The topology of toric symplectic manifolds

TL;DR

This paper explores the topology of toric symplectic manifolds, establishing finiteness of toric structures, uniqueness in certain products, and constructing examples with multiple structures using advanced geometric techniques.

Contribution

It introduces new topological results, including bounds on the number of toric structures and methods to construct manifolds with multiple toric structures, expanding understanding of symplectic topology.

Findings

A closed symplectic manifold supports finitely many toric structures.

The product of two complex projective spaces (dimension ≥ 2) has a unique toric structure.

Constructed examples of manifolds with multiple toric structures using bundle and blow-up techniques.

Abstract

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 ring 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.