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Integrable systems on singular symplectic manifolds: from local to global

Authors :
Cardona Aguilar, Robert
Miranda Galcerán, Eva|||0000-0001-9518-5279
Universitat Politècnica de Catalunya. Doctorat en Matemàtica Aplicada
Universitat Politècnica de Catalunya. Departament de Matemàtiques
Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions
Source :
UPCommons. Portal del coneixement obert de la UPC, Universitat Politècnica de Catalunya (UPC)
Publication Year :
2021

Abstract

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 [36] and [37] 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 bsymplectic forms in [36]. 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. Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Both authors are supported by the grants reference number 2017SGR932 (AGAUR) and PID2019-103849GB-I00 / AEI / 10.13039/501100011033.

Details

Language :
English
Database :
OpenAIRE
Journal :
UPCommons. Portal del coneixement obert de la UPC, Universitat Politècnica de Catalunya (UPC)
Accession number :
edsair.dedup.wf.001..031a7a9814f8d64f357fe2ae51b65a69