Geometry & Topology

The topology of toric symplectic manifolds

Dusa McDuff

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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.

Article information

Geom. Topol., Volume 15, Number 1 (2011), 145-190.

Received: 9 June 2010
Revised: 29 September 2010
Accepted: 15 November 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 53D05: Symplectic manifolds, general
Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] 57S15: Compact Lie groups of differentiable transformations

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


McDuff, Dusa. The topology of toric symplectic manifolds. Geom. Topol. 15 (2011), no. 1, 145--190. doi:10.2140/gt.2011.15.145.

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