Polytopes with mass linear functions II: the 4-dimensional case

Abstract

This paper continues the analysis begun in {\it Polytopes with mass linear functions, Part I} of the structure of smooth moment polytopes $\Delta\subset \ft^*$ that support a mass linear function $H \in \ft$. As explained there, besides its purely combinatorial interest, this question is relevant to the study of the homomorphism $\pi_1(T^n)\to \pi_1(\Symp(M_\Delta, \omega_\Delta))$ from the fundamental group of the torus $T^n$ to that of the group of symplectomorphisms of the $2n$-dimensional symplectic toric manifold $(M_\Delta, \omega_\Delta)$ associated to $\Delta$. In Part I, we made a general investigation of this question and classified all mass linear pairs $(\Delta, H)$ in dimensions up to three. The main result of the current paper is a classification of all 4-dimensional examples. Along the way, we investigate the properties of general constructions such as fibrations, blowups and expansions (or wedges), describing their effect both on moment polytopes and on mass linear functions. We end by discussing the relation of mass linearity to Shelukhin's notion of full mass linearity. The two concepts agree in dimensions up to and including 4. However full mass linearity may be the more natural concept when considering the question of which blow ups preserve mass linearity.