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)


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].


Manifold Reso Kirwan 


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  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

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