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- Title
Desingularizing b<sup>m</sup>-Symplectic Structures.
- Authors
Guillemin, Victor; Miranda, Eva; Weitsman, Jonathan
- 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.
- Subjects
POISSON manifolds; SYMPLECTIC geometry; DARBOUX transformations; BIVECTORS; QUANTIZATION (Physics)
- Publication
IMRN: International Mathematics Research Notices, 2019, Vol 2019, Issue 10, p2981
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnx126