---
date: 2022-04-07 20:17
modification date: Thursday 7th April 2022 20:17:03
title: "GIT quotient"
aliases: [GIT quotients, categorical quotient, GIT, geometric quotient, quotient, linearize, linearization]
created: 2023-03-28T18:09
updated: 2023-03-28T18:09
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- Kirwan's paper: <https://www.jstor.org/stable/pdf/1971369.pdf>
- Links:
	- [VGIT](VGIT.md)
	- [Hilbert-Mumford](Hilbert-Mumford.md)

---

# GIT quotient

Main tools:

- [[the numerical criterion]], 
- [[Unsorted/Luna slice theorem]], and 
- [[Kirwan’s desingularization]]

# Motivation
![](attachments/Pasted%20image%2020220605122450.png)

![](attachments/Pasted%20image%2020220523002234.png)

![](attachments/Pasted%20image%2020220914195435.png)

![](attachments/2023-03-11git.png)
![](attachments/2023-03-11git2.png)

# Summary

![](attachments/Pasted%20image%2020220605121607.png)
![](attachments/Pasted%20image%2020220605121634.png)

# Construction

![](attachments/Pasted%20image%2020220523002247.png)


![](attachments/Pasted%20image%2020220411231808.png)
![](attachments/Pasted%20image%2020220914193932.png)

![](attachments/Pasted%20image%2020220407201712.png)
![](attachments/Pasted%20image%2020220407201809.png)

## Constructing the ring of invariants

Uses a [Reynolds operator](Reynolds%20operator)
![](attachments/Pasted%20image%2020220407201836.png)
![](attachments/Pasted%20image%2020220407201845.png)
![](attachments/Pasted%20image%2020220407202015.png)

# Geometric quotients

![](attachments/Pasted%20image%2020221120153740.png)

![](attachments/Pasted%20image%2020220407201931.png)
![](attachments/Pasted%20image%2020220407201953.png)

![](attachments/Pasted%20image%2020220914194136.png)

# Linearizations

![](attachments/Pasted%20image%2020221119111422.png)
![](attachments/Pasted%20image%2020221119111558.png)
![](attachments/Pasted%20image%2020221119111610.png)
![](attachments/Pasted%20image%2020221119111739.png)

# Projective GIT quotients

![](attachments/Pasted%20image%2020221119112033.png)


# Stack definition


![](attachments/Pasted%20image%2020221119194911.png)
# Examples

![](attachments/Pasted%20image%2020220914195303.png)
![](attachments/Pasted%20image%2020220914195315.png)

# Misc


Sometimes dual to a [VHS](Unsorted/VHS.md) description:
![](attachments/Pasted%20image%2020220523011857.png)