---
date: 2022-03-18 20:48
modification date: Friday 18th March 2022 20:48:53
title: abelian variety
aliases: ["abelian variety", "PPAVs", "A_g", "abelian varieties", "moduli space of abelian varieties", "moduli stack of abelian varieties", "moduli stack of abelian varieties"]
---


---
^7cc3c7

- Tags 
	- #AG 
- Refs:
	- Notes from Haiyang  	
		- [Learning/Abelian Varieties ac30f83986fe4dbc9a2c2f14b036550a/EllipticCurve_notes1.pdf](attachments/EllipticCurve_notes1.pdf)
		- [Learning/Abelian Varieties ac30f83986fe4dbc9a2c2f14b036550a/EllipticCurve_notes2.pdf](attachments/EllipticCurve_notes2.pdf)
		- [Learning/Abelian Varieties ac30f83986fe4dbc9a2c2f14b036550a/EllipticCurve_notes3.pdf](attachments/EllipticCurve_notes3.pdf)
	- [Notion site](https://www.notion.so/Abelian-Varieties-lccs-c04a95cd6aee4d3f865927b3e6adde00)
	- <https://www.jmilne.org/math/CourseNotes/AV.pdf>
- Links:
	- [polarization](polarization)
	- [abelian-by-finite](Unsorted/Kodaira-Parshin.md)
	- [Shimura variety](Unsorted/Shimura%20variety.md)

---

# abelian variety

See Satake compactification constructed by Faltings.

![](attachments/2023-03-14-11.png)
![](attachments/2023-03-14-12.png)

# Principally polarized

Construction: $\Ag = \HH_g/\Sp_{2g}(\ZZ)$ where $\HH_g \da \ts{\tau \in \Mat_{g\times g}(\CC) \st \tau = \tau^t,\,\, \Im(\tau) > 0 }$.
Smooth, separated, non-proper (for $g>0$) [DM stack](Unsorted/Deligne-Mumford%20stack.md) of dimension ${g(g+1)\over 2}$.
![](attachments/Pasted%20image%2020220502190201.png)

# The tautological

$\Ag$ carries the [Hodge bundle](Hodge%20bundle):

![](attachments/Pasted%20image%2020220501132916.png)
![](attachments/Pasted%20image%2020220501132953.png)

# Duals

See relative Jacobian.
![](attachments/Pasted%20image%2020220502190103.png)
![](attachments/Pasted%20image%2020220503155933.png)
![](attachments/Pasted%20image%2020220503155956.png)

# CM 

![2023-01-09-7](attachments/2023-01-09-7.png)