An Equivariant Version of the Euler Obstruction



For a complex analytic variety with an action of a finite group and for an invariant 1-form on it, we give an equivariant version (with values in the Burnside ring of the group) of the local Euler obstruction of the 1-form and describe its relation with the equivariant radial index defined earlier. This leads to equivariant versions of the local Euler obstruction of a complex analytic space and of the global Euler obstruction.


Group action Euler obstruction Burnside ring 

Mathematics Subject Classification

32S05 58E40 19A22 58A10 


  1. Atiyah, M., Segal, G.: On equivariant Euler characteristics. J. Geom. Phys. 6(4), 671–677 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. Brasselet, J.P., Tráng, D.L., Seade, J.: Euler obstruction and indices of vector fields. Topology 39, 1193–1208 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. Brasselet, J.-P., Massey, D., Parameswaran, A.J., Seade, J.: Euler obstruction and defects of functions on singular varieties. J. Lond. Math. Soc. 70(1), 59–76 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. Brasselet, J.-P., Schwartz, M.-H.: Sur les classes de Chern d’un ensemble analytique complexe. In: Caractéristique d’Euler-Poincaré, Astérisque 82–83, 93–147 (1981)Google Scholar
  5. Brasselet, J.-P., Seade, J., Suwa, T.: Proportionality of indices of 1-forms on singular varieties. In: Singularities in geometry and topology 2004. Adv. Stud. Pure Math., vol. 46, pp. 49–65. Math. Soc. Japan, Tokyo (2007)Google Scholar
  6. Brasselet, J.-P., Seade, J., Suwa, T.: Vector fields on singular varieties. Lecture Notes in Mathematics, vol. 1987. Springer, Berlin (2009)Google Scholar
  7. Dixon, L., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds. Nucl. Phys. B 261, 678–686 (1985)MathSciNetCrossRefGoogle Scholar
  8. Dixon, L., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds. II. Nucl. Phys. B 274, 285–314 (1986)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  10. Ebeling, W., Gusein-Zade, S.M.: Indices of vector fields and 1-forms on singular varieties. In: Global aspects of complex geometry, pp. 129–169. Springer, Berlin (2006)Google Scholar
  11. Ebeling, W., Gusein-Zade, S.M.: Equivariant indices of vector fields and 1-forms. Eur. J. Math. 1, 286–301 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. Hall Jr., M.: Combinatorial theory. Second edition. Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York (1986)Google Scholar
  13. Lück, W., Rosenberg, J.: The equivariant Lefschetz fixed point theorem for proper cocompact \(G\)-manifolds. In: High-dimensional manifold topology, pp. 322–361. World Sci. Publ., River Edge, NJ (2003)Google Scholar
  14. MacPherson, R.: Chern classes for singular varieties. Ann. Math. 100, 423–432 (1974)MathSciNetCrossRefMATHGoogle Scholar
  15. Seade, J.A., Tibar, M., Verjovsky, A.: Global Euler obstruction and polar invariants. Math. Ann. 333, 393–403 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. tom Dieck, T.: Transformation groups and representation theory. Lecture Notes in Mathematics, vol. 766. Springer, Berlin (1979)Google Scholar
  17. Verdier, J.-L.: Caractéristique d’Euler-Poincaré. Bull. Soc. Math. Fr. 101, 441–445 (1973)CrossRefMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2016

Authors and Affiliations

  1. 1.Leibniz Universität Hannover, Institut für Algebraische GeometrieHannoverGermany
  2. 2.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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