Algebroid Desingularizable Poisson Structures

Abstract

We introduce algebroid desingularizable Poisson manifolds, those Poisson manifolds induced by a symplectic Lie algebroid with an almost-injective anchor. This class coincides with that of $E$-symplectic manifolds, and contains log-symplectic manifolds and $b^m$-symplectic manifolds. We give an infinitesimal obstruction to desingularizability: at a zero of the Poisson structure the isotropy Lie algebra must contain an abelian ideal of at least half the dimension of the manifold, so no Poisson structure with a semisimple isotropy Lie algebra can be desingularized. We then characterize the linear case, showing that the dual of a real $n$-dimensional Lie algebra $\mathfrak{g}$, equipped with the KKS Poisson structure, is desingularizable if and only if $\mathfrak{g}$ possesses an abelian ideal of dimension $n-r$, where $2r$ is the maximal coadjoint orbit dimension. In this case, we construct a desingularizing algebroid explicitly.

arXiv comment12 pages, updated to reflect the version to appear in the Journal of Geometry and Physics