On localization and Riemann-Roch numbers for symplectic quotients
| dc.creator | Jeffrey, Lisa C. | |
| dc.creator | Kirwan, Frances C. | |
| dc.date | 1995-06-04 | |
| dc.date.accessioned | 2026-07-07T09:06:33Z | |
| dc.date.available | 2026-07-07T09:06:33Z | |
| dc.description | Suppose $(M,ω)$ is a compact symplectic manifold acted on by a compact Lie group $K$ in a Hamiltonian fashion, with moment map $μ: M \to \Lie(K)^*$ and Marsden-Weinstein reduction $M_{red} = μ^{-1}(0)/K$. In this paper, we assume that $M$ has a $K$-invariant Kähler structure. In an earlier paper, we proved a formula (the residue formula) for $η_0 e^{ω_0}[M_{red}]$ for any $η_0 \in H^*(M_{red})$, where $ω_0$ is the induced symplectic form on $M_{red}$. Here we apply the residue formula in the special case $η_0 = Td(M_{red})$; when $K$ acts freely on $μ^{-1}(0)$ this yields a formula for the Riemann-Roch number $RR (L_{red})$ of a holomorphic line bundle $L_{red}$ on $M_{red}$ that descends from a holomorphic line bundle $L$ on $M$ for which $c_1(L) = ω$. Using the holomorphic Lefschetz formula we similarly obtain a formula for the $K$-invariant Riemann-Roch number $RR^K(L) $ of $L$. In the case when the maximal torus $T$ of $K$ has dimension one (except in a few special circumstances), we show the two formulas are the same. Thus in this special case the residue formula is equivalent to the result of Guillemin and Sternberg that $RR(L_{red}) = RR^K(L)$. (The residue formula was proved under the assumption that 0 is a regular value of $μ$, and was given in terms of the restrictions of classes in the equivariant cohomology $H^*_T(M) $ of $M$ to the | |
| dc.description | 19 pages; September 1994, revised March 1995, LaTeX v. 2.09 | |
| dc.identifier | https://arxiv.org/abs/alg-geom/9506007 | |
| dc.identifier | http://arxiv.org/abs/alg-geom/9506007 | |
| dc.identifier | Quart. J. Math. 47 (1996) 165-185 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/150048 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 58F05 | |
| dc.title | On localization and Riemann-Roch numbers for symplectic quotients | |
| dc.type | text |