Non-Compact Symplectic Toric Manifolds

TL;DR

This paper extends Delzant's classification of symplectic toric manifolds to include non-compact cases, introducing new tools like unimodular local embeddings and degree two cohomology classes to classify these more general manifolds.

Contribution

It generalizes the classification framework to non-compact symplectic toric manifolds using unimodular local embeddings and cohomology classes, and constructs a new correspondence with symplectic toric bundles.

Findings

Classification of non-compact symplectic toric manifolds achieved

Introduction of unimodular local embeddings as a key concept

Establishment of an equivalence between manifolds with corners and symplectic toric bundles

Abstract

A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular ("Delzant") polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map on the quotient need not be an embedding. Moreover, even when the map on the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class on the quotient makes an appearance. Nevertheless, the quotient is a manifold with corners, and the induced map from the quotient to the dual of the Lie algebra is what we call a unimodular local embedding. We classify non-compact symplectic toric manifolds in terms of manifolds with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the Lie algebra of the corresponding torus. The main new ingredient is the construction of a symplectic toric manifold from such data. The proof passes through an equivalence of categories between symplectic toric manifolds and symplectic toric bundles over a fixed unimodular local embedding. This equivalence also gives a geometric interpretation of the degree two cohomology class.