Acta Mathematica

Constructing integrable systems of semitoric type

Álvaro Pelayo and San Vũ Ngọc

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Abstract

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.

Note

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).

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892529

Digital Object Identifier
doi:10.1007/s11511-011-0060-4

Zentralblatt MATH identifier
1225.53074

Rights
2011 © Institut Mittag-Leffler

Citation

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. https://projecteuclid.org/euclid.acta/1485892529.


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