From Lie groupoids to resolutions of singularities. Applications to symplectic and Poisson resolutions

TL;DR

This paper develops methods using symplectic Lie groupoids to construct resolutions of singularities in Poisson and symplectic geometry, providing a new approach to desingularization that leverages groupoid integration techniques.

Contribution

It introduces a novel framework for resolving singularities in Poisson manifolds via symplectic Lie groupoids, extending the application of groupoid integration to desingularization problems.

Findings

Constructed symplectic resolutions of Poisson leaf closures.

Demonstrated how Lie groupoids can lift and resolve singularities.

Applied techniques to cases with additional multi-vector field structures.

Abstract

We use the techniques of integration of Poisson manifolds into symplectic Lie groupoids to build symplectic resolutions (= desingularizations) of the closure of a symplectic leaf. More generally, we show how Lie groupoids can be used to lift singularities, in particular when one imposes a compatibility condition with an additional structure given by a multi-vector field.