Toric structures on bundles of projective spaces

Abstract

Recently, extending work by Karshon et al., Borisov and McDuff showed in The topology of toric symplectic manifolds that a given closed symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, in The topology of toric symplectic manifolds McDuff also showed that a product of two projective spaces $\mathbb{C}P^r \times \mathbb{C}P^s$ with any given symplectic form has a unique toric structure provided that $r, s \geq 2$. In contrast, the product $\mathbb{C}P^r \times \mathbb{C}P^1$ can be given infinitely many distinct toric structures, although 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 $\mathbb{C}P^r$ bundles over $\mathbb{C}P^s$ for some $r, s$. We show that there is a unique toric structure if $r \lt s$, and also that if $r, s \geq 2$, $M$ 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.