Toric Structures on Symplectic Bundles of Projective Spaces

TL;DR

This paper investigates the number of toric structures on symplectic manifolds with second cohomology of dimension two, showing finiteness results and uniqueness conditions for certain classes of these manifolds.

Contribution

It extends previous results by classifying toric structures on $C P^r$ bundles over $C P^s$ with $ ext{dim } H^2=2$, establishing conditions for uniqueness and finiteness.

Findings

Unique toric structure if r < s.

At most finitely many toric structures if r,s ≥ 2.

Finiteness of toric structures independent of specific symplectic form.

Abstract

Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces $\bC P^r\times \bC P^s$ with any given symplectic form has a unique toric structure provided that $r,s\geq 2$. In contrast, the product $\bC P^r \times \bC P^1$ can be given infinitely many distinct toric structures, though only a finite number of these are compatible with each given symplectic form $\omega$. In this paper we extend these results by considering the possible toric structures on a toric symplectic manifold $(M,\omega)$ with $\dim H^2(M)=2$. In particular, all such manifolds are $\bC P^r$ bundles over $\bC P^s$ for some $r,s$. We show that there is a unique toric structure if $r<s$, and also that if $r,s\geq 2$ then $M$ has at most finitely many distinct toric structures that are compatible with any symplectic structure on $M$. Thus, in this case the finiteness result does not depend on fixing the symplectic structure. We will also give other examples where $(M,\omega)$ has a unique toric structure, such as the case where $(M,\omega)$ is monotone.