A characteristic number of bundles determined by mass linear pairs

Abstract

Let $Δ$ 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 $(Δ, {\bf b})$. We prove the equivalence between the fact that $(Δ, {\bf b})$ 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 $Δ$ is a $Δ_{n-1}$ bundle over $Δ_1$; when $Δ$ is the polytope associated with the one point blow up of ${\mathbb C}P^n$; and when $Δ$ is the polytope associated with a Hirzebruch surface.