Toric structures on bundles of projective spaces (Q2347212)

Recently, extending work by \textit{Y. Karshon} et al. [ibid. 5, No. 2, 139--166 (2007; Zbl 1136.53060)], A. V. Borisov, and \textit{D. McDuff} [Geom. Topol. 15, No. 1, 145--190 (2011; Zbl 1218.14045)] showed that a given symplectic manifold \((M,\omega)\) has a finite number of distinct toric structures. Moreover, McDuff [ibid.] 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 \(r,s\geq2\). In contrast, the product \(\mathbb{C}P^r\times\mathbb{C}P^1\) can be given infinitely many distinct toric structures, though only a finite number of these are compatible with every given symplectic form \(\omega\). The aim of this paper is to 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 \(r,s\). The author shows that there is a unique toric structure if \(r<s\), and also that if \(r,s\geq2\) 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. Some examples where \((M,\omega)\) has a unique toric structure such as the case where \((M,\omega)\) is monotone, are given.