Desingularizing $b^m$-symplectic structures

A $2n$-dimensional Poisson manifold $(M,\mathrm{\Pi })$ 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 $\mathrm{\Pi }$ into a family of symplectic forms ${\omega }_{ϵ}$ having the property that ${\omega }_{ϵ}$ is equal to the $b^m$-symplectic form dual to $\mathrm{\Pi }$ outside an $ϵ$-neighborhood of $Z$ and, in addition, converges to this form as $ϵ$ 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 ${\omega }_{ϵ}$'s. We will also prove versions of these results for $m$ odd; however, in the odd case the family ${\omega }_{ϵ}$ has to be replaced by a family of folded symplectic forms.