Frances Kirwan

Geometric invariant theory

“Geometric Invariant Theory” by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory to the construction of moduli spaces. This third, revised edition has been long awaited for by the mathematical community. It is now appearing in a completely updated and enlarged version with an additional chapter on the moment map by Prof. Frances Kirwan (Oxford) and a fully updated bibliography of work in this area. The book deals firstly with actions of algebraic groups on algebraic varieties, separating orbits by invariants and construction quotient spaces; and secondly with applications of this theory to the construction of moduli spaces. It is a systematic exposition of the geometric aspects of the classical theory of polynomial invariants.

Localization for nonabelian group actions

Suppose X is a compact symplectic manifold acted on by a compact Lie group K (which may be nonabelian) in a Hamiltonian fashion, with moment map µ : X → Lie(K) ∗ and Marsden-Weinstein reduction MX = µ −1 (0)/K. There is then a natural surjective map �0 from the equivariant cohomology H ∗ K (X) of X to the cohomology H ∗ (MX). In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of MX of any �0 ∈ H ∗ (MX) whose degree is the dimension of MX, provided that 0 is a regular value of the moment map µ on X. This formula is given in terms of any class � ∈ H ∗ (X) for which �0(�) = �0, and involves ∗

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

Let n and d be coprime positive integers, and define M(n,d) to be the moduli space of (semi)stable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann surface E. This moduli space is a compact Kifhler manifold which has been studied from many different points of view for more than three decades (see for instance Narasimhan and Seshadri [41]). The subject of this article is the characterization of the intersection pairings in the cohomology ring1 H* (M (n, d)). A set of generators of this ring was described by Atiyah and Bott in their seminal 1982 paper [2] on the YangMills equations on Riemann surfaces (where in addition inductive formulas for the Betti numbers of M (n, d) obtained earlier using number-theoretic methods [13], [25] were rederived). By Poincare duality, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring. In 1991 Donaldson [15] and Thaddeus [47] gave formulas for the intersection pairings between products of these generators in H* (M (2, 1)) (in terms of Bernoulli numbers). Then using physical methods, Witten [50] found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in H* (M (n, d)) for general rank n. These generalized his (rigorously proved) formulas [49] for the symplectic volume of M (n, d): For instance, the symplectic volume of M (2, 1) is given by

Localization and the quantization conjecture

Let (M, ω) be a compact symplectic manifold with a Hamiltonian action of a compact Lie group K. Suppose that 0 is a regular value of the moment map μ: M → Lie(K)∗, so that the Marsden-Weinstein reduction Mred = μ−(0)K is a symplectic orbifold. In our earlier paper (Quart. J. Math., 47, 1996) we proved a formula (the residue formula) for η0eω0[Mred] for any η0 ϵ H∗(Mred), where ω0 is the induced symplectic form on Mred. This formula is given in terms of the restrictions of classes in the equivariant cohomology H∗T(M) of M to the components of the fixed point set of a maximal torus T in M. In this paper, we consider a line bundle L on Mfor which c1(L) = ω. If M is given a K-invariant complex structure compatible with ω we may apply the residue formula when η0 is the Todd class of Mred to obtain a formula for the Riemann-Roch number RR(Lred) of the induced line bundle Lred on Mred when K acts freely on μ−1(0). More generally when 0 is a regular value of μ, so that Mred is an orbifold and Lred is an orbifold bundle, Kawasakis Riemann-Roch theorem for orbifolds can be applied, in combination with the residue formula. Using the holomorphic Lefschetz formula we similarly obtain a formula for the K-invariant Riemann-Roch number RRK(L) of L. We show that the formulae obtained for RR(Lred) and RRK(L) are almost identical and in many circumstances (including when K is a torus) are the same. Thus in these circumstances a special case of the residue formula is equivalent to the conjecture of Guillemin and Sternberg (Invent. Math. 67 (1982), 515–538) (proved in various degrees of generality by Guillemin and Sternberg themselves and others including Sjamaar, Guillemin, Vergne and Meinrenken) that RR(Lred) = RRK(L).

Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface

are coprime. This space had long been studied by algebraicgeometers (see for instance Narasimhan and Seshadri 1965 [22]), but a new view-point on it was revealed by the seminal 1982 paper [1] of Atiyah and Bott on theYang-Mills equations on Riemann surfaces. In this paper a set of generators forthe (rational) cohomology ring of

Momentum maps and reduction in algebraic geometry

Abstract This survey article discusses how the geometry and topology of symplectic reductions at coadjoint orbits vary as the orbit varies, and what happens when the symplectic reductions acquire singularities, with applications including moduli spaces in algebraic geometry.

Complete Sets of Relations in the Cohomology Rings of Moduli Spaces of Holomorphic Bundles and Parabolic Bundles Over a Riemann Surface

The cohomology ring of the moduli space of stable holomorphic vector bundles of rank n and degree d over a Riemann surface of genus g>1 has a standard set of generators when n and d are coprime. When n=2 the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumfords relations to the cases when n>2 and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.

Implosion for hyperkähler manifolds

We introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of SU(n) actions. Our space is a stratified hyperkahler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

Symplectic implosion and nonreductive quotients

To a Hamiltonian action of a compact Lie group K on a symplectic manifold X, the symplectic implosion construction of Guillemin, Jeffrey and Sjamaar associates a stratified symplectic space X impl with a Hamiltonian action of the maximal torus T of K such that, if ζ lies in a fixed positive Weyl chamber in the dual of the Lie algebra of T, then the symplectic reduction of X by K at level ζ can be canonically identified with the symplectic reduction of Ximpl by T at level ζ. Moreover Ximpl can be obtained as the symplectic quotient by K of the product of X and the universal symplectic implosion (T*K)impl, and (T*K)impl can be naturally identified with the canonical affine completion of G/U max where G is the complexification of K and U max is a maximal unipotent subgroup of G (or equivalently the unipotent radical of a Borel subgroup of G). Thus if X is a projective variety with a linear G-action, the symplectic implosion X impl can be identified with the nonreductive GIT quotient G//Umax. In this paper the symplectic implosion construction is generalised so that if U is the unipotent radical of any parabolic subgroup P of G then the associated generalised symplectic implosion of T*K can be naturally identified with the canonical affine completion of G/U, and hence when G acts linearly on a projective variety X the nonreductive GIT quotient X//U can be identified with the associated generalised symplectic implosion of X.

Yang–Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SL_n. This article surveys this link between Yang-Mills theory and Tamagawa numbers, and explains how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth complex projective curve can be adapted to the setting of A^1-homotopy theory to study the motivic cohomology of these moduli spaces.