---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: Hodge theory
aliases:
  - Hodge theory
  - Hodge
  - de Rham theorem
  - interpreting Hodge numbers
created: 2023-04-02T12:06
updated: 2024-01-16T14:53
---

---
- Tags: 
	- #MOC
- Refs:
	- REU project: <http://math.uchicago.edu/~may/REU2018/REUPapers/Sheagren.pdf#page=1>
	- More detailed notes: <https://scholar.harvard.edu/files/pspark/files/harvardminorthesis.pdf#page=3>
	- [Class notes in Hodge theory](Hodge.pdf)
	- [Review of classical Hodge theory](mhm_chap2.pdf)
- Links:
	- [Hodge decomposition](Hodge%20decomposition.md)
	- [Hodge symmetry](Hodge%20symmetry)
	- Hodge weight
	- [Hodge filtration](Hodge%20filtration)
	- [Hodge diamond](Unsorted/Hodge%20diamonds.md)
	- [elliptic operator](Unsorted/elliptic%20operator.md)
	- [VHS](Unsorted/VHS.md)
	- [Unsorted/Hodge-Tate theory](Unsorted/Hodge-Tate%20theory.md)
	- [tangent bundle](tangent%20bundle.md)
	- [Sobolev space](Unsorted/elliptic%20operator.md)
	- [examples of computations of Hodge numbers](examples%20of%20computations%20of%20Hodge%20numbers.md)
	- [period](Unsorted/period.md)
	- [deformation theory](deformation%20theory.md)

---

Hodge numbers:

- $h^{0, 1} = \dim H^1(\OO_X) = \dim \T_{[L]} \Pic(X)$, so this measures how line bundles can be deformed.

# Hodge theory

![](attachments/Pasted%20image%2020220503103628.png)
![](attachments/Pasted%20image%2020220503103640.png)
![](attachments/Pasted%20image%2020220503103754.png)

![](attachments/Pasted%20image%2020220424180556.png)


# de Rham theorem

![](attachments/Pasted%20image%2020220424181417.png)

# Hodge star

![](attachments/Pasted%20image%2020220424181509.png)
![](attachments/Pasted%20image%2020220424181529.png)
![](attachments/Pasted%20image%2020220424181843.png)

![](attachments/Pasted%20image%2020220424183028.png)
![](attachments/Pasted%20image%2020220424182905.png)

# Hodge Laplacian

An [elliptic operator](Unsorted/elliptic%20operator.md).

![](attachments/Pasted%20image%2020220424181605.png)
![](attachments/Pasted%20image%2020220424181614.png)
![](attachments/Pasted%20image%2020220424181625.png)

# Hodge decomposition

![](attachments/Pasted%20image%2020220424181652.png)
![](attachments/Pasted%20image%2020220424181659.png)
![](attachments/Pasted%20image%2020220424181732.png)

# Bar duality

![](attachments/Pasted%20image%2020220424181803.png)

# Dolbeaut derivative

![](attachments/Pasted%20image%2020220424181824.png)
![](attachments/Pasted%20image%2020220424181855.png)
![](attachments/Pasted%20image%2020220424181911.png)

See [kahler manifold](Unsorted/Kahler.md)

# Hodge theorem and Hodge diamond

![](attachments/Pasted%20image%2020220424182022.png)