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].


Line Bundle Maximal Torus Cohomology Ring Equivariant Cohomology Symplectic Quotient 
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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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