On the density function on moduli spaces of toric 4-manifolds

Abstract

AbstractThe optimal density function assigns to each symplectic toric manifoldMa positive numberd ≤1 obtained by considering the ratio between the maximum volume ofMwhich can be filled by symplectically embedded disjoint balls and the total symplectic volume ofM. In the toric version,Mis toric and the balls need to be embedded respecting the toric action onM. We give a brief survey of toric symplectic manifolds and the recent constructions of moduli space structure on them, and then we recall how to define a natural density function on this moduli space. We review previous work which explains how the study of the density function can be reduced to a problem in convex geometry, and use this correspondence to to give a simple description of the regions of continuity of the maximal density function when the dimension is 4.