MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $A$ be an abelian variety defined over an algebraically closed field, say over $\mathbb{C}$. There is a dual abelian variety $\hat{A}$, along with a Poincare line bundle $L$ on $A\times \hat{A}$. Is there any relation between $\widehat {A\times A} $ and $\hat{A}\times \hat{A}$, for instance are they isogenous. What happens when $A$ is principally polarized, can we say relate the Poincare bundles in this case.

share|cite|improve this question
@Marty: You should write this as an answer. – J.C. Ottem Jan 16 '11 at 23:22
Ok.. Done. I've deleted my comments accordingly. – Marty Jan 16 '11 at 23:34

You can define $\hat{A}$ as $\underline{\mathrm{Pic}}^0(A)$. Now, over any algebraically closed field $k$, let $V$ and $W$ be proper (irreducible, reduced) varieties. Then $\underline{\mathrm{Pic}}^0(V)\times \underline{\mathrm{Pic}}^0(W)\to\underline{\mathrm{Pic}}^0(V\times W)$ is an isomorphism. Injectivity is immediate, and surjectivity follows from the theorem of the cube (see e.g. Mumford, Abelian Varieties, section 6 in chapter II).

share|cite|improve this answer

Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.

share|cite|improve this answer

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

Indeed let us write $X_i=V_i/\Gamma_i$, where $\Gamma_i$ is a lattice in the complex vector space $V_i$. Then

$X_1 \times X_2 \cong V_1 \times V_2/ \Gamma_1 \times \Gamma_2$

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.