nLab symplectic realization

Definition

For $(X, \pi)$ a Poisson manifold, a symplectic realization of it is a symplectic manifold $(Y, \omega)$ and a Poisson map $(Y, \omega) \to (X, \pi)$ such that $Y \to X$ is a surjective submersion $Y \to X$.

Properties

Solution in terms of symplectic groupoids

For any symplectic groupoid $\Sigma$ with base a Poisson manifold $P$ the target map is a symplectic realization of $P$ and the source map is a symplectic realization of the opposite structure. Thus $\Sigma$ with its symplectic structure may be regarded as a desingularization of $P$ with its Poisson structure. Since the symplectic groupoid is the Lie integration of the Poisson Lie algebroid of the Poisson manifold, symplectic realization has been reduced to a problem in Lie theory.