.: Chern classes for singular algebraic varieties. Ann. Math. 100, 423–432 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Matsui, Y., Takeuchi, K.: A geometric degree formula for A-discriminants and Euler obstructions of toric varieties. Adv. Math. 226, 2040–2064 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Piene, R.: Polar classes of singular varieties. Ann. Sci. École Norm. Sup 11(4), 247–276 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Piene, R.: Cycles polaires et classes de Chern pour les varoétés projectives singulières, Introduction à la théorie des singularités, II, Travaux en Cours, (37), pp. 4–34. Hermann, Paris (1988)Google Scholar
  32. 32.
    Schurmann, J., Tibar, M.: Index formula for MacPherson cycles of affine algebraic varieties. Tohoku Math. J. (2) 62(1), 29–44 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Siersma, D.: A bouquet theorem for the Milnor fibre. J. Algebraic Geom. 4(1), 51–66 (1995)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Siesquén, N.C.C.: Euler obstruction of essentially isolated determinantal singularities. Topol. Appl. 234, 166–177 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Serre, J.P.: Algebre Locale. Multiplicities. Lecture Notes in Mathematics. 11 Springer, Berlin-New York (1965)Google Scholar
  36. 36.
    Teissier, B.: Variétés polaires, II, Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Rábida, 1981), pp. 314–491. Lecture Notes in Math, vol. 961. Springer, Berlin (1982)Google Scholar
  37. 37.
    Thom, R.: Un lemme sur les applications differentiables. Bol. Soc. Mat. Mexicana 2(1), 59–71 (1956)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Tibăr, M.: Bouquet decomposition of the Milnor fibre. Topology 35(1), 227–241 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Trivedi, S.: Stratified transversality of holomorphic maps. Int. J. Math 24(13), 1350106–12 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zhang, X.: Chern–Schwartz–MacPherson class of determinantal varieties. arXiv:1605.05380 [math.AG]

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Terence Gaffney
    • 1
  • Nivaldo G. GrulhaJr.
    • 2
    Email author
  • Maria A. S. Ruas
    • 2
  1. 1.Department of MathematicsNortheastern UniversityBostonUSA
  2. 2.Instituto de Ciências Matemáticas e de Computação-USPSão CarlosBrazil

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