Integrable systems on singular symplectic manifolds: From local to global

Abstract

In this article we consider integrable systems on manifolds endowed with singular symplectic structures of order one. By singular symplectic structures of order one we mean structures which are symplectic away from an hypersurface along which the symplectic volume either goes to infinity or to zero in a transversal way (singularity of order one) resulting either in a b-symplectic form or a folded symplectic forms. The hypersurface where the form degenerates is called critical set. In this article we give a new impulse to the investigation of action-angle coordinates for this structures initiated in [30] and [29] by proving an action-angle theorem for folded symplectic integrable systems, establishing new cotangent models for these systems and investigating duality with b-integrable systems via desingularization. We provide global constructions of integrable systems and investigate obstructions for global existence of action-angle coordinates in both scenarios. The new topological obstructions found emanate from the topology of the critical set of the singular symplectic manifold Z. The existence of these obstructions in turn implies the existence of singularities for the integrable system on Z.