---
tag: HZXSESHC8P
kind: concept
name: Dual abelian variety
msc: [14K30, 14K05]
sources:
  - url: https://en.wikipedia.org/wiki/Dual_abelian_variety
same_as:
  wikidata: Q5310181
  wikipedia: Dual abelian variety
---

## Summary

The dual of an [[abelian variety|R7W03TXRS9]] $A$ is an abelian variety $A^\vee$ of the same dimension.
Its points are the line bundles on $A$ that are algebraically equivalent to zero.
It is the identity component of the [[Picard group|3Z9DCWYXQZ]] of $A$, made into a variety.
Duality is an involution: the dual of $A^\vee$ is $A$ again.
Polarizations of $A$ are certain maps $A \to A^\vee$, and the Weil pairing comes from duality.

## Explanations

### L1 human

On an abelian variety one can look at all line bundles, the algebraic analogues of twisted strips over the space.
Those that can be deformed continuously to the trivial one form a group that is itself an abelian variety.
This new abelian variety is the dual.
For an elliptic curve $E$ with origin $O$, the point $P$ corresponds to the line bundle of the divisor $P - O$, so $E$ is its own dual.
In higher dimension $A$ and $A^\vee$ are isogenous but need not be isomorphic.