Blow-ups of Lie groupoids and Lie algebroids

TL;DR

This thesis thoroughly explains and compares various methods for constructing blow-ups of Lie groupoids and Lie algebroids, providing new insights and unifying different approaches within a geometric framework.

Contribution

It offers a detailed exposition of existing blow-up constructions for Lie groupoids and algebroids, and introduces a general geometric blow-up construction that unifies previous methods.

Findings

Unified blow-up construction for Lie groupoids and algebroids.

Demonstrated Morita invariance of the blow-up construction.

Provided explicit examples including blow-ups along submanifolds and leaves.

Abstract

In this master's thesis, we will go into the (projective) blow-up construction for Lie groupoids and Lie algebroids. In the literature, there are different methods to be found on how to do this, especially for Lie groupoids. The main goal of the thesis is to explain, in detail, the Lie groupoid and the Lie algebroid blow-up constructions, but also to examine and compare different points of view. More explicitly, we will rigorously explain the blow-up construction for Lie groupoids by Claire Debord and Georges Skandalis. Moreover, we will show that the blow-up construction for Lie groupoids by Songhao Li and Marco Gualtieri, and the construction by Kirsten Wang, fit into this setting. Also, we will show that, analogously, we obtain a general geometric blow-up construction for Lie algebroids. This construction for Lie algebroids coincides with the construction of lower elementary modification in the codimension one case (by e.g. Songhao Li and Marco Gualtieri, Melinda Lanius, or Ralph Klaasse). Examples that are discussed include the blow-up of a pair groupoid (resp. tangent bundle) along a pair groupoid (resp. tangent bundle), the blow-up of a groupoid (resp. algebroid) along the groupoid (resp. algebroid) restricted to a saturated submanifold, and the blow-up of a regularly foliated manifold along a leaf. The blow-up construction by Debord and Skandalis uses the theory of deformation to the normal cones. We will give a broad introduction to this theory and go into some of its known applications. We also discuss the observation made by Debord and Skandalis about Morita invariance of the construction of blow-up and of deformation to the normal cone.