A characteristic number of bundles determined by mass linear pairs

Abstract

Let $\Delta$ be a Delzant polytope in ${\mathbb R}^n$ and ${\bf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(\Delta, {\bf b})$. We prove the equivalence between the fact that $(\Delta, {\bf b})$ is a mass linear pair (D. McDuff, S. Tolman, {\em Polytopes with mass linear functions, part I.} {\tt arXiv:0807.0900 [math.SG]}) and the vanishing of a characteristic number of $E$ in the following cases: When $\Delta$ is a $\Delta_{n-1}$ bundle over $\Delta_1$; when $\Delta$ is the polytope associated with the one point blow up of ${\mathbb C}P^n$; and when $\Delta$ is the polytope associated with a Hirzebruch surface.