---
created: 2023-04-28T11:18
updated: 2024-01-09T14:00
aliases:
  - pure Hodge structure, mixed Hodge structure, Hodge structure
---

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Torelli](Unsorted/Torelli.md)
	- [vanishing cycles](Unsorted/vanishing%20cycles.md)
	- [Gauss-Manin](Unsorted/Gauss-Manin.md)
	- [horizontal forms](Unsorted/Riemannian%20Geometry.md)
	- [VHS](Unsorted/VHS.md)

---

# Hodge structures

Important idea: the grading may not vary holomorphically in families, but the *filtrations* do. This is because $\Fil^p \VV \intersect \bar{\Fil}^q \VV$ has a part that varies anti-holomorphically. Parameterized by [Hermitian symmetric spaces](Hermitian%20symmetric%20domains.md).
 
Example: take $V(y^2=x(x-1)(x-\lambda)) \to \PP^1\smts{0,1,\infty}$; then $\dim \VV = 2$ and $\Fil^1 \VV = \gens{dx/2y}$; trivializing on a small $\DD$ and integrating yields $\omega_1,\omega_2$ holomorphic, but doing this for $\Gr_{0, 1} \VV$ yields antiholomorphic periods $\bar\omega_i$.

Hodge tensor: for $V=\bigoplus V^{p, q}$ a Hodge decomposition, a map $V\tensorpower\CC k \to \RR$ which is $h\dash$invariant for $h$ the Hermitian metric.

![](attachments/Pasted%20image%2020220807160306.png)

# pure Hodge structure

![](attachments/Pasted%20image%2020220410230002.png)
![](attachments/Pasted%20image%2020220410230022.png)


# Weight filtration

![](attachments/2023-03-06monodrom-1.png)

![](attachments/2023-03-06exell.png)

# Limiting Hodge structure


![](attachments/2023-03-06limhodge-1.png)

![](attachments/2023-03-06limiimim.png)


# Mixed Hodge structures

For a complex variety Y the cohomology with compact support has a natural mixed Hodge structure.

![](2023-10-04.png)

The Hodge data that occurs in the cohomology of possibly non-smooth, non-proper algebraic varieties. Deligne constructs a functorial MHS for any complex such variety. Any family for which cohomology groups form a local system will automatically yield a VMHS.

![](attachments/2023-03-06mhs.png)

![](attachments/Pasted%20image%2020220528005228.png)

![](attachments/Pasted%20image%2020220424192912.png)

![](attachments/Pasted%20image%2020220410230039.png)
![](attachments/Pasted%20image%2020220410230131.png)

![](attachments/2023-03-08mixedhs.png)
![](attachments/2023-03-08splitting.png)

## Main theorem

![](attachments/Pasted%20image%2020220410230231.png)

## Nilpotent orbit theorem

![](attachments/2023-03-06nilp.png)

![](attachments/2023-03-08inlp.png)
![](attachments/2023-03-08assume.png)
![](attachments/2023-03-08lim.png)

Limit filtration formulation:
![](attachments/2023-03-08limfil.png)

Why nilpotent orbit:
![](attachments/2023-03-08nilp1.png)
![](attachments/2023-03-08nilp2.png)

Yields a MHS:
![](attachments/2023-03-08mhs.png)

## Deligne's theorem for quasiprojective varieties

![](attachments/2023-03-08deligne.png)
![](attachments/2023-03-08direction.png)

## Example

![](attachments/Pasted%20image%2020220410230210.png)

![](attachments/Pasted%20image%2020220422224136.png)

# Polarizations

Motivations:

- If $(V, J)\in \mods{\CC}$, a polarization $\psi$ is given by $\psi = \Im h$ for $h$ a positive definite [Hermitian metric](Hermitian%20symmetric%20domains.md).

- $X \da \CC^g/\Lambda$ is compact Kahler, thus $H^1(X;\ZZ)$ carries a HS which is polarized iff $X\in \Ab\Var$.
- If $\mcl\in \Locsys(X)$ underlies a polarizable [VHS](Unsorted/VHS.md) on $X$ quasiprojective, then the corresponding point in $\Rep(\pi_1 X, \GL_n)$ is semisimple.

![](attachments/Pasted%20image%2020220807162606.png)

![](attachments/2023-03-11polar.png)
![](attachments/2023-03-11polar-1.png)

# Misc

![2023-01-09](attachments/2023-01-09.png)
![2023-01-09-1](attachments/2023-01-09-1.png)