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The Euler obstruction of a function on a determinantal variety and on a curve

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Abstract

Given an analytic function germ f: (X, 0) → C on an isolated determinantal singularity or on a reduced curve, we present formulas relating the local Euler obstruction of f to the vanishing Euler characteristic of the fiber X ∩ f -1(0) and to the Milnor number of f. Restricting ourselves to the case where X is a complete intersection, we obtain an easy way to calculate the local Euler obstruction of f as the difference between the dimension of two algebras.

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References

  1. E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris. Geometry of Algebraic Curves. Springer-Verlag (1985).

    Book  MATH  Google Scholar 

  2. C. Biviá-Ausina and J.J. Nuño-Ballesteros. The deformation multiplicity of a map germ with respect to a Boardman symbol. Proc. Roy. Soc. Edinburgh Sect., 131A(5) (2001), 1003–1022.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.-P. Brasselet, D.T. Lê and J. Seade. Euler obstruction and indices of vector fields. Topology, 39(6) (2000), 1193–1208.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.-P. Brasselet, D. Massey, A.J. Parameswaran and J. Seade. Euler obstruction and defects of functions on singular varieties. J. London Math. Soc. (2), 70(1) (2004), 59–76.

    Article  MathSciNet  MATH  Google Scholar 

  5. J.-P. Brasselet and M.-H. Schwartz. Sur les classes de Chern d’un ensemble analytique complexe. Astérisque, 82-83 (1981), 93–147.

    MathSciNet  MATH  Google Scholar 

  6. E. Brieskorn and G.M. Greuel. Singularities of complete intersections. Manifolds- Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), Univ. Tokyo Press, (1975), 123–129.

    MATH  Google Scholar 

  7. R.O. Buchweitz and G.M. Greuel. The Milnornumber and deformations of complex curve singularities. Invent. Math., 58(3) (1980), 241–248.

    Article  MathSciNet  MATH  Google Scholar 

  8. T.M. Dalbelo, N.G. Grulha Jr. and M.S. Pereira. Toric surfaces, vanishing Euler characteristic and Euler obstruction of a function. Ann. Fac. Sci. Toulouse Math. (6), 24(1) (2015), 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Damon and B. Pike. Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology, Geom. Topol. 18(2) (2014), 911–962.

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Dutertre and N.G. Grulha Jr. Lê-Greuel type formula for the Euler obstruction and applications. Adv. Math., 251 (2014), 127–146.

    Article  MathSciNet  MATH  Google Scholar 

  11. W. Ebeling and S. M. Gusein-Zade, On indices of 1-forms on determinantal singularities, Proc. Steklov Inst. Math., 267 (2009), 113–124.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Frühbis-Krüger and A. Neumer. Simple Cohen-Macaulay codimension 2 singularities. Comm. Algebra, 38(2) (2010), 454–495.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. Gaffney. Multiplicities and equisingularity of ICIS germs. Invent. Math., 123(2) (1996), 209–220.

    Article  MathSciNet  MATH  Google Scholar 

  14. T. Gaffney and A. Rangachev. Pairs of modules and determinantal isolated singularities, arXiv:1501.00201.

  15. G. González-Sprinberg. L’obstruction locale d’Euler et le théorème de MacPherson. Astérisque, 83 (1981), 7–32.

    MATH  Google Scholar 

  16. V.V. Goryunov. Functions on space curves. J. London Math. Soc. (2), 61 (2000), 807–822.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Grauert and R. Remmert. Theory of Stein spaces, Springer (1979).

    Book  MATH  Google Scholar 

  18. N.G. Grulha Jr. The Euler Obstruction and Bruce–Roberts’ Milnor Number. Quarterly Journal of Mathematics, 60 (2009), 291–302.

    Article  MathSciNet  MATH  Google Scholar 

  19. H.A. Hamm. Lokale topologische Eigenschaften komplexer Raume. Math. Ann.. 191 (1971), 235–252.

    Article  MathSciNet  MATH  Google Scholar 

  20. Lê Dũng Tráng. Computation of the Milnor number of an isolated singularity of a complete intersection. Funkcional. Anal. i Priložen., 8(2) (1974), 45–49.

    MathSciNet  Google Scholar 

  21. Lê Dũng Tráng and B. Teissier. Variétés polaires locales et classes de Chern des variétés singulières. Ann. of Math. (2), 114(3) (1981), 457–491.

    Article  MathSciNet  MATH  Google Scholar 

  22. Lê Dũng Tráng. Complex analytic functions with isolated singularities. J. Algebraic Geom., 1 (1992), 83–100.

    MathSciNet  MATH  Google Scholar 

  23. R.D. MacPherson. Chern classes for singular algebraic varieties. Ann. of Math., 100 (1974), 423–432.

    Article  MathSciNet  MATH  Google Scholar 

  24. D. Mond and D. van Straten. Milnor number equals Tjurina number for functions on space curves. J. London Math. Soc. (2), 63 (2001), 177–187.

    Article  MathSciNet  MATH  Google Scholar 

  25. J.J. Nuño-Ballesteros, B. Oréfice-Okamoto and J.N. Tomazella. The vanishing Euler characteristic of an isolated determinantal variety. Israel Journal of Mathematics, 197(1) (2013), 475–495.

    Article  MathSciNet  MATH  Google Scholar 

  26. J.J. Nuño-Ballesteros and J.N. Tomazella. The Milnor number of a function on a space curve germ. Bull. Lond. Math. Soc., 40(1) (2008), 129–138.

    Article  MathSciNet  MATH  Google Scholar 

  27. M.S. Pereira and M.A.S. Ruas. Codimension Two Determinantal Varieties with Isolated Singularities. Mathematica Scandinavic, 115 (2014), 161–172.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Seade, M. Tibar and A. Verjovsky. Milnor numbers and Euler obstruction.Bull. Braz. Math. Soc. (N.S.), 36 (2005), 275–283.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, 13560-905, São Carlos, SP, Brazil

    D. A. H. Ament, B. Oréfice-Okamoto & J. N. Tomazella

  2. Departament de Geometria i Topologia, Universitat de València Campus de Burjassot, 46100, Burjassot, Spain

    J. J. Nuño-Ballesteros

Authors
  1. D. A. H. Ament
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  2. J. J. Nuño-Ballesteros
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  3. B. Oréfice-Okamoto
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  4. J. N. Tomazella
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Corresponding author

Correspondence to J. N. Tomazella.

Additional information

The author has been supported by CAPES.

The author has been partially supported by CAPES-PVE Grant 88881.062217/2014-01.

The author has been partially supported by FAPESP Grant 2013/14014-3.

The author has been partially supported by FAPESP Grant 2013/10856-0 and CNPq Grant 309626/2014-5.

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Ament, D.A.H., Nuño-Ballesteros, J.J., Oréfice-Okamoto, B. et al. The Euler obstruction of a function on a determinantal variety and on a curve. Bull Braz Math Soc, New Series 47, 955–970 (2016). https://doi.org/10.1007/s00574-016-0198-y

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  • DOI: https://doi.org/10.1007/s00574-016-0198-y

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