Journal of Symplectic Geometry

Volume 13 (2015)

Number 2

Fiber connectivity and bifurcation diagrams of almost toric integrable systems

Pages: 343 – 386



Álvaro Pelayo (Department of Mathematics, University of California at San Diego)

Tudor S. Ratiu (Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Switzerland)

San Vũ Ngoc (Institut Universitaire de France, & Institut de Recherches Mathématiques de Rennes, Université de Rennes 1, Rennes, France)


We describe the bifurcation diagrams of almost toric integrable Hamiltonian systems on a four dimensional symplectic manifold $M$, not necessarily compact. We prove that, under a weak assumption, the connectivity of the fibers of the induced singular Lagrangian fibration $M \to \mathbb{R}^2$ can be detected from the bifurcation diagram alone. In this case, it is possible to give a detailed description of the image of the fibration.


integrable systems, Lagrangian fibrations, singularities, symplectic geometry, quantum systems, affine structures

2010 Mathematics Subject Classification

Primary 37J35, 53D05. Secondary 14D06, 53D20, 53D50.

Published 13 April 2015