Avinguda del Doctor Maranon 44-50 08028, Barcelona, Spain
Universitat Politècnica de Catalunya
Miranda E., Planas A.
Equivariant Classification of $b^m$-symplectic Surfaces
2018, vol. 23, no. 4, pp. 355-371
Inspired by Arnold’s classification of local Poisson structures  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 bm-symplectic structures. In this paper we extend the classification scheme in  for $b^m$-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  is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.