The topology of toric symplectic manifolds

McDuff, Dusa¹

¹ Mathematics Department, Barnard College, Columbia University, MC4410, 3009 Broadway, New York NY 10027, USA

Geometry & topology, Tome 15 (2011) no. 1, pp. 145-190.

Résumé

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.

Détail

DOI : 10.2140/gt.2011.15.145

Keywords: toric symplectic manifold, monotone symplectic manifold, Fano polytope, monotone polytope, mass linear function, Delzant polytope, center of gravity, cohomological rigidity

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