Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 055, 37 pages      arXiv:0907.2891
Contribution to the Special Issue on Poisson Geometry in Mathematics and Physics

Non-Compact Symplectic Toric Manifolds

Yael Karshon a and Eugene Lerman b
a) Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
b) Department of Mathematics, The University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA

Received August 15, 2014, in final form July 10, 2015; Published online July 22, 2015

A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular (''Delzant'') polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map on the quotient need not be an embedding. Moreover, even when the map on the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class on the quotient makes an appearance. Nevertheless, the quotient is a manifold with corners, and the induced map from the quotient to the dual of the Lie algebra is what we call a unimodular local embedding. We classify non-compact symplectic toric manifolds in terms of manifolds with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the Lie algebra of the corresponding torus. The main new ingredient is the construction of a symplectic toric manifold from such data. The proof passes through an equivalence of categories between symplectic toric manifolds and symplectic toric bundles over a fixed unimodular local embedding. This equivalence also gives a geometric interpretation of the degree two cohomology class.

Key words: Delzant theorem; symplectic toric manifold; Hamiltonian torus action; completely integrable systems.

pdf (625 kb)   tex (48 kb)


  1. Atiyah M.F., Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15.
  2. Bates L.M., Examples for obstructions to action-angle coordinates, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), 27-30.
  3. Boucetta M., Molino P., Géométrie globale des systèmes hamiltoniens complètement intégrables: fibrations lagrangiennes singulières et coordonnées action-angle à singularités, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), 421-424.
  4. Condevaux M., Dazord P., Molino P., Géométrie du moment, in Travaux du Séminaire Sud-Rhodanien de Géométrie, I, Publ. Dép. Math. Nouvelle Sér. B, Vol. 88, Univ. Claude-Bernard, Lyon, 1988, 131-160.
  5. Davis M.W., Januszkiewicz T., Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), 417-451.
  6. Dazord P., Delzant T., Le problème général des variables actions-angles, J. Differential Geom. 26 (1987), 223-251.
  7. Delzant T., Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339.
  8. Duistermaat J.J., On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706.
  9. Duistermaat J.J., Pelayo A., Reduced phase space and toric variety coordinatizations of Delzant spaces, Math. Proc. Cambridge Philos. Soc. 146 (2009), 695-718, arXiv:0704.0430.
  10. Guillemin V., Sternberg S., Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513.
  11. Guillemin V., Sternberg S., A normal form for the moment map, in Differential Geometric Methods in Mathematical Physics (Jerusalem, 1982), Math. Phys. Stud., Vol. 6, Reidel, Dordrecht, 1984, 161-175.
  12. Haefliger A., Salem É., Actions of tori on orbifolds, Ann. Global Anal. Geom. 9 (1991), 37-59.
  13. Jänich K., On the classification of $O(n)$-manifolds, Math. Ann. 176 (1968), 53-76.
  14. Joyce D., On manifolds with corners, in Advances in Geometric Analysis, Adv. Lect. Math. (ALM), Vol. 21, Int. Press, Somerville, MA, 2012, 225-258, arXiv:0910.3518.
  15. Karshon Y., Appendix (to D. McDuff and L. Polterovich), Invent. Math. 115 (1994), 431-434.
  16. Lerman E., Symplectic cuts, Math. Res. Lett. 2 (1995), 247-258.
  17. Lerman E., Contact toric manifolds, J. Symplectic Geom. 1 (2002), 785-828, math.SG/0107201.
  18. Lerman E., Meinrenken E., Tolman S., Woodward C., Nonabelian convexity by symplectic cuts, Topology 37 (1998), 245-259, dg-ga/9603015.
  19. Lerman E., Tolman S., Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), 4201-4230, dg-ga/9511008.
  20. Lukina O.V., Takens F., Broer H.W., Global properties of integrable Hamiltonian systems, Regul. Chaotic Dyn. 13 (2008), 602-644.
  21. Marle C.-M., Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), 227-251.
  22. Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
  23. Meinrenken E., Symplectic geometry, Unpublished lecture notes, 2000, available from
  24. Meyer K.R., Symmetries and integrals in mechanics, in Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 259-272.
  25. Michor P.W., Manifolds of differentiable mappings, Shiva Mathematics Series, Vol. 3, Shiva Publishing Ltd., Nantwich, 1980.
  26. Montaldi J., Persistence and stability of relative equilibria, Nonlinearity 10 (1997), 449-466.
  27. Moser J., On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294.
  28. Nielsen L.T., Transversality and the inverse image of a submanifold with corners, Math. Scand. 49 (1981), 211-221.
  29. Schwarz G.W., Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63-68.
  30. Sjamaar R., Lerman E., Stratified symplectic spaces and reduction, Ann. of Math. 134 (1991), 375-422.
  31. Yoshida T., Local torus actions modeled on the standard representation, Adv. Math. 227 (2011), 1914-1955, arXiv:0710.2166.
  32. Zung N.T., Symplectic topology of integrable Hamiltonian systems. II. Topological classification, Compositio Math. 138 (2003), 125-156, math.DG/0010181.

Previous article  Next article   Contents of Volume 11 (2015)