Poisson structures with compact support

TL;DR

This paper constructs explicit Poisson structures with compact support, demonstrating their existence on various manifolds and showing how to modify existing structures to have compact support or vanish at boundaries.

Contribution

It provides explicit methods to create Poisson structures with compact support and extends their existence to a broad class of manifolds with boundary conditions.

Findings

Poisson structures with polynomial coefficients of degree at most two can be modified to have compact support.

Manifolds with boundary admit Poisson structures that vanish at the boundary and match the original structure elsewhere.

Any even-dimensional manifold admits a Poisson structure that is symplectic outside a codimension one subset.

Abstract

We explicitly construct several Poisson structures with compact support. For example, we show that any Poisson structure on $\R^n$ with polynomial coefficients of degree at most two can be modified outside an open ball, such that it becomes compactly supported. We also show that a symplectic manifold with either contact or cosymplectic boundary admits a Poisson structure which vanishes to infinite order at the boundary and agrees with the original symplectic structure outside an arbitrarily small tubular neighbourhood of the boundary. As a consequence, we prove that any even-dimensional manifold admits a Poisson structure which is symplectic outside a codimension one subset.