A characteristic number of bundles determined by mass linear pairs

TL;DR

This paper establishes a link between mass linear pairs in symplectic geometry and the vanishing of a specific characteristic number of associated symplectic fibrations, in particular cases involving certain polytopes.

Contribution

It proves the equivalence between mass linear pairs and the vanishing of a characteristic number for specific classes of Delzant polytopes.

Findings

Equivalence holds for $ ext{Delta}_{n-1}$ bundles over $ ext{Delta}_1$

Equivalence holds for polytopes from one point blow-ups of $ ext{CP}^n$

Equivalence holds for polytopes from Hirzebruch surfaces

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.