---
created: 2022-02-23T18:45
updated: 2024-04-13T20:34
title: period
aliases:
  - periods
  - period map
  - period domain
tags:
  - dissertation
---

---

- Tags 
	- #arithmetic-geometry/p-adic-hodge-theory 
- Refs:
	- <https://home.mathematik.uni-freiburg.de/arithgeom/preprints/friedrich-periods.pdf>
	- Colmez-Fontaine, *Construction des reprsentations p-adiques semistables*
- Links:
	- [p-adic Hodge theory](Unsorted/p-adic%20Hodge%20theory.md)
	- [Algebraic de Rham](Unsorted/algebraic%20de%20Rham%20cohomology.md)
	- [Hodge-Tate conjecture](Hodge-Tate%20conjecture)
	- [period ring](period%20ring.md)

---

# period

![](attachments/Pasted%20image%2020220503104315.png)

![](attachments/Pasted%20image%2020220502142601.png)
---

![](2023-03-31-111.png)


# Period domains

![](2024-04-13-1.png)
![](2024-04-13-2.png)

![](2024-04-13-4.png)

![](2024-04-13-3.png)


- It is well known that the only cases when a period domain is Hermitian symmetric are weight 1 Hodge structures and weight 2 Hodge structures with $h^{2,0}=$ 1.

# Motivations

![](attachments/Pasted%20image%2020220318192305.png)
![](attachments/Pasted%20image%2020220318192322.png)

See [Fontaine-Mazur conjecture](Unsorted/Fontaine-Mazur%20conjecture.md).
![](attachments/Pasted%20image%2020220318192414.png)
![](attachments/Pasted%20image%2020220318192935.png)


Related to [alteration](Unsorted/alteration.md), [potentially semistable](Unsorted/potentially%20semistable.md) reduction.
![](attachments/Pasted%20image%2020220410155447.png)

![](attachments/Pasted%20image%2020220410155457.png)

# Examples

![](attachments/Pasted%20image%2020220503104549.png)

# Periods

![Pasted image 20211106004513.png](Pasted%20image%2020211106004513.png)

![Pasted image 20211106004934.png](Pasted%20image%2020211106004934.png)

![Pasted image 20211106005059.png](Pasted%20image%2020211106005059.png)

## In comparisons

![](attachments/Pasted%20image%2020220318201305.png)


# CYs

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

![](attachments/2023-03-06cy3.png)

# Period domains and period maps

![](attachments/2023-03-07dd-1.png)


Generally, given a fixed vector space $V$ with a familiy of filtrations parameterized by simply connected $S$, the modui space of filtrations $\tilde D$ is a partial flag variety and one gets a map $S\to \tilde D$ whose image lies in the period domain $D$.

For $\pi_1 S\neq 1$, pass to the universal cover, the monodromy representation is $\pi_1 S\to \Gamma$ a discrete group where $\Gamma\actson D$ and one gets $S\to \dcosetl{\Gamma}{D}$ instead.

Example: $S = \PP^1\smts{0,1,\infty}$ and the Legendre family of elliptic curves $V(y^2 = x(x-1)(x-\lambda))$; then $\VV$ is a weight 1 HS coming from varying $H^1$, $D = \HH$ and $\Gamma = \Gamma(2)\leq \SL_2(\ZZ)$ is a [congruence subgroup](Unsorted/congruence%20subgroup.md) of level 2, and one gets an isomorphism $$\PP^1\smts{0,1,\infty}\iso \dcosetl{\Gamma(2)}{\HH}$$

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

$$d \Phi_s\left(T_{S, 0}\right) \subseteq T_{\mathrm{hor}, \Phi(s)}(D)$$

Note that $\HH\to \Delta\smz$ is the universal cover.