Desingularizing b^m-symplectic structures
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A 2n-dimensional Poisson manifold (M; ) 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 paper we will discuss a desingularization procedure which, for m even, converts into a family of symplectic forms ! having the property that ! is equal to the bm-symplectic form dual to outside an -neighborhood of Z and, in addition, converges to this form as 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 ! 's. We will also prove versions of these results for m odd; however, in the odd case the family ! has to be replaced by a family of \folded
CitationMiranda, E., Guillemin, V., Weitsman, J. Desingularizing b^m-symplectic structures. "International mathematics research notices", 2017.