Moduli spaces of toric manifolds

TL;DR

This paper introduces a new metric on the moduli space of four-dimensional symplectic toric manifolds, analyzing its topological properties such as connectedness, compactness, and completeness.

Contribution

It constructs a distance based on the Duistermaat-Heckman measure and Hausdorff metric specifically for four-dimensional cases, providing new insights into the space's topology.

Findings

The moduli space is path-connected.

The space is compact.

The space is complete.

Abstract

We construct a distance on the moduli space of symplectic toric manifolds of dimension four. Then we study some basic topological properties of this space, in particular, path-connectedness, compactness, and completeness. The construction of the distance is related to the Duistermaat-Heckman measure and the Hausdorff metric. While the moduli space, its topology and metric, may be constructed in any dimension, the tools we use in the proofs are four-dimensional, and hence so is our main result.