On the Volume Elements of a Manifold with Transverse Zeroes
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Moser proved in 1965 in his seminal paper  that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from a hypersurface where they fulfill a transversality assumption (b-Poisson structures). We do this using the desingularization technique introduced in  and extend it to bm-Nambu structures.
KeywordsMoser path method volume forms singularities b-symplectic manifolds
MSC2010 numbers53D05 53D17
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