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

TL;DR

This paper surveys the moduli space of toric 4-manifolds, explores the density function related to symplectic embeddings, and characterizes its continuity regions using convex geometry.

Contribution

It provides a concise overview of the moduli space structure of toric 4-manifolds and describes the continuity regions of the density function through convex geometric methods.

Findings

The density function's continuity regions are characterized in 4-dimensional cases.

A reduction of the density function study to convex geometry problems is established.

The paper offers a survey of the moduli space and the density function's properties in toric symplectic manifolds.

Abstract

The optimal density function assigns to each symplectic toric manifold $M$ a number $0 < d \leq 1$ obtained by considering the ratio between the maximum volume of $M$ which can be filled by symplectically embedded disjoint balls and the total symplectic volume of $M$. In the toric version of this problem, $M$ is toric and the balls need to be embedded respecting the toric action on $M$. The goal of this note is first to give a brief survey of the notion of toric symplectic manifold and the recent constructions of moduli space structure on them, and recall how to define a natural density function on this moduli space. Then we review previous works which explain 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 $4$.