Symplectic toric stratified spaces with isolated singularities

Abstract

Let G be a torus with Lie algebra g. We provide a classification of two types of toric objects: symplectic toric cones and symplectic toric stratified spaces with isolated singularities. Both types of object are classified via orbital moment map and a second degree cohomology class.As symplectic toric stratified spaces with isolated singularities are locally modeled on symplectic toric cones, we first focus on classifying symplectic toric cones. We show that symplectic toric cones have a certain type of map ψ : W → g * (called homogeneous unimodular local embeddings) as orbital moment maps. Conversely, every ψ has a symplectic toric cone for which it is an orbital moment map. We classify the symplectic toric cones with orbital moment map ψ by showing that their isomorphism classes are in bijective correspondence with the first Chern classes H 2 (W ; Z G ) of principal G-bundles over W , for Z G the integral lattice ker(exp : g → G). This generalizes Lerman's classification of compact connected contact toric manifolds.Symplectic toric stratified spaces with isolated singularities are spaces with neighborhoods of singularities modeled on symplectic cones. We first show their quotients W are space stratified by manifolds with corners and their moment maps are a particular type of map ψ : W → g * called stratified unimodular local embeddings. Every stratified unimodular local embedding ψ is the orbital moment map of a symplectic toric stratified space. Finally, we show that, for any stratified unimodular local embedding ψ and for W reg the top stratum of W , the isomorphism classes of symplectic toric stratified spaces with isolated singularities with orbital moment map ψ are in bijective correspondence with the cohomology classes H 2 (W reg ; Z G ) × C, for C ⊂ H 2 (W reg ; R) a subspace dependent on the topology of W . This generalizes Burns, Guillemin, and Lerman's classification of the compact connected symplectic toric stratified spaces with isolated singularities.