---
date: 2021-10-28 19:35
modification date: Thursday 28th October 2021 19:35:10
title: nonabelian Hodge correspondence
aliases: [nonabelian Hodge theory]

---

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

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Higgs bundle](Unsorted/Higgs%20bundle.md)
	- [stable vector bundle](stable%20vector%20bundle)
	- [Kahler manifold](Kahler%20manifold)
	- [flat connection](flat%20connection.md)
	- [Dolbeaut moduli space](Dolbeaut%20moduli%20space)
	- [Hodge structure](Hodge%20structure)
	- [P=W conjecture](P=W%20conjecture)
	- [Hitchin moduli space](Hitchin%20moduli%20space.md)

---

# Motivations

- $H^*_{\dR}(X)\leadsto \mcm_{\dR}(X, G)$.
- $H^*(X; \Omega^*_X)\leadsto \mcm_{\mathrm{Dol}}(X, G)$
	- ![](attachments/2023-03-12exp.png)
- $H_\sing^*(X; \CC) \leadsto \mcm_{\mathrm{Betti}}(X; G)$.
	- ![](attachments/2023-03-12betti.png)

![](attachments/2023-03-12dobl.png)

Toward classifying holomorphic vector bundles:

![](attachments/2023-03-12reductive.png)


# nonabelian Hodge correspondence

![](attachments/2023-03-12.png)

![](attachments/Pasted%20image%2020220503233616.png)

![Pasted image 20211105101150.png](Pasted%20image%2020211105101150.png)
![Pasted image 20211105101223.png](Pasted%20image%2020211105101223.png)
![Pasted image 20211105101304.png](Pasted%20image%2020211105101304.png)
![Pasted image 20211105101323.png](Pasted%20image%2020211105101323.png)
![Pasted image 20211105101349.png](Pasted%20image%2020211105101349.png)
![Pasted image 20211105101432.png](Pasted%20image%2020211105101432.png)
![Pasted image 20211105101457.png](Pasted%20image%2020211105101457.png)

![Pasted image 20211105102033.png](Pasted%20image%2020211105102033.png)

See [Higgs bundle](Unsorted/Higgs%20bundle.md)

![Pasted image 20211105102529.png](Pasted%20image%2020211105102529.png)
![Pasted image 20211105102619.png](Pasted%20image%2020211105102619.png)

See [lambda connection](Unsorted/lambda%20connection.md)

![](attachments/Pasted%20image%2020220407004217.png)
![](attachments/Pasted%20image%2020220407004250.png)