# On the volume elements of a manifold with transverse zeroes

## TL;DR

This paper extends Moser's classical volume form classification to forms with transversal zeroes, introducing a relative cohomology framework and comparing it with $b$-Poisson structures and desingularization techniques.

## Contribution

It generalizes volume form classification to forms with zeroes using relative cohomology and relates it to $b$-Poisson structures and desingularization methods.

## Key findings

- Introduces a cohomology classification for volume forms with transversal zeroes.
- Extends desingularization techniques to $b^m$-Nambu structures.
- Provides a comparison between volume form classification and $b$-Poisson structures.

## Abstract

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 set of 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 by Guillemin-Miranda-Weitsman and extend it to $b^m$-Nambu structures.