Toric symplectic singular spaces I: isolated singularities

Abstract

We generalize a theorem of Delzant classifying compact connected symplectic manifolds with completely integrable torus actions to certain singular symplectic spaces. The assumption on singularities is that if they are not finite quotient then they are isolated.

Symplectic toric stratified spaces with isolated singularities

“…In Section 13, we provide some examples of the use of all our main classification theorems. We also provide some explanation of how our classification of contact toric manifolds is an extension of the result of Lerman [15] in the compact connected case as well as an explanation of how our classification of symplectic toric stratified spaces is an extension of the result of Burns, Guillemin, and Lerman [5] again in the compact connected case.…”

“…In the case of symplectic toric stratified spaces, the precedent in the compact case comes from Burns, Guillemin, and Lerman [5]. Their definition agrees with ours with the exceptions that (1) The open dense piece of a symplectic toric stratified space with isolated singularities is allowed to be a symplectic toric orbifold; and (2) The symplectic toric cones on which neighborhoods of singularities are modelled must be Reeb type: for (M, ω) such a symplectic toric cone with homogeneous moment map µ modeling an open neighborhood of singular point x α , we assume there exists vector Y ∈ g such that µ − µ(x α ), Y > 0.…”

“…In 1991, Sjamaar and Lerman [23] showed that, in general, symplectic reduction results in a stratified space and, furthermore, that each strata inherits a symplectic form from the original manifold. In 2005, Burns, Guillemin, and Lerman [5] defined symplectic toric stratified spaces with isolated singularities and classified these in the compact connected case using the images of their moment maps.…”

Toric Geometry of the Regular Convex Polyhedra

“…When the polytope is rational, these quotients are examples of the symplectic stratified spaces described by Sjamaar-Lerman in [17]; in particular, the strata are either manifolds or orbifolds [2, Remark 6.6]. We remark that the nonsimple rational case was addressed also by Burns-Guillemin-Lerman in [7,8]; they gave an in-depth treatment and a classification theorem in the case of isolated singularities.…”

Cone decompositions of non-simple polytopes

“…These singular toric varieties appear in many contexts. They can arise, for instance, as a result of singular symplectic reduction on toric manifolds (see for example [8] for the case of isolated singularities) and as Calabi Yau manifolds, with implications in mirror symmetry and string theory [11]. It is then interesting to extend the existing simple-polytope decompositions to non-simple polytopes.…”