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The topology of toric symplectic manifolds

Dusa McDuff

Geometry & Topology 15 (2011) 145–190

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 algebra 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.

toric symplectic manifold, monotone symplectic manifold, Fano polytope, monotone polytope, mass linear function, Delzant polytope, center of gravity, cohomological rigidity
Mathematical Subject Classification 2000
Primary: 14M25, 53D05
Secondary: 52B20, 57S15
Received: 9 June 2010
Revised: 29 September 2010
Accepted: 15 November 2010
Published: 30 January 2011
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Frances Kirwan
Dusa McDuff
Mathematics Department
Barnard College
Columbia University
3009 Broadway
New York NY 10027