Equivariant resolutions of singularities as differential substacks

TL;DR

This paper establishes a correspondence between equivariant resolutions of singularities in manifolds and certain substacks of differential stacks associated with Lie groupoids, providing a new geometric framework for understanding singularities.

Contribution

It introduces a novel correspondence between equivariant resolutions and substacks of differential stacks, linking singularity resolution to Lie subgroupoids.

Findings

Equivariant resolutions correspond to specific substacks of differential stacks.

Construction of resolutions from Lie subgroupoids is demonstrated.

Framework bridges singularity resolution with differential stack theory.

Abstract

We show that there is an one-to-one correspondence between resolutions (equivariant w.r.t. a Lie groupoid action) of a singular subset of a manifold, and substacks (of a certain type) of the differential stack associated to the Lie groupoid in question. In particular, we show how to build an equivariant resolution out of Lie subgroupoids (of a certain type).