共查询到20条相似文献,搜索用时 687 毫秒
1.
Yves Benoist 《Geometriae Dedicata》2002,89(1):177-241
For any symplectic action of a compact connected group on a compact connected symplectic manifold, we show that the intersection
of the Weyl chamber with the image of the moment map is a closed convex polyhedron. This extends Atiyah–Guillemin–Sternberg–Kirwan's
convexity theorems to non-Hamiltonian actions. As a consequence, we describe those symplectic actions of a torus which are
coisotropic (or multiplicity free), i.e. which have at least one coisotropic orbit: they are the product of an Hamiltonian
coisotropic action by an anhamiltonian one. The Hamiltonian coisotropic actions have already been described by Delzant thanks
to the convex polyhedron. The anhamiltonian coisotropic actions are actions of a central torus on a symplectic nilmanifold.
This text is written as an introduction to the theory of symplectic actions of compact groups since complete proofs of the
preliminary classical results are given.
An erratum to this article is available at . 相似文献
2.
The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G-manifold is a stratified symplectic space. Suppose 1→A→G→T→1 is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G-manifold with respect to A yields a Hamiltonian T-space. We show that if the A-moment map is proper, then the convexity theorem holds for such a Hamiltonian T-space, even when it is singular. We also prove that if, furthermore, the T-space has dimension and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case. This paper is a concise version of a companion paper [B. Lian. B. Song, A convexity theorem and reduced Delzant spaces, math.DG/0509429]. 相似文献
3.
Mikhail Kogan 《Annals of Global Analysis and Geometry》1997,15(6):543-553
This paper concerns with symplectic topology of compact completely integrable Hamiltonian systems with only stable nondegenerate elliptic singularities. We describe all systems whose universal coverings admit actian-angle coordinates and, in particular, prove that some finite cover of a base space is diffeomorphic to a product of a convex polytope and a solvmanifold. We also construct an obstruction, vanishing of which guarantees splitting of some finite cover of a phase space as a toric variety and a torus fibering over a solvmanifold. 相似文献
4.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(10):887-890
We consider a topological space which is locally isomorphic to the quotient of Rk by the action of a discrete group and we call it quasifold of dimension k. Quasifolds generalize manifolds and orbifolds and represent the natural framework for performing symplectic reduction with respect to the induced action of any Lie subgroup, compact or not, of a torus. We define quasitori, Hamiltonian actions of quasitori and the moment mapping for symplectic quasifolds, and we show that every simple convex polytope, rational or not, is the image of the moment mapping for the action of a quasitorus on a quasifold. 相似文献
5.
In this paper we generalize to coisotropic actions of compact Lie groups a theorem of Guillemin on deformations of Hamiltonian
structures on compact symplectic manifolds. We show how one can reconstruct from the moment polytope the symplectic form on
the manifold.
Received: 21 March 2006 相似文献
6.
We consider compact symplectic manifolds acted on effectively by a compact connected Lie group K in a Hamiltonian fashion. We prove that the squared moment map ∥μ∥2 is constant if and only if K is semisimple and the manifold is K-equivariantly symplectomorphic to a product of a flag manifold and a compact symplectic manifold which is acted on trivially by K. In the almost-Kähler setting the symplectomorphism turns out to be an isometry. 相似文献
7.
Qi Lin Yang 《数学学报(英文版)》2008,24(4):555-564
Let a connected compact Lie group G act on a connected symplectic orbifold of orbifold fundamental group Г. If the action preserves the symplectic structure and there is a G-equivariant and mod-Г proper momentum map for the lifted action on the universal branch covering orbifold, and if the lifted G-action commutes with that of Г, then the symplectic convexity theorem is still true for this kind of lifted Hamiltonian action. 相似文献
8.
R.F. Goldin 《Geometric And Functional Analysis》2002,12(3):567-583
Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of is generated by a small number of classes satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T . We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan
map is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman
[TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two , quotiented by the diagonal 2-torus action.
Submitted: September 2001, Revised: December 2001, Revised: February 2002. 相似文献
9.
We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence. 相似文献
10.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1997,324(5):491-496
Using an idea of Witten (see [8]), we give a localization formula in equivariant cohomology in the case of an Hamiltonian torus action on a compact symplectic manifold χ. The integral over χ of an equivariant closed form can be written as an integral over the submanifold of critical points of the square of the moment map. 相似文献
11.
Christopher Allday Volker Hauschild Volker Puppe 《Transactions of the American Mathematical Society》2002,354(7):2971-2982
We prove that, under certain conditions, if a compact connected Lie group acts effectively on a closed manifold, then there is no fixed point. Because two of the main conditions are satisfied by any Hamiltonian action on a closed symplectic manifold, the theorem applies nicely to such actions. The method of proof, however, is cohomological; and so the result applies more generally.
12.
13.
本文证明了单连通Poisson紧李群切作用及约化Poisson作用于Poisson流形,若带有等动量映射,则可通过调整Poisson流形的Poisson结构,变成保Poisson结构的Poisson作用,并且该作用限制到Poisson流形的辛叶片上,相对于新Poisson结构是Hamiltion作用。我们把Meyer-Marsden-Weinstein约化从Hamiltion作用推广到切Poisson作用,包括正则值和非正则值两种形式。 相似文献
14.
15.
L. Charles 《Journal of Functional Analysis》2006,236(1):299-350
This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kähler manifold M with a Hamiltonian torus action. In the seminal paper [V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (3) (1982) 515-538], Guillemin and Sternberg introduced an isomorphism between the invariant part of the quantum space associated to M and the quantum space associated to the symplectic quotient of M, provided this quotient is non-singular. We prove that this isomorphism is a Fourier integral operator and that the Toeplitz operators of M descend to Toeplitz operators of the reduced phase space. We also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced Toeplitz operator, a result related to the Riemann-Roch-Kawasaki theorem. 相似文献
16.
I. Mundet i Riera 《Advances in Mathematics》2007,214(1):469-493
Suppose that an algebraic torus G acts algebraically on a projective manifold X with generically trivial stabilizers. Then the Zariski closure of the set of pairs {(x,y)∈X×X|y=gx for some g∈G} defines a nonzero equivariant cohomology class . We give an analogue of this construction in the case where X is a compact symplectic manifold endowed with a Hamiltonian action of a torus, whose complexification plays the role of G. We also prove that the Kirwan map sends the class [ΔG] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map. 相似文献
17.
Eugene Lerman Susan Tolman 《Transactions of the American Mathematical Society》1997,349(10):4201-4230
In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly, we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each open facet and that all such orbifolds are algebraic toric varieties.
18.
Let Ham(M) be the group of Hamiltonian symplectomorphisms of a quantizable, compact, symplectic manifold (M, ω). We prove the existence of an action integral around loops in Ham(M), and determine the value of this action integral on particular loops when the manifold is a coadjoint orbit. 相似文献
19.
For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkähler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville’s. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt’s theory of subnormal subgroups. 相似文献
20.
Shintar? Kuroki 《Transformation Groups》2011,16(2):481-536
The purpose of this paper is to classify torus manifolds (M
2n
, T
n
) with codimension one extended G-actions (M
2n
, G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T
n
. For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds.
As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold
is a complex projective bundle over a product of complex projective spaces. 相似文献