Poisson structures on smooth 4-manifolds
Differential Geometry
2016-09-23 v3 Mathematical Physics
math.MP
Symplectic Geometry
Abstract
We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely many circles and isolated points. The Poisson bivector has rank 0 on the singularities, where we give its local form explicitly.
Cite
@article{arxiv.1406.7105,
title = {Poisson structures on smooth 4-manifolds},
author = {Luis C. García-Naranjo and Pablo Suárez-Serrato and Ramón Vera},
journal= {arXiv preprint arXiv:1406.7105},
year = {2016}
}
Comments
v3: 17pgs. We shortened, and streamlined, both the proof and the exposition. The main result now follows from a formula used by Damianou-Petalidou, attributed to Flaschka-Ratiu