Let T × M → M be a Hamiltonian torus action on a connected symplectic manifold M for which the associated moment map Φ : M → t∗ is proper as a map into a convex open set ρ ⊆ t∗. We consider a closed submanifold Q of M and show that under certain local conditions on Q one has Φ(Q) = Φ(M). We apply this result in the special case that Q arises as the fixed point set of some involution σ on M whic...

For a flag manifold M = G/B with the canonical torus action, the T−equivariant cohomology is generated by equivariant Schubert classes, with one class τu for every element u of the Weyl group W . These classes are determined by their restrictions to the fixed point set M ≃ W , and the restrictions are polynomials with nonnegative integer coefficients in the simple roots. The main result of this...

The main results of this manuscript concern the Morse theory associated to the norm-square of a Kähler moment map f = ‖Φ − α‖2 on the space of representations Rep(Q,v) of a quiver; these are the first steps in a larger research program concerning the hyperkähler analogue of the well-known Kirwan surjectivity theorem in symplectic geometry. The first main result is that, although ‖Φ − α‖2 is not...

In this paper we extend the results of Kirwan et alii on convexity properties of the moment map for Hamiltonian group actions, and on the connectedness of the fibers of the moment map, to the case of non-compact orbifolds. Our motivation is twofold. First, the category of orbifolds is important in symplectic geometry because, generically, the symplectic quotient of a symplectic manifold is an o...

In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a com...

We show that the image cone of a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be considered as a version of the Atiyah Guillemin Sternberg convexity theorem for torus action...

The dynamics of Vlasov kinetic moments is shown to be Lie-Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are ...

We consider symplectic manifolds with Hamiltonian torus actions which are “almost but not quite completely integrable”: the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are “centered” (see below) and the moment map is proper. In particular, this classifies the moment map preimages of all sufficientl...

(i) Gromov’s Compactness Theorem for pseudo-holomorphic curves in symplectic manifolds ([23]) and the topology of symplectomorphism groups of rational ruled surfaces (sections 2 and 3, references [1, 2]). (ii) Atiyah-Guillemin-Sternberg’s Convexity Theorem for the moment map of Hamiltonian torus actions ([9, 25]) and Kähler geometry of toric orbifolds in symplectic coordinates (sections 4 and 5...

Let K be a compact connected Lie group and M a compact Hamiltonian K-manifold, i.e., a symplectic K-manifold equipped with a moment map μ : M → k. In this paper, we determine Col(M): the set of all functions on M which Poisson commute with all Kinvariant functions. For this, we construct a finite reflection group WM and show that Col(M) is completely determined by μ(M) and WM . More precisely, ...