Poisson Structures in Dimensions Two and Three

Abstract

In the two-dimensional case the Jacobi identity is trivially satisfied, so it is easy to describe all Poisson structures on, for example, the affine plane. Yet, the local classification of Poisson structures in two dimensions is non-trivial, and has up to now only been accomplished under quite strong regularity assumptions on the singular locus of the Poisson structure, which can be identified with the zero locus of a local function on the manifold. This classification will be treated in detail in the first part of this chapter.

In the three-dimensional case, the Jacobi identity can be stated as the integrability condition of a distribution, which eventually leads to the symplectic foliation, or as the integrability condition of a differential one-form, dual to the Poisson structure with respect to a volume form (assuming that the manifold is orientable). Many results about three-dimensional Poisson manifolds are essentially true because the Poisson structure is of rank two (or zero). We therefore also present some general results about rank two Poisson structures. Combining dimensions two and three, we discuss Poisson surfaces which have a Du Val singularity at the origin.