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Abstract: This is a collection of results on the topology of toric symplecticmanifolds. Using an idea of Borisov, we show that a closed symplectic manifoldsupports at most a finite number of toric structures. Further, the product oftwo projective spaces of complex dimension at least two and with a standardproduct symplectic form has a unique toric structure. We then discuss variousconstructions, using wedging to build a monotone toric symplectic manifoldwhose center is not the unique point displaceable by probes, and bundles andblow ups to form manifolds with more than one toric structure. The bundleconstruction uses the McDuff-Tolman concept of mass linear function. UsingTimorin-s description of the cohomology ring via the volume function we developa cohomological criterion for a function to be mass linear, and explain itsrelation to Shelukhin-s higher codimension barycenters.

Autor: Dusa McDuff


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