In this article we consider a generalization of manifolds and orbifolds which we call quasifolds; quasifolds of dimension k are locally isomorphic to the quotient of the space R k by the action of a discrete group -typically they are not Hausdorff topological spaces. The analogue of a torus in this geometry is a quasitorus. We define Hamiltonian actions of quasitori on symplectic quasifolds and we show that any simple convex polytope, rational or not, is the image of the moment mapping for a family of effective Hamiltonian actions on symplectic quasifolds having twice the dimension of the corresponding quasitorus.
Simple Non-Rational Convex Polytopes via Symplectic Geometry
QuasifoldsWe begin by introducing the local model for quasifolds.Definition 1.1 (Model) LetŨ be a connected, simply connected manifold of dimension k and let Γ be a discrete group acting smoothly on the manifoldŨ so that the set of points,Ũ 0 , where the action is free, is connected and dense. Consider the space of orbits,Ũ /Γ, of the action of the group Γ on the manifoldŨ , endowed with the quotient
The purpose of this article is to view the Penrose kite from the perspective
of symplectic geometry.Comment: 24 pages, 7 figures, minor changes in last version, to appear in
Comm. Math. Phys
In this article we extend cutting and blowing up to the nonrational symplectic toric setting. This entails the possibility of cutting and blowing up for symplectic toric manifolds and orbifolds in nonrational directions.
Nous considérons un espace topologique qui est localement isomorphe au quotient de R k par l'action d'un groupe discret et nous l'appelons quasi-variété de dimension k. Les quasivariétés généralisent les variétés et les V -variétés et représentent le cadre naturel pour la réduction symplectique par rapportà l'action induite d'un sous-groupe de Lie, compact ou non, d'un tore. Nous définissons les quasi-tores, les actions hamiltoniennes de quasi-tores et l'application moment sur une quasi-variété symplectique, et nous montrons que tout polytope convexe simple, rationnel ou non, est l'image de l'application moment pour l'action d'un quasi-tore sur une quasi-variété.
On a generalization of the notion of orbifoldAbstract. We consider a topological space which is locally isomorphic to the quotient of R k 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.
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