Internal absolute geometry I: desingularization

TL;DR

This paper introduces a new axiomatization of Grothendieck sites with additional structure, enabling the reconstruction of internal groupoids and providing a framework for desingularization in geometric contexts such as Nash blowups and moduli spaces.

Contribution

It presents a novel axiomatization of structured Grothendieck sites and describes sheaves that reconstruct internal groupoids, facilitating desingularization processes in complex geometric spaces.

Findings

Sheaves encode candidate holonomy groupoids for desingularization.

Applicable to Nash blowups resulting in almost regular foliations.

Potentially useful for various moduli space analyses.

Abstract

We introduce an axiomatization of Grothendieck sites with additional structure, and we describe sheaves that reconstruct groupoids which are internal to the site structure. This setting applies to various concrete situations, where a Nash blowup of a singular space results in an almost regular foliation. It also potentially applies to various types of moduli spaces. The sheaf can encode candidate holonomy groupoids that desingularize such spaces.