# Moduli spaces of toric manifolds

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## 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.

## Keywords

Toric manifold Delzant polytope Moduli space Metric space## Mathematics Subject Classification (2000)

MSC 53D20 MSC 53D05## Notes

### Acknowledgments

We would like to thank the anonymous referee who made many useful comments and clarifications which have significantly improved an earlier version of the paper. AP is grateful to Helmut Hofer for discussions and support. He also thanks Isabella Novik for discussions concerning general polytope theory, and Problem 4, during a visit to the University of Washington in 2010. The authors are also grateful to Victor Guillemin and Allen Knutson for helpful advice.

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