Proper Moment Map Research Articles

Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications

The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration.We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic.As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

The equivariant Riemann–Roch theorem and the graded Todd class

Let G be a torus with Lie algebra g and let M be a G-Hamiltonian manifold with Kostant line bundle L and proper moment map. Let Λ⊂g⁎ be the weight lattice of G. We consider a parameter k≥1 and the multiplicity m(λ,k) of the quantized representation RRG(M,Lk). Define 〈Θ(k),f〉=∑λ∈Λm(λ,k)f(λ/k) for f a test function on g⁎. We prove that the distribution Θ(k) has an asymptotic development 〈Θ(k),f〉∼kdim⁡M/2∑n=0∞k−n〈DHn,f〉 where the distributions DHn are the twisted Duistermaat–Heckman distributions associated with the graded equivariant Todd class of M. When M is compact, and f polynomial, the asymptotic series is finite and exact.

Classification of affine vortices

We prove a Hitchin-Kobayashi correspondence for affine vortices generalizing a result of Jaffe-Taubes for the action of the circle on the affine line. Namely, suppose a compact Lie group K has a Hamiltonian action on a Kaehler manifold X which is either compact or convex at infinity with a proper moment map, and so that stable=semistable for the action of the complexified Lie group G. Then, for some sufficiently divisible integer n, there is a bijection between gauge equivalence classes of K-vortices with target X modulo gauge and isomorphism classes of maps from the weighted projective line P(1,n) to X/G that map the stacky point at infinity P(n) to the semistable locus in X. The results allow the construction and partial computation of the quantum Kirwan map in Woodward, and play a role in the conjectures of Dimofte, Gukov, and Hollande relating vortex counts to knot invariants.

Geometric quantization for proper moment maps: the Vergne conjecture

We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.

The Fundamental Group ofS1-manifolds

AbstractWe address the problem of computing the fundamental group of a symplecticS1-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a HamiltonianS1-action. Several examples are presented to illustrate our main results.

Witten's nonabelian localization for noncompact Hamiltonian spaces

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, show that certain integrals of equivariant cohomology classes localize as a sum of contributions from these compact critical sets, and bound the contribution from each critical set. In the case (1) that the contribution from higher critical sets grows slowly enough that the overall integral converges rapidly and (2) that 0 is a regular value of the moment map, we recover Witten's result [E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303–368; http://xxx.lanl.gov/abs/hep-th/9204083] identifying the polynomial part of these integrals as the ordinary integral of the image of the class under the Kirwan map to the symplectic quotient.

THE KIRWAN MAP FOR SINGULAR SYMPLECTIC QUOTIENTS

Let M be a Hamiltonian K -space with proper moment map $\mu$ . The symplectic quotient $X=\mu^{-1}(0)/K$ is a singular stratified space with a symplectic structure on the strata. In this paper we generalise the Kirwan map, which maps the K equivariant cohomology of $\mu^{-1}(0)$ to the middle perversity intersection cohomology of X , to this symplectic setting. The key technical results which allow us to do this are Meinrenken's and Sjamaar's partial desingularisation of singular symplectic quotients and a decomposition theorem, proved in Section 2 of this paper, exhibiting the intersection cohomology of a ‘symplectic blowup’ of the singular quotient X along a maximal depth stratum as a direct sum of terms including the intersection cohomology of X .

INTERSECTION COHOMOLOGY OF SYMPLECTIC QUOTIENTS BY CIRCLE ACTIONS

Let T = U(1) and M be a Hamiltonian T-space with proper moment map μ : M → R. When 0 is not a regular value of μ, the symplectic quotient X = μ−1(0)/T is a singular stratified space. A description is provided of the middle perversity intersection cohomology of X as a subspace of the equivariant cohomology H T * ( μ − 1 ( 0 ) ) . The approach is sheaf theoretic.

The equivariant cohomology of hypertoric varieties and their real loci

Let M be a Hamiltonian T space with a proper moment map, bounded below in some component. In this setting, we give a combinatorial description of the T -equivariant cohomology of M, extending results of Goresky, Kottwitz and MacPherson and techniques of Tolman and Weitsman. Moreover, when M is equipped with an antisymplectic involution σ anticommuting with the action of T , we also extend to this noncompact setting the “mod 2” versions of these results to the real locus Q := M of M. We give applications of these results to the theory of hypertoric varieties.

Non-Abelian Symplectic Cuts and the Geometric Quantization of Noncompact Manifolds

Let (M, ω) be a Hamiltonian U(n)-space with proper moment map. In the case where n = 1, Lerman constructed a one-parameter family of Hamiltonian U(1)-spaces M ξ called the symplectic cuts of M. We generalize this construction to Hamiltonian U(n) spaces. Motivated by recent theorems that show that 'quantization commutes with reduction,' we next give a definition of geometric quantization for noncompact Hamiltonian G-spaces with proper moment map, and use our cutting technique to illustrate the proof of existence of such quantizations in the case of U(n) spaces. We then show (Theorem 1) that such quantizations exist in general.

The moment map and equivariant cohomology with generalized coefficients

Let M be a symplectic manifold acted on by a compact Lie group G in a Hamiltonian fashion, with proper moment map. In this situation we introduce a pushforward morphism P : H ∗ G(M)→ M −∞( g ∗) G , from the equivariant cohomology of M to the space of G-invariant distributions on g ∗ , which gives rise to symplectic invariants, in particular the pushforward of the Liouville measure. For the study of this pushforward morphism we make an intensive use of equivariant forms with generalized coefficients.

Kählerian potentials and convexity properties of the moment map

Since Kostant proved his convexity theorems for torus actions on ag varieties ([Ko]), there have been numerous contributions to this subject in the more general symplectic and K ahlerian setting. For example, let K be a compact Lie group acting in a Hamiltonian fashion on a connected compact symplectic manifold X . Then the intersection (X )+ of the image of the moment map : X → (LieK)∗ with a positive Weyl chamber t∗ + is convex. Thus (X ) = K · (X )+, where (X )+ is a natural convex section. This was proved by Atiyah and Guillemin–Sternberg ([A], [G-S]) in the abelian case, i.e., where K = T is a compact torus, by Mumford [M]) for X projective algebraic with an integral K ahlerian structure and K not necessarily abelian and in its nal form in the compact symplectic case by Kirwan ([K]). A precise description of (X )+ := (X ) ∩ t∗ + in the compact K ahlerian framework can be given in terms of the image of the components of the set of T xed points in X . In the projective case, where the K ahlerian structure comes from a projective embedding, there are explicit connections to the representations theory of G = KC which are due to Brion and to Mumford–Ness ([B], [N]). The purpose of the present paper is to prove convexity properties of in the setting of non-compact K ahlerian spaces. Note that, by removing an appropriate K-invariant subset from X one can easily construct non-convex -images from convex ones. On the other hand Hilgert–Neeb–Plank proved ([H-N-P]) that convexity holds for a proper moment map : X → (LieK)∗. Another convexity result has been obtained by Sjamaar for X a ne and the restriction of the moment map of a representation ([S2]). Here the representation theoretical approach of Brion is used. Now, let X be an irreducible complex space endowed with a holomorphic action of a complex reductive group G. Assume that a maximal