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 bm-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 g, equipped with the KKS Poisson structure, is desingularizable if and only if 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.
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BibTeX
@misc{rankin2026algebroid,
title = {{Algebroid Desingularizable Poisson Structures}},
author = {Shane Rankin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.22519},
eprint = {2605.22519},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.22519},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebroid Desingularizable Poisson Structures
AU - Shane Rankin
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2605.22519
UR - https://arxiv.org/abs/2605.22519
AB - 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 g, equipped with the KKS Poisson structure, is desingularizable if and only if 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.
ER -