  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.
