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This article is cited in 11 scientific papers (total in 11 papers)
Equivariant Classification of $b^m$-symplectic Surfaces
Eva Mirandaab, Arnau Planasb a IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL University, Sorbonne Université, 77 Avenue Denfert-Rochereau, 75014 Paris, France
b Universitat Politècnica de Catalunya and Barcelona Graduate School of Mathematics, BGSMath, Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Edifici P, UPC, Avinguda del Doctor Marañon 44-50 08028, Barcelona, Spain
Abstract:
Inspired by Arnold’s classification of local Poisson structures [1] in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of $b^m$-Poisson structures which can be also visualized using differential forms with singularities as $b^m$-symplectic structures. In this paper we extend the classification scheme in [24] for bm-symplectic surfaces to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct $b^m$-symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.
Keywords:
Moser path method, singularities, $b^m$-symplectic manifolds, group actions.
Received: 27.10.2017 Accepted: 28.05.2018
Citation:
Eva Miranda, Arnau Planas, “Equivariant Classification of $b^m$-symplectic Surfaces”, Regul. Chaotic Dyn., 23:4 (2018), 355–371
Linking options:
https://www.mathnet.ru/eng/rcd328 https://www.mathnet.ru/eng/rcd/v23/i4/p355
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