Acta Mathematica

, 206:93 | Cite as

Constructing integrable systems of semitoric type



Let (M, ω) be a connected, symplectic 4-manifold. A semitoric integrable system on (M, ω) essentially consists of a pair of independent, real-valued, smooth functions J and H on M, for which J generates a Hamiltonian circle action under which H is invariant. In this paper we give a general method to construct, starting from a collection of five ingredients, a symplectic 4-manifold equipped a semitoric integrable system. Then we show that every semitoric integrable system on a symplectic 4-manifold is obtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors, this gives a classification of semitoric integrable systems on 4-manifolds, in terms of five invariants.


Manifold Integrable System Smooth Manifold Symplectic Manifold Regular Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    Atiyah, M. F., Convexity and commuting Hamiltonians. Bull. London Math. Soc., 14 (1982), 1–15.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    Bourbaki, N., Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7). Actualités Scientifiques et Industrielles, 1333. Hermann, Paris, 1967.Google Scholar
  3. [3]
    Brandsma, H., Paracompactness, covers and perfect maps, in Topology Explained. Topology Atlas, York University, Toronto, ON, 2003. Available at
  4. [4]
    Daverman, R. J., Decompositions of Manifolds. AMS Chelsea Publishing, Providence, RI, 2007.MATHGoogle Scholar
  5. [5]
    Delzant, T., Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France, 116 (1988), 315–339.MATHMathSciNetGoogle Scholar
  6. [6]
    Dufour, J.-P. & Molino, P., Compactification d’actions de R n et variables action-angle avec singularités, in Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., 20, pp. 151–167. Springer, New York, 1991.Google Scholar
  7. [7]
    Duistermaat, J. J., On global action-angle coordinates. Comm. Pure Appl. Math., 33 (1980), 687–706.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Eliasson, L. H., Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case. Comment. Math. Helv., 65 (1990), 4–35.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    Gross, M. & Siebert, B., Affine manifolds, log structures, and mirror symmetry. Turkish J. Math., 27 (2003), 33–60.MATHMathSciNetGoogle Scholar
  10. [10]
    — Mirror symmetry via logarithmic degeneration data. I. J. Differential Geom., 72 (2006), 169–338.MATHMathSciNetGoogle Scholar
  11. [11]
    — Mirror symmetry via logarithmic degeneration data, II. J. Algebraic Geom., 19 (2010), 679–780.MATHMathSciNetGoogle Scholar
  12. [12]
    — From real affine geometry to complex geometry. Preprint, 2007. arXiv:math/0703822 [math.AG].Google Scholar
  13. [13]
    Guillemin, V. & Sternberg, S., Convexity properties of the moment mapping. Invent. Math., 67 (1982), 491–513.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    Leung, N. C. & Symington, M., Almost toric symplectic four-manifolds. J. Symplectic Geom., 8 (2010), 143–187.MATHMathSciNetGoogle Scholar
  15. [15]
    Miranda, E. & Zung, N. T., Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems. Ann. Sci. École Norm. Sup., 37 (2004), 819–839.MATHGoogle Scholar
  16. [16]
    Pelayo, A. & Vũ Ngọc, S., Semitoric integrable systems on symplectic 4-manifolds. Invent. Math., 177 (2009), 571–597.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    Symington, M., Four dimensions from two in symplectic topology, in Topology and Geometry of Manifolds (Athens, GA, 2001), Proc. Sympos. Pure Math., 71, pp. 153–208. Amer. Math. Soc., Providence, RI, 2003.Google Scholar
  18. [18]
    Vũ Ngọc, S., On semi-global invariants for focus-focus singularities. Topology, 42 (2003), 365–380.CrossRefMathSciNetGoogle Scholar
  19. [19]
    — Moment polytopes for symplectic manifolds with monodromy. Adv. Math., 208 (2007), 909–934.CrossRefMathSciNetGoogle Scholar
  20. [20]
    Willard, S., General Topology. Dover, Mineola, NY, 2004.MATHGoogle Scholar
  21. [21]
    Wloka, J. T., Rowley, B. & Lawruk, B., Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge, 1995.CrossRefMATHGoogle Scholar
  22. [22]
    Ziegler, G. M., Lectures on Polytopes. Graduate Texts in Mathematics, 152. Springer, New York, 1995.MATHGoogle Scholar
  23. [23]
    Zung, N. T., Symplectic topology of integrable Hamiltonian systems. I. Arnold–Liouville with singularities. Compositio Math., 101 (1996), 179–215.MATHMathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2011

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonU.S.A.
  2. 2.Washington University in St. Louis, Department of MathematicsSt. LouisU.S.A.
  3. 3.Institut de Recherches Mathématiques de Rennes, Université de Rennes 1Rennes cedexFrance

Personalised recommendations