Dual abelian variety

Definition

Let $A$ be an abelian variety over a field $k$ , with unit $e \in A(k)$ . For a point $x$ of $A$ , let $t_x \colon A \to A$ be translation by $x$ . Let $\bar{k}$ be an algebraic closure of $k$ . Let $\operatorname{Pic}^0(A)$ be the subgroup of the Picard group of classes of line bundles $L$ with $t_x^{*} L \cong L$ for all $x \in A(\bar{k})$ . The dual abelian variety of $A$ is an abelian variety $A^\vee$ over $k$ with a line bundle $P$ on $A \times_k A^\vee$ , the Poincaré bundle, such that:

$P$ restricted to $\set{e} \times A^\vee$ is trivial;
$P$ restricted to $A \times \set{y}$ lies in $\operatorname{Pic}^0$ for every point $y$ of $A^\vee$ ;
$(A^\vee, P)$ is universal, in the following sense. Let $T$ be a $k$ -scheme and $L$ a line bundle on $A \times_k T$ . Assume $L$ is trivial on $\set{e} \times T$ and in $\operatorname{Pic}^0$ on each fibre $A \times \set{t}$ . Then there is a unique morphism $f \colon T \to A^\vee$ with $L \cong (\id_A \times f)^{*} P$ .

Remarks

The pair $(A^\vee, P)$ exists and is unique up to unique isomorphism. When $k$ is algebraically closed, clause 3 with $T = \operatorname{Spec} k$ gives a bijection $A^\vee(k) \cong \operatorname{Pic}^0(A)$ . The dimension of $A^\vee$ equals the dimension of $A$ , and $(A^\vee)^\vee \cong A$ .

Example: let $E$ be an elliptic curve over an algebraically closed field with origin $O$ . The map $P \mapsto \mathcal{O}_E(P - O)$ is a bijection from $E(k)$ onto the line bundles of degree $0$ , and these are exactly the translation-invariant ones. So $E^\vee \cong E$ .

Non-example: the full Picard group of $E$ is not the dual. The line bundle $\mathcal{O}_E(O)$ has degree $1$ , and $t_x^{*} \mathcal{O}_E(O) \cong \mathcal{O}_E(O)$ fails for every $x \ne O$ , so it is not in $\operatorname{Pic}^0(E)$ .

Over $\C$ , if $A = V / \Lambda$ , then $A^\vee$ is the quotient of the space of $\C$ -antilinear forms $f$ on $V$ by the lattice of those $f$ with $\operatorname{Im} f(\Lambda) \subseteq \Z$ .

Tag	`JQWXE54TVM`
URI	`m2://JQWXE54TVM`
Kind	definition
Writing	W0 stub
MSC	14K30, 14K05
Defines	Dual abelian variety
Uses	Abelian variety, Invertible sheaf, Picard group
Changed	2 November 2026

Dual abelian variety

Definition

Remarks

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