Acta Mathematica

Constructing integrable systems of semitoric type

Álvaro Pelayo and San Vũ Ngọc

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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.


The first author was partially supported by an NSF post-doctoral fellowship. This work was done while the first author was at the Massachusetts Institute of Technology (2007–2008) and at the University of California, Berkeley (2008–2010).

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Acta Math., Volume 206, Number 1 (2011), 93-125.

Received: 3 April 2009
First available in Project Euclid: 31 January 2017

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2011 © Institut Mittag-Leffler


Pelayo, Álvaro; Vũ Ngọc, San. Constructing integrable systems of semitoric type. Acta Math. 206 (2011), no. 1, 93--125. doi:10.1007/s11511-011-0060-4.

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  • A tiyah, M. F., Convexity and commuting Hamiltonians. Bull. London Math. Soc., 14 (1982), 1–15.
  • B ourbaki, 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.
  • B randsma, H., Paracompactness, covers and perfect maps, in Topology Explained. Topology Atlas, York University, Toronto, ON, 2003. Available at
  • D averman, R. J., Decompositions of Manifolds. AMS Chelsea Publishing, Providence, RI, 2007.
  • D elzant, T., Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France, 116 (1988), 315–339.
  • D ufour, J.-P. & M olino, P., Compactification d’actions de Rn 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.
  • D uistermaat, J. J., On global action-angle coordinates. Comm. Pure Appl. Math., 33 (1980), 687–706.
  • E liasson, L. H., Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case. Comment. Math. Helv., 65 (1990), 4–35.
  • G ross, M. & S iebert, B., Affine manifolds, log structures, and mirror symmetry. Turkish J. Math., 27 (2003), 33–60.
  • — Mirror symmetry via logarithmic degeneration data. I. J. Differential Geom., 72 (2006), 169–338.
  • — Mirror symmetry via logarithmic degeneration data, II. J. Algebraic Geom., 19 (2010), 679–780.
  • — From real affine geometry to complex geometry. Preprint, 2007. arXiv:math/0703822 [math.AG].
  • G uillemin, V. & S ternberg, S., Convexity properties of the moment mapping. Invent. Math., 67 (1982), 491–513.
  • L eung, N. C. & S ymington, M., Almost toric symplectic four-manifolds. J. Symplectic Geom., 8 (2010), 143–187.
  • M iranda, E. & Z ung, N. T., Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems. Ann. Sci. École Norm. Sup., 37 (2004), 819–839.
  • P elayo, A. & V ũ N gọc, S., Semitoric integrable systems on symplectic 4-manifolds. Invent. Math., 177 (2009), 571–597.
  • S ymington, 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.
  • V ũ N gọc, S., On semi-global invariants for focus-focus singularities. Topology, 42 (2003), 365–380.
  • — Moment polytopes for symplectic manifolds with monodromy. Adv. Math., 208 (2007), 909–934.
  • W illard, S., General Topology. Dover, Mineola, NY, 2004.
  • W loka, J. T., R owley, B. & L awruk, B., Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge, 1995.
  • Z iegler, G. M., Lectures on Polytopes. Graduate Texts in Mathematics, 152. Springer, New York, 1995.
  • Z ung, N. T., Symplectic topology of integrable Hamiltonian systems. I. Arnold–Liouville with singularities. Compositio Math., 101 (1996), 179–215.