Equivariant motivic integration and proof of the integral identity conjecture for regular functions

  • Quy Thuong LêEmail author
  • Hong Duc Nguyen


We develop Denef–Loeser’s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions. In comparison with Hartmann’s work, the equivariant Grothendieck ring defined in this article is more elementary and it yields the application to the conjecture.

Mathematics Subject Classification

Primary 14E18 14G10 14L30 



This article was partially written during the authors’ visits to Department of Mathematics-KU Leuven in December 2017 and Vietnam Institute for Advanced Studies in Mathematics in January 2018. The authors thank sincerely these institutions for excellent atmospheres and warm hospitalities.


  1. 1.
    Batyrev, V.V.: Birational Calabi–Yau n-folds have equal Betti numbers. In: New Trends in Algebraic Geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser., vol. 264, pp. 1–11. Cambridge Univ. Press, Cambridge (1996)Google Scholar
  2. 2.
    Cluckers, R., Loeser, F.: Constructible motivic functions and motivic integration. Invent. Math. 173(1), 23–121 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebraic Geom. 7, 505–537 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Denef, J., Loeser, F.: Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135, 201–232 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Denef, J., Loeser, F.: Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Guibert, G.: Espaces d’arcs et invariants d’Alexander. Comment. Math. Helv. 77, 783–820 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Guibert, G., Loeser, F., Merle, M.: Iterated vanishing cycles, convolution, and a motivic analogue of the conjecture of Steenbrink. Duke Math. J. 132(3), 409–457 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Grothendieck, A.: Revêtements étales et groupe fondamental, Fasc. I: Exposés 1 à 5, volume 1960/61 of Séminaire de Géométrie Algébrique. Institut des Hautes Études Scientifiques, Paris (1963)Google Scholar
  9. 9.
    Grothendieck, A.: Éléments de géométrie algébrique. IV, Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math., t. 28 (1966)Google Scholar
  10. 10.
    Hartmann, A.: Equivariant motivic integration on formal schemes and the motivic zeta function. J. Commun. Algebra 47(4), 1423–1463 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hrushovski, E., Kazhdan, D.: Integration in valued fields. In: Algebraic and Number Theory, Progress in Mathematics, vol. 253, pp. 261–405. Birkhäuser (2006)Google Scholar
  12. 12.
    Hrushovski, E., Loeser, F.: Monodromy and the Lefschetz fixed point formula. Ann. Sci. École Norm. Sup. 48(2), 313–349 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donalson–Thomas invariants and cluster transformations. arXiv:0811.2435vl
  14. 14.
    Lê, Q.T.: On a conjecture of Kontsevich and Soibelman. Algebra Number Theory 6(2), 389–404 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lê, Q.T.: Proofs of the integral identity conjecture over algebraically closed fields. Duke Math. J. 164(1), 157–194 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lê, Q.T.: A short survey on the integral identity conjecture and theories of motivic integration. Acta Math. Vietnam 42(2), 289–310 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Loeser, F., Sebag, J.: Motivic integration on smooth rigid varieties and invariants of degenerations. Duke Math. J. 119(2), 315–344 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Looijenga, E.: Motivic measures. Astérisque 276, 267–297 (2002). (Séminaire Bourbaki 1999/2000, No. 874) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariants theory, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer, Berlin (1994)zbMATHCrossRefGoogle Scholar
  20. 20.
    Nicaise, J.: A trace formula for rigid varieties, and motivic Weil generating series for formal schemes. Math. Ann. 343, 285–349 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Nicaise, J., Sebag, J.: Motivic Serre invariants, ramification, and the analytic Milnor fiber. Invent. Math. 168(1), 133–173 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Nicaise, J., Payne, S.: A tropical motivic Fubini theorem with applications to Donaldson-Thomas theory. Duke Math. J. 168(10), 1843–1886 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Sebag, J.: Intégration motivique sur les schémas formels. Bull. Soc. Math. France 132(1), 1–54 (2004). (Séminaire Bourbaki 1999/2000, No. 874) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Stacks Project Authors.: Stacks Project (2019)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics, Mechanics and InformaticsVNU University of Science, Vietnam National University, HanoiHanoiVietnam
  2. 2.Basque Center for Applied MathematicsBilbaoSpain
  3. 3.TIMAS, Thang Long UniversityHanoiVietnam

Personalised recommendations