Integrable systems on singular symplectic manifolds: from local to global
In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a b-symplectic form...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2021 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/364971 |
| Online Access: | https://hdl.handle.net/2117/364971 https://dx.doi.org/10.1093/imrn/rnab253 |
| Access Level: | Open access |
| Keyword: | Symplectic geometry Geometria simplèctica Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria diferencial |
| Summary: | In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a b-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [34] and [35] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and b-symplectic forms in [34]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set Z of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on Z¿. |
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