Advertisement

A Note on Localization and the Riemann-Roch Formula

  • Lisa C. Jeffrey
  • Frances C. Kirwan
Part of the Progress in Mathematics book series (PM, volume 132)

Abstract

Let M be a compact symplectic manifold of (real) dimension 2m, equipped with the Hamiltonian action of a compact connected Lie group K with maximal torus T; we denote the moment map for this action by μ : M → k*. In this note, we shall treat some properties of the symplectic quotient M red = μ−1(0)/K, whose symplectic structure ω0 descends from the symplectic structure on M. (We assume that 0 is a regular value of μ, so that M red has at worst finite quotient singularities.) We shall describe some applications of the main result of [16] (Theorem 8.1, the residue formula): this formula specifies the evaluation on the fundamental class of M red of η0 e ω0, for any class η0H *(M red). The residue formula relates cohomology classes on M red to the equivariant cohomology H K * (M) M, via the natural ring homomorphism κ0 : H K * (M) → H *(M red) whose surjectivity was proved in [20].

Keywords

Line Bundle Maximal Torus Cohomology Ring Equivariant Cohomology Symplectic Quotient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M.F. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1–28.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    M.F. Atiyah, G.B. Segal, The index of elliptic operators II, Ann. Math. 87 (1968) 531–545.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    M.F. Atiyah, I.M. Singer, The index of elliptic operators III, Ann. Math. 87 (1968) 546–604.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag (Grundlehren vol. 298), 1992.MATHGoogle Scholar
  5. [5]
    N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539–541;MathSciNetMATHGoogle Scholar
  6. N. Berline, M. Vergne, Zéros d’un champ de vecteurs et classes caractéristiques équivariantes, Duke Math. J. 50 (1983) 539–549.MathSciNetMATHCrossRefGoogle Scholar
  7. [6]
    N. Berline, M. Vergne, The equivariant index and Kirillov’s character formula, Amer. J. Math. 107 (1985) 1159–1190.MathSciNetMATHCrossRefGoogle Scholar
  8. [7]
    T. Bröcker, T. Tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, 1985.MATHGoogle Scholar
  9. [8]
    H. Cartan, Notions d’algèbre différentielle; applications aux variétés où opère un groupe de Lie, in Colloque de Topologie (C.B.R.M., Bruxelles, 1950) 15–27; La transgression dans un groupe de Lie et dans un fibré principal, op. cit., 57–71.Google Scholar
  10. [9]
    R. De Souza, V. Guillemin, E. Prato, Consequences of quasi-free. Ann. Global Anal. Geom. 8 (1990) 77–85.MathSciNetMATHCrossRefGoogle Scholar
  11. [10]
    M. Duflo, M. Vergne, Orbites coadjointes et cohomologie équivariante, in M. Duflo, N.V. Pedersen, M. Vergne (ed.), The Orbit Method in Representation Theory (Progress in Mathematics, vol. 82), Birkhäuser, (1990) 11–60.CrossRefGoogle Scholar
  12. [11]
    J.J Duistermaat, G. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259–268; Addendum, 72 (1983) 153–158.MathSciNetMATHCrossRefGoogle Scholar
  13. [12]
    V. Guillemin, Reduced phase spaces and Riemann-Roch, in Lie Groups and Geometry, in honor of B. Kostant, J.-L Brylinski, R. Brylinski, V. Guillemin, V. Kac (eds), Progress in Mathematics, Vol. 123, Birkhäuser, 1994, 305–334.Google Scholar
  14. [13]
    V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982) 515–538.MathSciNetMATHCrossRefGoogle Scholar
  15. [14]
    V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984.MATHGoogle Scholar
  16. [15]
    R. Hartshorne, Residues and Duality (Lecture Notes in Mathematics v. 20), Springer, 1966.MATHGoogle Scholar
  17. [16]
    L.C. Jeffrey, F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995), 291–327.MathSciNetMATHCrossRefGoogle Scholar
  18. [17]
    L.C. Jeffrey, F.C. Kirwan, On localization and Riemann-Roch numbers for symplectic quotients, Quart. J. Math., to appear.Google Scholar
  19. [18]
    J. Kalkman, Cohomology rings of symplectic quotients, J. Reine Angew. Math. 485 (1995), 37–52.MathSciNetCrossRefGoogle Scholar
  20. [18]
    T. Kawasaki, The Riemann-Roch theorem for complex V-manifolds, Osaka J. Math. 16 (1979) 151–159.MathSciNetMATHGoogle Scholar
  21. [20]
    F.C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry Princeton University Press, 1984.MATHGoogle Scholar
  22. [21]
    C.-M. Marie, Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique, in: Rendiconti del Seminario Matematico, Universitàe Politechnico, Torino 43 (1985) 227–251.Google Scholar
  23. [22]
    E. Meinrenken, On Riemann-Roch formulas for multiplicities, J. Amer. Math. Soc., to appear.Google Scholar
  24. [23]
    M. Vergne, Quantification géométrique et multiplicités, C.R. Acad. Sci. Paris Sér. 1. Math. 319 (1994), 327–332; Geometric quantization and multiplicities I, Duke Math. J., to appear.MathSciNetMATHGoogle Scholar
  25. [24]
    M. Vergne, A note on Jeffrey-Kirwan-Witten localization, Topology, to appear.Google Scholar
  26. [25]
    E. Witten, Two dimensional gauge theories revisited, preprint hepth/9204083; J. Geom. Phys. 9 (1992) 303–368.MathSciNetMATHCrossRefGoogle Scholar
  27. [26]
    S. Wu, An integral formula for the squares of moment maps of circle actions, Lett. Math. Phys. 29 (1993), 311–328.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

  • Lisa C. Jeffrey
    • 1
  • Frances C. Kirwan
    • 2
  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Balliol CollegeOxfordUK

Personalised recommendations