---
date: 2022-04-06 00:07
modification date: Wednesday 6th April 2022 00:07:11
title: VHS
aliases:
  - VHS
  - period domain
  - polarizable VHS
  - Griffiths transversality
created: 2023-04-06T12:06
updated: 2024-02-21T18:00
---

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [[Kodaira-Parshin trick]]
	- [Hodge structure](Unsorted/pure%20Hodge%20structure.md)
	  
---

# VHS

Parameterized by [Hermitian symmetric space](Hermitian%20symmetric%20domains.md).

Families varying holomorphically: for $\VV$ over $S$, need a holomorphic morphism $\psi: S\to \Gr_{\vector d}(V)$ where $\vector d$ corresponds to the the flag $F_p$ defining $\VV_p$ at $p\in S$.
Differentiate to get $d\psi: \T_p S\to \T_{F_p}\Gr_{\vector d}(V) \subseteq \bigoplus_{k\geq 0} \Hom(F_p^k, V/F_p^k)$; if $\im d\psi \subset \bigoplus_{k\geq 0}\Hom(F_p^{k-1}, V/F_p^{k-1})$ then this defines a VHS.
Every irreducible Hermitian symmetric domain is a moduli of VHS for some vector space $V$.

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

![](attachments/Pasted%20image%2020220523003245.png)
![](attachments/Pasted%20image%2020220523003236.png)
![](attachments/Pasted%20image%2020220523003440.png)

![](attachments/Pasted%20image%2020220406000713.png)
![](attachments/Pasted%20image%2020220406000723.png)

![](attachments/Pasted%20image%2020220429213927.png)

Use in Ellenberg-Lawrence-Venkatesh: see [[repulsion]].

![](attachments/Pasted%20image%2020220514202713.png)

![](2024-02-21-4.png)

See [nonabelian Hodge theory](nonabelian%20Hodge%20correspondence.md):
![](2024-02-21-5.png)


# Polarizable VHSs

![](attachments/Pasted%20image%2020220526003107.png)

## Facts

![](attachments/Pasted%20image%2020220526003257.png)
![](attachments/Pasted%20image%2020220526003307.png)

# Examples

![](attachments/Pasted%20image%2020220523003457.png)

# Misc

Relation to [local system](Unsorted/local%20system.md), [monodromy](Unsorted/monodromy.md), [Picard-Fuchs](Unsorted/Picard-Fuchs.md).
![](attachments/2023-03-05pf.png)

# Transversality

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

Griffiths transversality says that the period map is transverse to certain distributions on the period domain.

# Motivating Example

![](attachments/2023-03-07.png)
![](attachments/2023-03-07vhs.png)

Summary:

- A family $f: \mcx\to S$,
- $\mch^k_\ZZ = \RR^k f_* \ul \ZZ$ a local system on $S$
- $\mch^k = \RR^k f_* \Omega_{\mcx/S}$ a locally free $\OO_S\dash$module.
	- On fibers, $\mch^k_x = H^k(X; \CC)$.
- Hodge filtration: $\ts{ \Fil^n \mch^k}_{n\leq k}$ a decreasing filtration by sub-bundles where $\Fil^p \intersect \bar\Fil^{q+1} = \emptyset$ when $p+q=k$.
	- On fibers, $\Fil^p_x$ is the Hodge filtration on $H^k(X_x; \CC)$.
- $\omega \in H^0(S; \RR^2 f_* \ul \ZZ)$ inducing $\omega_x \in H^{1, 1}(X_x) \intersect H^2(X; \ZZ)$ an integral $(1,1)$ form which polarizes  the fiber $X_x$.
- Hodge-Riemann form: a locally constant non-degenerate form $Q: (\mch_\ZZ^k)\tensorpower{\ZZ}{2}\to \ZZ$ 
- Gauss-Manin: $\nabla\in \Hom(\mch_k, \mch_k \tensor \Omega_S)$ where the local system of horizontal sections is $\mch^k_\CC$.
- Griffiths Transversality: $\nabla \Fil^p \injects \Fil^{p-1} \tensor \Omega_S$.
- Lefschetz operator: $L(\wait) = \omega \cupprod(\wait)$, so $L^{n-k+1}: \mch^k \to \mch^{2n-k+2}$ with $\mch^k_\prim \da \ker L^{n-k-1}$ whose fibers are $H^k_\prim(X_x; \CC)$.

# Definition

![](attachments/2023-03-07vhsdef.png)

Get a monodromy representation $T: \pi_1(X, x) \to \Orth((\mch_\ZZ)_x, Q)$.

# Siegel Upper Half Space

![](attachments/2023-03-07sigel.png)

![](attachments/2023-03-07siegel.png)

# Infinitesimal VHS (IVHS)

![](attachments/2023-03-08vhs1.png)
![](attachments/2023-03-08ivhs1.png)

# Borel's theorem

The monodromy action is [quasi-unipotent](quasi-unipotent.md): ![](2023-04-06-2.png)