The topology of toric symplectic manifolds

Abstract

This is a collection of results on the topology of toric symplectic manifolds. Using an idea of Borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Further, the product of two projective spaces of complex dimension at least two (and with a standard product symplectic form) has a unique toric structure. We then discuss various constructions, using wedging to build a monotone toric symplectic manifold whose center is not the unique point displaceable by probes, and bundles and blow ups to form manifolds with more than one toric structure. The bundle construction uses the McDuff--Tolman concept of mass linear function. Using Timorin's description of the cohomology ring via the volume function we develop a cohomological criterion for a function to be mass linear, and explain its relation to Shelukhin's higher codimension barycenters.

36 pages, one figure; v2: proofs improved, small changes to some statements

References in corpus (3)

• Classification problems of toric manifolds via topology
• Remarks on invariants of hamiltonian loops
• Symmetry of a symplectic toric manifold

Cited by in corpus (9)

• The Kähler geometry of Bott manifolds
• Simply Connected Manifolds with Infinitely Many Toric Contact Structures and Constant Scalar Curvature Sasaki Metrics
• Remarks on the classification of quasitoric manifolds up to equivariant homeomorphism
• Remarks on Lagrangian intersections in toric manifolds
• Symplectic cohomological rigidity via toric degnerations
• Rigidity problems in toric topology, a survey
• Counting toric actions on symplectic four-manifolds
• On the Hofer-Zehnder conjecture for semipositive symplectic manifolds
• Toric Structures on Symplectic Bundles of Projective Spaces