TY - JOUR

T1 - Polytopes with mass linear functions II

T2 - The four-dimensional case

AU - McDuff, Dusa

AU - Tolman, Susan

N1 - Funding Information:
This research was partially supported by NSF grant DMS 0905191 (to D.M.), and by NSF grant DMS

PY - 2013

Y1 - 2013

N2 - This paper continues the analysis begun in Polytopes with mass linear functions, Part I of the structure of smooth moment polytopes that support a mass linear function. As explained there, besides its purely combinatorial interest, this question is relevant to the study of the homomorphism π1(Tn)→π1(Symp(MΔ, ωΔ)) from the fundamental group of the torus T n to that of the group of symplectomorphisms of the 2n-dimensional symplectic toric manifold (MΔ,ωΔ) associated to Δ. In Part I, we made a general investigation of this question and classified all mass linear pairs (Δ,H) in dimensions up to 3. The main result of the current paper is a classification of all four-dimensional examples. Along the way, we investigate the properties of general constructions such as fibrations, blow ups, and expansions (or wedges), describing their effect on both moment polytopes and 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 blowups preserve mass linearity.

AB - This paper continues the analysis begun in Polytopes with mass linear functions, Part I of the structure of smooth moment polytopes that support a mass linear function. As explained there, besides its purely combinatorial interest, this question is relevant to the study of the homomorphism π1(Tn)→π1(Symp(MΔ, ωΔ)) from the fundamental group of the torus T n to that of the group of symplectomorphisms of the 2n-dimensional symplectic toric manifold (MΔ,ωΔ) associated to Δ. In Part I, we made a general investigation of this question and classified all mass linear pairs (Δ,H) in dimensions up to 3. The main result of the current paper is a classification of all four-dimensional examples. Along the way, we investigate the properties of general constructions such as fibrations, blow ups, and expansions (or wedges), describing their effect on both moment polytopes and 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 blowups preserve mass linearity.

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U2 - 10.1093/imrn/rns147

DO - 10.1093/imrn/rns147

M3 - Article

AN - SCOPUS:84881155044

SN - 1073-7928

VL - 2013

SP - 3509

EP - 3599

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 15

ER -