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