On Nash resolution of (singular) Lie algebroids

TL;DR

This paper introduces a Nash-type blow-up for Lie algebroids, providing a new resolution method that generalizes to singular subalgebroids, with concrete examples and extensions of existing constructions.

Contribution

It develops a Nash resolution for Lie algebroids, extending Mohsen's construction to singular subalgebroids, and offers explicit examples.

Findings

Constructed a Nash-type blow-up for Lie algebroids with a short exact sequence.

Extended the Nash resolution to singular subalgebroids following Mohsen's approach.

Provided concrete examples illustrating the resolution process.

Abstract

Any Lie algebroid $A$ admits a Nash-type blow-up $\mathrm{Nash}(A)$ that sits in a nice short exact sequence of Lie algebroids $0\rightarrow K\rightarrow \mathrm{Nash}(A)\rightarrow \mathcal{D}\rightarrow 0$ with $K$ a Lie algebra bundle and $\mathcal{D}$ a Lie algebroid whose anchor map is injective on an open dense subset. The base variety is a blowup determined by the singular foliation of $A$. We provide concrete examples. Moreover, we extend the construction following Mohsen's to singular subalgebroids in the sense of Androulidakis-Zambon.