Moduli spaces of toric manifolds
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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.
KeywordsToric 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.
- 1.Audin, M., Cannas da Silva, A., Lerman, E.: Symplectic geometry of integrable systems. Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser, Basel (2003)Google Scholar
- 3.Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)Google Scholar
- 4.Cannas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764, Corrected 2nd Printing, Springer, Berlin (2008)Google Scholar
- 7.Cox, D.: Toric varieties and toric resolutions. In: Resolution of Singularities, Progress in Math. 181, pp. 259–284. Birkhäuser, Basel, Boston, Berlin (2000)Google Scholar
- 11.Fulton, W.: Introduction to Toric Varieties. Annals of Mathematical Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton (1993)Google Scholar
- 23.Pelayo, Á., Schmidt, B.: Maximal ball packings of symplectic-toric manifolds. Int. Math. Res. Not., ID rnm139, 24 (2008)Google Scholar
- 27.Pelayo, Á., Ratiu, T.S., Vũ Ngọc, S.: Symplectic bifurcation theory for integrable systems, arXiv: 1108.0328Google Scholar