Desingularizing $b^m$-symplectic structures

Abstract

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.

new version, 13 pages, 3 figures, final version accepted at IMRN, International Mathematics Research Notices

References in corpus (3)

• Examples of integrable and non-integrable systems on singular symplectic manifolds
• Cotangent models for integrable systems
• Action-angle variables and a KAM theorem for b-Poisson manifolds

Cited by in corpus (12)

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• Reduction theory for singular symplectic manifolds and singular forms on moduli spaces
• On the volume elements of a manifold with transverse zeroes
• Constructions of b-semitoric systems
• Classical KMS Functionals and Phase Transitions in Poisson Geometry
• Obstructions for Symplectic Lie Algebroids
• Non-exactness of toric Poisson structures
• The Arnold conjecture for singular symplectic manifolds
• Integrable systems on singular symplectic manifolds: From local to global
• The singular Weinstein conjecture
• E-structures and almost regular Poisson manifolds
• Folded symplectic forms in contact topology