Skip to main content

Moduli spaces of toric manifolds

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Notes

  1. 1.

    In the literature, these manifolds are usually called equivariantly symplectomorphic. However, the same name is sometimes also used for the notion in Definition 5, and so we use different names to distinguish the two.

References

  1. 1.

    Audin, M., Cannas da Silva, A., Lerman, E.: Symplectic geometry of integrable systems. Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser, Basel (2003)

  2. 2.

    Atiyah, M.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14, 1–15 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)

  4. 4.

    Cannas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764, Corrected 2nd Printing, Springer, Berlin (2008)

  5. 5.

    Čech, E.: On bicompact spaces. Ann. Math. (2) 38, 823–844 (1937)

    Article  Google Scholar 

  6. 6.

    Coffey, J., Kessler, L., Pelayo, Á.: Symplectic geometry on moduli spaces of \(J\)-holomorphic curves. Ann. Glob. Anal. Geom. 41, 265–280 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Cox, D.: Toric varieties and toric resolutions. In: Resolution of Singularities, Progress in Math. 181, pp. 259–284. Birkhäuser, Basel, Boston, Berlin (2000)

  8. 8.

    Delzant, T.: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France 116, 315–339 (1988)

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Duistermaat, J.J., Heckman, G.J.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69, 259–268 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Duistermaat, J.J., Pelayo, Á.: Reduced phase space and toric variety coordinatizations of Delzant spaces. Math. Proc. Cambr. Phil. Soc. 146, 695–718 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Fulton, W.: Introduction to Toric Varieties. Annals of Mathematical Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton (1993)

  12. 12.

    Guillemin, V.: Kaehler structures on toric varieties. J. Diff. Geom. 40, 285–309 (1994)

    MATH  MathSciNet  Google Scholar 

  13. 13.

    Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Harada, M., Holm, T.S., Jeffrey, L.C., Mare, A.-L.: Connectivity properties of moment maps on based loop groups. Geom. Topol. 10, 1607–1634 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Hausmann, J.-C., Knutson, A.: Polygon spaces and Grassmannian. L’Enseign. Math. 43, 173–198 (1997)

    MATH  MathSciNet  Google Scholar 

  16. 16.

    Hausmann, J.-C., Knutson, A.: The cohomology ring of polygon spaces. Ann. l’Inst. Fourier 48, 281–321 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Kapovich, M., Millson, J.: On the moduli space of polygons in the Euclidean plane. J. Diff. Geom. 42(1), 133–164 (1995)

    MATH  MathSciNet  Google Scholar 

  18. 18.

    Karshon, Y., Kessler, L., Pinsonnault, M.: A compact symplectic four-manifold admits only finitely many inequivalent toric actions. J. Symplectic Geom. 5, 139–166 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Krein, M., Milman, D.: On extreme points of regular convex sets. Studia Math. 9, 133–138 (1940)

    MathSciNet  Google Scholar 

  20. 20.

    Lang, R.: A note on the measurability of convex sets. Arch. Math. 47, 90–92 (1986)

    Article  MATH  Google Scholar 

  21. 21.

    Lerman, E., Tolman, S.: Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Am. Math. Soc. 349, 4201–4230 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Pelayo, Á.: Toric symplectic ball packing. Topol. Appl. 153, 3633–3644 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Pelayo, Á., Schmidt, B.: Maximal ball packings of symplectic-toric manifolds. Int. Math. Res. Not., ID rnm139, 24 (2008)

  24. 24.

    Pelayo, Á., Vũ Ngọc, S.: Semitoric integrable systems on symplectic 4-manifolds. Invent. Math. 177, 571–597 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Pelayo, Á., Vũ Ngọc, S.: Constructing integrable systems of semitoric type. Acta Math. 206, 93–125 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Pelayo, Á., Vũ Ngọc, S.: First steps in symplectic and spectral theory of integrable systems. Discret. Cont. Dyn. Syst. Ser. A 32, 3325–3377 (2012)

    Article  MATH  Google Scholar 

  27. 27.

    Pelayo, Á., Ratiu, T.S., Vũ Ngọc, S.: Symplectic bifurcation theory for integrable systems, arXiv: 1108.0328

  28. 28.

    Shephard, G.C., Webster, R.J.: Metrics for sets of convex bodies. Mathematika 12, 73–88 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  29. 29.

    Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481 (1937)

    Article  Google Scholar 

  30. 30.

    Ngọc, Vũ: S.: Moment polytopes for symplectic manifolds with monodromy. Adv. Math. 208, 909–934 (2007)

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. Sabatini.

Additional information

Á.P. was partly supported by NSF Grants DMS-0965738 and DMS-0635607, an NSF CAREER Award, a Leibniz Fellowship, Spanish Ministry of Science Grant MTM 2010-21186-C02-01, and by the Spanish National Research Council. A.R.P. was partly supported by an AMS-Simons Travel Grant. T.S.R. was partly supported by a MSRI membership, Swiss NSF grant 200021-140238, a visiting position at IHES, and by the government grant of the Russian Federation for support of research projects implemented by leading scientists, Lomonosov Moscow State University under the agreement No. 11.G34.31.0054.

Appendix: Polytopes

Appendix: Polytopes

Let \(V\) be a finite dimensional real vector space. A convex polytope \(S\) in \(V\) is the closed convex hull of a finite set \(\{v_1, \ldots , v_n\}\), i.e., the smallest convex set containing \(S\) or, equivalently,

$$\begin{aligned} \text{ Conv }\{v_1, \ldots , v_n\} : = \left\{ \sum _{i=1}^n a_i v_i\;\left| \; a_i \in [0,1], \;\sum _{i=1}^n a_i = 1 \right\} \right. . \end{aligned}$$

The dimension of \(\text{ Conv }\{v_1, \ldots , v_n\}\) is the dimension of the vector space \(\text{ span }_ \mathbb R \{v_1, \ldots , v_n\}\). A polytope is full dimensional if its dimension equals the dimension of \(V\).

Note that the definition implies that a convex polytope is a compact subset of \(V\). An extreme point of a convex subset \(C \subseteq V\) is a point of \(C\) which does not lie in any open line segment joining two points of \(C\). Thus, a convex polytope is the closed convex hull of its extreme points (by the Krein–Milman  [19] Theorem) called vertices. In particular, the set of vertices is contained in \(\{v_1, \ldots , v_n\}\). Clearly, there are infinitely many descriptions of the same polytope as a closed convex hull of a finite set of points. However, the description of a polytope as the convex hull of its vertices is minimal and unique.

There is another description of convex polytopes in terms of intersections of half-spaces. Let \(V^*\) be the dual of \(V\) and denote by \(\left\langle \,, \right\rangle :V ^*\times V \rightarrow \mathbb R \) the natural non-degenerate duality pairing. The positive (negative) half-space defined by \(\alpha \in V^*\) and \(a \in \mathbb R \) is defined by

$$\begin{aligned} V_{\alpha , a}^{\pm }: =\left\{ v\in V \,\left| \,\left\langle \alpha ,v\right\rangle \gtreqless a\right\} \right. . \end{aligned}$$

Traditionally, in the theory of convex polytopes, the half spaces are chosen to be of the form \(V_{\alpha , a}^{-}\). With these definitions, a convex polytope is given as a finite intersection of half-spaces. As for the convex hull representation, there are infinitely many representations of the same convex polytope as a finite intersection of half-spaces, but unlike it, a distinguished one that is minimal exists only for full dimensional polytopes, we will describe it in the next paragraph.

A face of a convex polytope is an intersection with a half-space satisfying the following condition: the boundary of the half-space does not contain any interior point of the polytope. Thus the faces of a convex polytope are themselves polytopes (and hence compact sets). Let \(m\) be the dimension of a convex polytope. Then the whole polytope is the unique \(m\)-dimensional face, or body, the \((m-1)\)-dimensional faces are called facets, the \(1\)-dimensional faces are the edges, and the \(0\)-dimensional faces are the vertices of the polytope. If the convex polytope is full-dimensional, its minimal and unique description as an intersection of half-spaces is given when the boundary of those half-spaces contain the facets.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pelayo, Á., Pires, A.R., Ratiu, T.S. et al. Moduli spaces of toric manifolds. Geom Dedicata 169, 323–341 (2014). https://doi.org/10.1007/s10711-013-9858-x

Download citation

Keywords

  • Toric manifold
  • Delzant polytope
  • Moduli space
  • Metric space

Mathematics Subject Classification (2000)

  • MSC 53D20
  • MSC 53D05