Constructing integrable systems of semitoric type

Álvaro Pelayo School of Mathematics, Institute for Advanced Study San Vũ Ngọc Institut de Recherches Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, Rennes cedex, France

Differential Geometry mathscidoc:1701.10002

Acta Mathematica, 206, (1), 93-125, 2009.4
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.
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@inproceedings{álvaro2009constructing,
  title={Constructing integrable systems of semitoric type},
  author={Álvaro Pelayo, and San Vũ Ngọc},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203356184710736},
  booktitle={Acta Mathematica},
  volume={206},
  number={1},
  pages={93-125},
  year={2009},
}
Álvaro Pelayo, and San Vũ Ngọc. Constructing integrable systems of semitoric type. 2009. Vol. 206. In Acta Mathematica. pp.93-125. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203356184710736.
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