Desingularizing $b^m$-symplectic structures

A $2n$-dimensional Poisson manifold $(M ,\Pi)$ is said to be $b^m$-symplectic if it is symplectic on the complement of a hypersurface $Z$ and has a simple Darboux canonical form at points of $Z$ which we will describe below. In this paper we will discuss a desingularization procedure which, for $m$ even, converts $\Pi$ into a family of symplectic forms $\omega_{\epsilon}$ having the property that $\omega_{\epsilon}$ is equal to the $b^m$-symplectic form dual to $\Pi$ outside an $\epsilon$-neighborhood of $Z$ and, in addition, converges to this form as $\epsilon$ tends to zero in a sense that will be made precise in the theorem below. We will then use this construction to show that a number of somewhat mysterious properties of $b^m$-manifolds can be more clearly understood by viewing them as limits of analogous properties of the $\omega_{\epsilon}$'s. We will also prove versions of these results for $m$ odd; however, in the odd case the family $\omega_{\epsilon}$ has to be replaced by a family of folded symplectic forms.