Non-Compact Symplectic Toric Manifolds

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.

References in corpus (1)

• On manifolds with corners

Cited by in corpus (24)

• Integrable systems, symmetries and quantization
• Convexity and Thimm's Trick
• The tropical momentum map: a classification of toric log symplectic manifolds
• Toric Vaisman Manifolds
• Toric geometry of $G_2$-manifolds
• Semitoric systems of non-simple type
• Seidel Representations and Quantum Cohomology of Toric Orbifolds
• Faithful Semitoric Systems
• Symplectic Origami
• Quasimap Floer cohomology and varying symplectic quotients
• A completely integrable system on $G_2$ coadjoint orbits
• The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry
• ECH embedding obstructions for rational surfaces
• From semi-toric systems to Hamiltonian S^1-spaces
• Scalar-flat Kahler metrics on non-compact symplectic toric 4-manifolds
• Classification of polytope metrics and complete scalar-flat Kähler 4-Manifolds with two symmetries
• Seidel Representation for Symplectic Orbifolds
• Packing Densities of Delzant and Semitoric Polygons
• Symplectic toric stratified spaces with isolated singularities
• Removing a ray from a noncompact symplectic manifold
• Canonical forms for perturbations of the harmonic oscillator
• The cutting construction of toric symplectic and contact manifolds
• Shifts of prepotentials (with an appendix by Michele Vergne)
• Categories of symplectic toric manifolds as Picard stack torsors