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2011 Constructing integrable systems of semitoric type
Álvaro Pelayo, San Vũ Ngọc
Author Affiliations +
Acta Math. 206(1): 93-125 (2011). DOI: 10.1007/s11511-011-0060-4


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.

Funding Statement

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).


Download Citation

Álvaro Pelayo. San Vũ Ngọc. "Constructing integrable systems of semitoric type." Acta Math. 206 (1) 93 - 125, 2011.


Received: 3 April 2009; Published: 2011
First available in Project Euclid: 31 January 2017

zbMATH: 1225.53074
MathSciNet: MR2784664
Digital Object Identifier: 10.1007/s11511-011-0060-4

Rights: 2011 © Institut Mittag-Leffler

Vol.206 • No. 1 • 2011
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