On the Volume Elements of a Manifold with Transverse Zeroes

International audienceMoser proved in 1965 in his seminal paper [15] that two volume forms ona compact manifold can be conjugated by a diffeomorphism, that is to saythey are equivalent, if and only if their associated cohomology classesin the top cohomology group of a manifold coincide. In particular, thisyields a classification of compact symplectic surfaces in terms of DeRham cohomology. In this paper we generalize these results for volumeforms admitting transversal zeroes. In this case there is also acohomology capturing the classification: the relative cohomology withrespect to the critical hypersurface. We compare this classificationscheme with the classification of Poisson structures on surfaces whichare symplectic away from a hypersurface where they fulfill atransversality assumption ( b-Poisson structures). We do this using thedesingularization technique introduced in [10] and extend it to b m -Nambu structures