Poisson Structures With Compact Support

We construct several Poisson structures with compact support. For example, we show that any Poisson structure on ℝⁿ 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 that 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 that is symplectic outside a codimension one subset.