Desingularizing bᵐ-Symplectic Structures

A $2n$-dimensional Poisson manifold (M ,Π) is said to be bᵐ-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 article, we will discuss a desingularization procedure which, for m even, converts Π into a family of symplectic forms ωε having the property that ωε is equal to the bᵐ-symplectic form dual to Π 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 bᵐ-manifolds can be more clearly understood by viewing them as limits of analogous properties of the ωε’s. We will also prove versions of these results for m odd; however, in the odd case the family ωε has to be replaced by a family of “folded” symplectic forms.