---
date: 2022-02-21 00:52
modification date: Monday 21st February 2022 00:52:20
title: Hodge decomposition
aliases: [Hodge decomposition, Hodge-Tate decomposition]

---


---

- Tags 
	- #arithmetic-geometry #AG/complex
- Refs:
	- <https://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/hokudai.pdf> #resources/notes 
	- <https://people.math.harvard.edu/~mpopa/571/chapter1.pdf> #resources/notes 
- Links:
	- [Hodge theory MOC](Unsorted/Hodge%20theory%20MOC.md)
	- [Exercises in complex AG](Exercises%20in%20complex%20AG.md)
	- [families](families.md)
	- [Kodaira vanishing](Unsorted/Kodaira%20vanishing.md)
	- [pure Hodge structure](pure%20Hodge%20structure.md)

---


# Hodge decomposition

Can always choose a unique harmonic form $\omega \in \ker \laplacian^k: \Omega^k\selfmap$ the Laplace-Beltrami operator generalizing the Laplacian $\laplacian: C^\infty\selfmap$.

![](attachments/Pasted%20image%2020220401095010.png)
![](attachments/Pasted%20image%2020220401095235.png)

![](attachments/Pasted%20image%2020220318201927.png)

![](attachments/Pasted%20image%2020220221005225.png)
![](attachments/Pasted%20image%2020220221005251.png)
![](attachments/Pasted%20image%2020220221005308.png)



Hodge numbers are constant in [families](Unsorted/families.md):
![](attachments/Pasted%20image%2020220401095635.png)

Hodge numbers are **not** [birational](Unsorted/dominant%20morphism.md) invariants:
![](attachments/Pasted%20image%2020220401095936.png)


![](attachments/Pasted%20image%2020220410225835.png)
![](attachments/Pasted%20image%2020220410225853.png)

# Examples

For [kahler manifolds](Unsorted/Kahler.md):
![](attachments/Pasted%20image%2020220410225923.png)