Geometriae Dedicata

, Volume 169, Issue 1, pp 323–341 | Cite as

Moduli spaces of toric manifolds

  • Á. Pelayo
  • A. R. Pires
  • T. S. Ratiu
  • S. Sabatini
Original Paper


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.


Toric manifold Delzant polytope Moduli space  Metric space 

Mathematics Subject Classification (2000)

MSC 53D20 MSC 53D05 



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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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Á. Pelayo
    • 1
    • 2
  • A. R. Pires
    • 3
  • T. S. Ratiu
    • 4
  • S. Sabatini
    • 5
  1. 1.School of MathematicsInstitute of Advanced StudyPrincetonUSA
  2. 2.Mathematics DepartmentWashington UniversitySt. LouisUSA
  3. 3.Department of MathematicsCornell UniversityIthacaUSA
  4. 4.Section de Mathématiques and Bernoulli CenterLausanneSwitzerland
  5. 5.Section de MathématiquesLausanneSwitzerland

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