Toric symplectic stacks

Abstract

A. We give an intrinsic definition of compact toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of compact toric symplectic manifolds. As an application, we show that any compact toric symplectic stack can be deformed to an ineffective toric orbifold. . IA compact connected 2n-dimensional symplectic manifold (M, ω) is toric if there is an effective Hamiltonian action on M by an n-dimensional torus T. The image of M under the moment map µ is a convex polytope in t * , the linear dual of the Lie algebra t of T. The moment polytope has these three propertes:• It is simple, meaning there are n facets meeting at every vertex.• It is rational, meaning that normal vectors to the facets of µ(V) can be chosen in the cocharacter lattice Hom(S 1 , T) Z n in t R n .• It is smooth, meaning that at each vertex v of µ(M), the primitive normal vectors to the facets meeting at v form a Z-basis of Hom(S 1 , T).In [ ], Delzant showed that the assignment taking (M, ω, µ, T) to its moment polytope µ(M) ⊂ t * gives a one-to-one correspondence between the set of 2ndimensional toric symplecic manifolds (up to isomorphism), and the set of ndimensional polytopes which are simple, rational, and smooth (up to the action of Tang for their useful comments and discussions throughout this project.In this section we very briefly recall some background material on Hamiltonian actions on stacks. This was developed in detail in [ ], and the reader is referred there for more information. Some of the material is motivated by [ ]. For background on differentiable stacks in general we refer the reader to [ , , ].Lie groupoids. Lie groupoids X • have the source, target, identity, inverse, and multiplication structure maps denoted by s, t, u, (·) −1 , and •, respectively. We assume that X 0 and X 1 are both Hausdorff and second-countable. If G acts on X, then we write G ⋉ X ⇒ X for the action groupoid. Sometimes we will denote a groupoid by its manifold of arrows, so for instance G ⋉ X ⇒ X may be written G ⋉ X. A smooth manifold M will be identified with the trivial groupoid M ⇒ M.If X • is a Lie groupoid and x ∈ X 0 , then the isotropy group of x is the group s −1 (x) ∩ t −1 (x) of arrows which start and end at x.Let LieGpd denote the 2-category of Lie groupoids; see [ ] for a detailed description of its morphisms and -morphisms.Unless otherwise noted, the symbol ⋉ will be used exclusively for action groupoids, and not for semidirect products of groups.Morita equivalence. Let X be a manifold and Y • be a Lie groupoid, and let φ : X → Y 0 be a smooth map. Then φ is essentially surjective if t • pr 1 : Y 1 × s ,Y 0 ,φ X → Y 0 , which sends (g, x) → t(g) is a surjective submersion. More generally, a morphism φ • : X • → Y • of Lie groupoids is called essentially surjective if the map φ 0 is essentially surjective. A morphism φ • : X • → Y • is fully faithful if the square

Nonrational polytopes and fans in toric geometry

“…The definition is quite involved; for our purposes it is sufficient to recall that the datum of a stacky torus G is equivalent to their notion of quasilattice. In [27] Hoffman introduces the starting triple (∆, G, Λ f ∈F max ), where ∆ is a convex simple polytope, G is a stacky torus, F max is the set of facets of ∆ and Λ f is a free subgroup of rank 1, given by the intersection of ∂(Q) with the straight line normal to the facet f . Notice that there is a unique inward pointing vector that generates Λ f .…”

“…Viceversa, any triple (∆, G, Λ f ∈F max ) uniquely determines a fundamental triple. Now, let us recall the following observation in [27]: assigning the triple (∆, G, Λ f ∈F max ) is equivalent to assigning the triple (∆, G, Λ f ∈F ), where F is the set of all faces of ∆ and the groups Λ f are subgroups of ∂( Q) satisfying certain conditions. The triple (∆, G, Λ f ∈F ) is said to be a decorated stacky moment polytope, for which a natural notion of isomorphism is given.…”

“…Notice that we can always construct, from a given fundamental triple, an object that lies in the family above. The articles [12,27] and, in the calibrated case, [34,14] can be viewed as instances of the mapping ≈ Ψ, [32] can be viewed as an instance of the mapping Ψ, and finally [45,46,4], and [34,14] can be viewed as instances of the mapping Ψ. In our understanding, in each of the above-mentioned constructions relative to the mappings ≈ Ψ and Ψ, there is a mapping that projects any space of the family to the toric quasifold.…”

Toric Quasifolds

“…We conclude by remarking that, in recent years, there has been a surge of interest in nonrational toric geometry, and several alternate approaches have been introduced, involving either foliations [14,15,26], presymplectic manifolds [29,35] or stacks [23,22,27]. An account on how several of these different points of view connect with ours and with each other can be found in [12].…”

Quasifold groupoids and diffeological quasifolds

“…Recent work involving quasifolds and symplectic geometry includes [2], [3], [4], and [15]. Hoffman [7] works with not-necessarily-effective quasifold groupoids as stacks. Our results can also be written in terms of (effective) stacks, but we leave this for another paper.…”