Abstract

I will define a notion of resolution of a proper action (or, more generally, of a proper groupoid). Such resolutions always exist but are not canonical. However, for so-called polar actions (or, more generally, polar groupoids), I will describe a canonical construction of a resolution, which can be used to show that the leaf space carries a canonical orbifold structure. I will illustrate this construction with two main examples: (i) the adjoint action, where it allows one to identify the classical Weyl group with the orbifold fundamental group; and (ii) toric complex manifolds, where the resolution can be described in terms of the real part of the toric manifold. This talk is based on joint work with Marius Crainic and David Martínez Torres, as well as on discussions with Maarten Mol.