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The vanishing Euler characteristic of an isolated determinantal singularity

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An Erratum to this article was published on 11 April 2018

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Abstract

Let (X, 0) be a complex analytic isolated determinantal singularity. We will define the vanishing Euler characteristic of (X, 0) and the Milnor number of a holomorphic function germ with an isolated singularity on X, f: (X, 0) → ℂ.

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Change history

  • 11 April 2018

    "In [4, Theorem A.5], we present the following result."

  • 11 April 2018

    "In [4, Theorem A.5], we present the following result."

References

  1. E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of Algebraic Curves, Springer-Verlag, Berlin, 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, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 131 (2001), 1003–1022.

    Article  MathSciNet  MATH  Google Scholar 

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

  4. R. O. Buchweitz and G. M. Greuel, The Milnor number and deformations of complex curve singularities, Inventiones Mathematicae 58 (1980), 241–248.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Damon and B. Pike, Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology, arXiv:1201.1579.

  6. J. A. Eagon and M. Hochster, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, American Journal of Mathematics 93 (1971), 1020–1058.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Ebeling and S. M. Gusein-Zade, On indices of 1-forms on determinantal singularities, Proceedings of the Steklov Institute of Mathematics 267 (2009), 113–124.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Gaffney, Multiplicities and equisingularity of ICIS germs, Inventiones Mathematicae 123 (1996), 209–220.

    MathSciNet  MATH  Google Scholar 

  10. C. G. Gibson, Singular Points of Smooth Mappings, Pitman Publishing Limited, London, 1979.

    MATH  Google Scholar 

  11. M. Goresky and R. MacPherson, Stratified Morse Theory, Springer, Berlin, 1980.

    Google Scholar 

  12. H. Grauert and R. Remmert, Theory of Stein Spaces, Springer, Berlin, 1979.

    Book  MATH  Google Scholar 

  13. G. M. Greuel and G. Pfister, A singular introduction to commutative algebra, Springer-Verlag, Berlin-Heidelberg, 2008.

    Google Scholar 

  14. G. M. Greuel and J. Steenbrink, On the topology of smoothable singularities in Singularities, Part 1 (Arcata, Calif., 1981), Proceedings of Symposia in Pure Mathematics, Vol. 40, American Mathematical Society, Providence, RI, 1983, pp. 535–545.

    Google Scholar 

  15. T. de Jong and G. Pfister, Local Analytic Geometry. Basic Theory and Applications, Advanced Lectures in Mathematics, Friedr. Vieeweg & Sohn, Braunschweig, 2000.

    MATH  Google Scholar 

  16. V. H. Jorge Pérez and M. J. Saia, Euler Obstruction, polar multiplicities and equisingularity of map germs in O(n, p), n < p, Internatational Journal of Mathematics 17 (2006), 887–903.

    Article  MATH  Google Scholar 

  17. K. Kaveh, Morse theory and Euler characteristic of sections of spherical varieties, Tranformation Groups 9 (2004), 47–63.

    Article  MathSciNet  MATH  Google Scholar 

  18. Lê Dũng Tráng, Computation of the Milnor number of an isolated singularity of a complete intersection, Funkcional’nyi Analiz i ego Priloženija 8 (1974), 45–49.

    Google Scholar 

  19. Lê Dũng Tráng and B. Teissier, Variétés polaires locales et classes de Chern des variétés singuliéres, Annals of Mathematics 114 (1981), 457–491.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. J. N. Looijenga, Isolated Singular Points on Complete Intersections, London Mathematical Society Lecture Note Series, Vol. 77, Cambridge University Press, 1984.

  21. J. Milnor, Morse Theory, Princeton University Press, 1963.

  22. J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, 1968.

  23. D. Mumford, Algebraic Geometry. I. Complex Projective Varieties, A Series of Comprehensive Studies in Mathematics, Vol. 221, Springer, Berlin, 1976.

    MATH  Google Scholar 

  24. J. J. Nuño-Ballesteros and J. N. Tomazella, The Milnor number of a function on a space curve germ, The Bulletin of the London Mathematical Society 40 (2008), 129–138.

    Article  MATH  Google Scholar 

  25. R. S. Palais and S. Smale, A generalized Morse theory, Bulletin of the American Mathematical Society 70 (1964), 165–172.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. S. Pereira and M. A. S. Ruas, Codimension two determinantal varieties with isolated singularities, preprint. arXiv:1110.5580v3.

  27. C. T. C. Wall, Finite determinacy of smooth map germs, The Bulletin of the London Mathematical Society 13 (1981), 481–539.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    J. J. Nuño-Ballesteros

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

    B. Oréfice-Okamoto & J. N. Tomazella

Authors
  1. J. J. Nuño-Ballesteros
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  2. B. Oréfice-Okamoto
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  3. J. N. Tomazella
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Corresponding author

Correspondence to J. J. Nuño-Ballesteros.

Additional information

Partially supported by DGICYT Grant MTM2009-08933.

Partially supported by FAPESP Grant 2008/53944-8 and by CAPES Grant BEX 2820/10-2.

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Nuño-Ballesteros, J.J., Oréfice-Okamoto, B. & Tomazella, J.N. The vanishing Euler characteristic of an isolated determinantal singularity. Isr. J. Math. 197, 475–495 (2013). https://doi.org/10.1007/s11856-012-0188-8

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  • DOI: https://doi.org/10.1007/s11856-012-0188-8

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