Desingularizing b^m-symplectic structures

Abstract

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.