Inventiones mathematicae

, Volume 177, Issue 3, pp 571–597 | Cite as

Semitoric integrable systems on symplectic 4-manifolds

Open Access


Let (M,ω) be a symplectic 4-manifold. A semitoric integrable system on (M,ω) is a pair of smooth functions J,H∈C (M,ℝ) for which J generates a Hamiltonian S1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce.


  1. 1.
    Atiyah, M.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14, 1–15 (1982) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benoist, Y.: Actions symplectiques de groupes compacts. Geom. Dedic. 89, 181–245 (2002). Correction to “Actions symplectiques de groupes compacts”, MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Delzant, T.: Hamiltoniens périodiques et image convex de l’application moment. Bull. Soc. Math. France 116, 315–339 (1988) MATHMathSciNetGoogle Scholar
  4. 4.
    Dufour, J.P., Molino, P.: Compactification d’actions de ℝn et variables actions-angles avec singularités. In: Dazord, P., Weinstein, A., (eds.) Séminaire Sud-Rhodanien de Géométrie à Berkeley, vol. 20, pp. 151–167. MSRI, Berkeley (1989) Google Scholar
  5. 5.
    Duistermaat, J.J.: On global action-angle variables. Commun. Pure Appl. Math. 33, 687–706 (1980) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Duistermaat, J.J., Pelayo, A.: Symplectic torus actions with coisotropic principal orbits. Ann. Inst. Fourier 57 7, 2239–2327 (2007) MathSciNetGoogle Scholar
  7. 7.
    Dullin, H., Vũ Ngọc, S.: Symplectic invariants near hyperbolic-hyperbolic points. Regul. Chaotic Dyn. 12(6), 689–716 (2007) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Eliasson, L.H.: Hamiltonian systems with Poisson commuting integrals. Ph.D. thesis, University of Stockholm (1984) Google Scholar
  9. 9.
    Eliasson, L.H.: Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case. Comment. Math. Helv. 65(1), 4–35 (1990) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Guillemin, V., Sjamaar, R.: In: Convexity Properties of Hamiltonian Group Actions. CRM Monograph Series, vol. 26. American Mathematical Society, Providence (2005). iv+82 pp Google Scholar
  11. 11.
    Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984) MATHGoogle Scholar
  13. 13.
    Hirsch, M.: Differential Topology, vol. 33. Springer, Berlin (1980) Google Scholar
  14. 14.
    Karshon, Y.: Periodic Hamiltonian flows on four-dimensional manifolds. Mem. Am. Math. Soc. 141(672), viii+71 (1999) MathSciNetGoogle Scholar
  15. 15.
    Kirwan, F.C.: Convexity properties of the moment mapping, III. Invent. Math. 77, 547–552 (1984) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lerman, E., Tolman, S.: Hamiltonian torus actions on symplectic orbifolds. Trans. Am. Math. Soc. 349, 4201–4230 (1997) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lerman, E., Meinrenken, E., Tolman, S., Woodward, C.: Non-abelian convexity by symplectic cuts. Topology 37, 245–259 (1998) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Marle, C.-M.: Classification des actions hamiltoniennes au voisinage d’une orbite. C. R. Acad. Sci. Paris Sér. I Math. 299, 249–252 (1984) MATHMathSciNetGoogle Scholar
  19. 19.
    Marle, C.-M.: Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique. Rend. Semin. Mat. Univ. Politec. Torino 43, 227–251 (1985) MATHMathSciNetGoogle Scholar
  20. 20.
    Miranda, E., Zung, N.T.: Equivariant normal form for non-degenerate singular orbits of integrable Hamiltonian systems. Ann. Sci. Éc. Norm. Sup. (4) 37(6), 819–839 (2004) MATHGoogle Scholar
  21. 21.
    Ortega, J.-P., Ratiu, T.S.: A symplectic slice theorem. Lett. Math. Phys. 59, 81–93 (2002) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Pelayo, A.: Symplectic actions of 2-tori on 4-manifolds. Mem. Am. Math. Soc. (to appear) 91 pp. ArXiv:Math.SG/0609848
  23. 23.
    Sadovskií, D.A., Zhilinskií, B.I.: Monodromy, diabolic points, and angular momentum coupling. Phys. Lett. A 256(4), 235–244 (1999) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Sjamaar, R.: Convexity properties of the moment mapping re-examined. Adv. Math. 138(1), 46–91 (1998) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Vũ Ngọc, S.: On semi-global invariants for focus-focus singularities. Topology 42(2), 365–380 (2003) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Vũ Ngọc, S.: Moment polytopes for symplectic manifolds with monodromy. Adv. Math. 208(2), 909–934 (2007) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Zung, N.T.: Symplectic topology of integrable Hamiltonian systems. I. Arnold-Liouville with singularities. Compos. Math. 101(2), 179–215 (1996) MATHGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of California–BerkeleyBerkeleyUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA
  3. 3.Institut de Recherches Mathématiques de RennesUniversité de Rennes 1Rennes cedexFrance

Personalised recommendations