arXiv subjects

Frances C. Kirwan

Publications and source records attributed to Frances C. Kirwan.

3 recordsLinked to original sources

arXiv2006

Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations

This paper studies intersection theory on the compactified moduli space M(n,d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus g > 1 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of M(n,d). Based on our earlier work, we give a precise formula for the intersection cohomology pairing and provide a method to calculate pairings on the partial resolution of singularities of M(n,d). The case when n=2 is discussed in detail. Finally Witten's integral is considered for this singular case.

math.AG↗

arXiv1997

Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).

alg-geom↗

arXiv1995

On localization and Riemann-Roch numbers for symplectic quotients

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

alg-geom↗