Simple non-rational convex polytopes via symplectic geometry

Abstract

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

Toric symplectic singular spaces I: isolated singularities

“…In a series of papers, Prato [20,21] and Battaglia and Prato [4, 5] extended Delzant's construction of symplectic toric manifolds from simple unimodular polytopes. Namely, they found a way to attach spaces to arbitrary polytopes.…”

Counting Real Connected Components of Trinomial Curve Intersections and m -nomial Hypersurfaces

“…Since P is compact, ψ P (C) must therefore be compact and contain a point in ∂ P. So now let Q be the face of highest dimension d such that ψ P (C) intersects RelInt(Q). By assertion (1) (and the definition of a quasifold [Pr,Section 1]), ∂ P ∩ψ P (C) must be an (n −2)-dimensional quasifold with only finitely many connected components. Lemma 14 then tells us that d < n − 1 ⇒ dim(Q ∩ ψ P (C)) < d, since Init w ( f ) is not identically zero.…”

Symplectic Toric Geometry and the Regular Dodecahedron

“…For the definition and main properties of symplectic quasifolds and of Hamiltonian actions of quasitori on symplectic quasifolds, we refer the reader to [2,3]. On the other hand, the basic facts on polytopes that are needed can be found in Ziegler's book [4].…”

“…On the other hand, the basic facts on polytopes that are needed can be found in Ziegler's book [4]. We are now ready to recall from [2] the generalized Delzant construction. For the purposes of this paper, we will restrict our attention to the special case = 3.…”

“…The regular dodecahedron is one of the most remarkable examples of a simple polytope that is not rational. In this paper, we apply a generalization of Delzant's construction [2] to simple nonrational convex polytopes and we associate to the regular dodecahedron a symplectic toric quasifold. Quasifolds are a natural generalization of manifolds and orbifolds introduced by the author in [2]; they are not necessarily Hausdorff spaces and they are locally modeled by quotients of manifolds by the action of discrete groups.…”