Mackey Functors on Provarieties

  • Shoji Yokura
Conference paper
Part of the Trends in Mathematics book series (TM)


MacPherson’s Chern class transformation on complex algebraic varieties is a certain unique natural transformation from the constructible function covariant functor to the integral homology covariant functor, and it can be extended to a category of provarieties. In this paper, as further extensions of this we consider natural transformations among Mackey functors on provarieties and also on “indvarieties” and discuss some notions and examples related to these extensions.


provariety constructible function Chern class Mackey functor Grothendieck ring of varieties motivic measure bivariant theory ind-variety 


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  1. [AM]
    M. Artin and B. Mazur, Etale Homotopy, Springer Lecture Notes in Math. 100, Springer-Verlag, Berlin, 1969.zbMATHGoogle Scholar
  2. [Be]
    J. Benabou, Introduction to bicategories, Springer Lecture Notes in Math. 47 (1967), 1–77.MathSciNetGoogle Scholar
  3. [Bo]
    S. Bouc, Green functors and G-sets, Springer Lecture Notes in Math. 1671, 1997.Google Scholar
  4. [BrSc]
    J.-P. Brasselet and M.-H. Schwartz, Sur les classes de Chern d’une ensemble analytique complexe, Astérisque 82–83 (1981), 93–148.MathSciNetGoogle Scholar
  5. [Cr]
    A. Craw, An introduction to motivic integration, in Strings and Geometry, Clay Math. Proc., 3 (2004), Amer. Math. Soc., 203–225.Google Scholar
  6. [DL1]
    J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232.CrossRefMathSciNetGoogle Scholar
  7. [DL2]
    J. Denef and F. Loeser, Geometry on arc spaces of algebraic varieties, European Congress of Mathematicians (Barcelona, 2000), 1 (2001) Birkhäuser, 327–348.MathSciNetGoogle Scholar
  8. [Dim]
    A. Dimca, Sheaves in Topology, Springer-Verlag, 2004.Google Scholar
  9. [DP]
    I. Dimitrov and I. Penkov, Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups, IMRN 55 (2004), 2935–2953.CrossRefMathSciNetGoogle Scholar
  10. [DPW]
    I. Dimitrov, I. Penkov and J.A. Wolf, A Bott-Borel-Weil theory for direct limits of algebraic groups, Amer. J. Math. 124 (2002), 955–998.MathSciNetGoogle Scholar
  11. [Dr1]
    A.W.M. Dress, Notes on the theory of representations of finite groups, Part I, Mimeogrphed Lecture Notes. Univ. of Bielefeld, 1971.Google Scholar
  12. [Dr2]
    A.W.M. Dress, Contributions to the theory of induced representations, in Algebraic K-Theory II (H. Bass, ed.), Springer Lecture Notes in Math. 342, Springer-Verlag, 1973, 183–240.Google Scholar
  13. [Er]
    L. Ernström, Topological Radon transforms and the local Euler obstruction, Duke Math. J. 76 (1994), 1–21.CrossRefMathSciNetGoogle Scholar
  14. [Fu]
    W. Fulton, Intersection Theory, Springer-Verlag, 1981.Google Scholar
  15. [FM]
    W. Fulton and R. MacPherson, Categorical frameworks for the study of singular spaces, Memoirs of Amer. Math. Soc. 243, 1981.Google Scholar
  16. [Grot]
    A. Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique, II, Séminaire Bourbaki, 12ème année exposé 190–195, 1959–60.Google Scholar
  17. [KS]
    M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer-Verlag, Berlin, Heidelberg, 1990.zbMATHGoogle Scholar
  18. [Kum1]
    S. Kumar, Infinite Grassmannians and moduli spaces of G-bundles, in Vector Bundles on Curves — New Directions (M.S. Narasimhan, ed.), Springer Lecture Notes in Math. 1647 (1997), 1–49.Google Scholar
  19. [Kum2]
    S. Kumar, Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progress in Mathematics 204 (Birkhäuser), 2003.Google Scholar
  20. [Lin]
    H. Lindner, A remark on Mackey functors, Manuscripta Math. 18 (1976), 273–278.CrossRefMathSciNetGoogle Scholar
  21. [Lo]
    E. Looijenga, Motivic measures, Séminaire Bourbaki, Astérisque 276 (2002), 267–297.MathSciNetGoogle Scholar
  22. [Mac]
    R. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974), 423–432.CrossRefMathSciNetGoogle Scholar
  23. [MS]
    S. Mardesić and J. Segal, Shape Theory, North-Holland, 1982.Google Scholar
  24. [Po]
    B. Poonen, The Grothendieck ring of varieties is not a domain, Math. Res. Letters 9 (2002), 493–498.MathSciNetGoogle Scholar
  25. [Scha]
    P. Schapira, Operations on constructible functions, J. Pure Appl. Algebra, 72 (1991), 83–93.CrossRefMathSciNetGoogle Scholar
  26. [Sch1]
    J. Schürmann, A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes, math.AG/0202175.Google Scholar
  27. [Sch2]
    J. Schürmann, Topology of singular spaces and constructible sheaves, Monografie Matematyczne 63 (New Series), Birkhäuser, Basel, 2003.Google Scholar
  28. [Sc1]
    M.-H. Schwartz, Classes caractéristiques définies par une stratification d’une variété analytique complex, C. R. Acad. Sci. Paris t. 260 (1965), 3262–3264, 3535–3537.MathSciNetGoogle Scholar
  29. [Sc2]
    M.-H. Schwartz, Classes et caractères de Chern des espaces linéaires, C. R. Acad. Sci. Paris Sér. I. Math. 295(1982), 399–402.Google Scholar
  30. [Sha]
    I.R. Shafarevich, On some infinite-dimensional groups, II, Izv. Akad. Nauk USSR Ser. Mat. 45 (1981), 214–226.MathSciNetGoogle Scholar
  31. [Ve]
    W. Veys, Arc spaces, motivic integration and stringy invariants, Proceedings of 12th MSJ-IRI symposium “Singularity Theory and Its Applications” (Sapporo, Japan, 16–25 September, 2003), Advanced Studies in Pure Mathematics (to appear), (math.AG/0401374).Google Scholar
  32. [TW]
    J. Thevenaz and P. Webb, The structure of Mackey functors, Transactions Amer. Math. Soc. 347(6) (1995), 1865–1961.CrossRefMathSciNetGoogle Scholar
  33. [Y1]
    S. Yokura, On a Verdier-type Riemann-Roch for Chern-Schwartz-MacPherson class, Topology and Its Applications 94 (1999), 315–327.CrossRefMathSciNetGoogle Scholar
  34. [Y2]
    S. Yokura, Chern classes of proalgebraic varieties and motivic measures, math.AG/0407237.Google Scholar
  35. [Y3]
    S. Yokura, Characteristic classes of proalgebraic varieties and motivic measures, (preprint 2005).Google Scholar
  36. [Yo]
    T. Yoshida, Idempotents and Transfer Theorems of Burnside Rings, Character Rings and Span Rings, Algebraic and Topological Theories (to the memory of Dr.Takehiko MIYATA), (1985), 589–615.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Shoji Yokura
    • 1
  1. 1.Department of Mathematics and Computer Science, Faculty of ScienceUniversity of KagoshimaKagoshimaJapan

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