Algebroid Desingularizable Poisson Structures

Abstract

We introduce algebroid desingularizable Poisson manifolds, a class of Poisson manifolds induced by symplectic Lie algebroids with almost-injective anchors, generalizing structures including log-symplectic, $b^m$-symplectic, $E$-symplectic manifolds, and hypersurface algebroids. We give an infinitesimal obstruction to the existence of such an algebroid for a general Poisson manifold, and then characterize the linear case by showing that the dual of a real finite-dimensional Lie algebra, equipped with the KKS Poisson structure is desingularizable if and only if it possesses an abelian ideal of dimension $\dim(\mathfrak{g})-r$, where $2r$ is the maximal coadjoint orbit dimension.