The Dual Abelian Variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:18:52Z http://mathoverflow.net/feeds/question/52267 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52267/the-dual-abelian-variety The Dual Abelian Variety Rex 2011-01-16T22:12:15Z 2011-01-17T11:52:37Z <p>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.</p> http://mathoverflow.net/questions/52267/the-dual-abelian-variety/52271#52271 Answer by Francesco Polizzi for The Dual Abelian Variety Francesco Polizzi 2011-01-16T23:18:41Z 2011-01-17T00:20:33Z <p>Given two <em>complex tori</em> $X_1$ and $X_2$, there is always a canonical <em>isomorphism</em></p> <p>$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,</p> <p>see for instance Birkenhake-Lange's book <em>Complex Abelian Varieties</em>, Exercise 11 page 43.</p> <p>Indeed let us write $X_i=V_i/\Gamma_i$, where $\Gamma_i$ is a lattice in the complex vector space $V_i$. Then</p> <p>$X_1 \times X_2 \cong V_1 \times V_2/ \Gamma_1 \times \Gamma_2$</p> <p>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 $\Gamma_2$.</p> http://mathoverflow.net/questions/52267/the-dual-abelian-variety/52273#52273 Answer by Marty for The Dual Abelian Variety Marty 2011-01-16T23:34:26Z 2011-01-16T23:34:26Z <p>Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see <a href="http://mathoverflow.net/questions/38168/is-the-category-of-commutative-group-schemes-abelian" rel="nofollow">"Is the category of commutative group schemes abelian" here on MO</a>) 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.</p> http://mathoverflow.net/questions/52267/the-dual-abelian-variety/52306#52306 Answer by Laurent Moret-Bailly for The Dual Abelian Variety Laurent Moret-Bailly 2011-01-17T11:52:37Z 2011-01-17T11:52:37Z <p>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).</p>