Factorization of symplectic isomorphisms of abelian varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:08:36Z http://mathoverflow.net/feeds/question/25952 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25952/factorization-of-symplectic-isomorphisms-of-abelian-varieties Factorization of symplectic isomorphisms of abelian varieties unknown (google) 2010-05-25T23:44:18Z 2010-05-25T23:44:18Z <p><strong>Background</strong></p> <p>Let $A$ and $B$ be two abelian varieties with dual Abelian varieties $\widehat A$, $\widehat B$. An isomorphism of Abelian varieties $f\colon A\times\widehat A\to B\times\widehat B$ represented by the matrix <code>$$\left(\begin{array}{cc} a_{11} &amp; a_{12}\\ a_{21} &amp; a_{22}\end{array}\right)$$</code> (where $a_{11}\in Hom(A,B)$, $a_{12}\in Hom(\widehat A,B)$, etc.) is called <em>symplectic</em> if one has $f^{-1}=f^\dagger$ where $f^\dagger\colon B\times\widehat B\to A\times\widehat A$ is the morphism represented by the matrix <code>$$\left(\begin{array}{cc} \hat a_{22} &amp; -\hat a_{12}\\ -\hat a_{21} &amp; \hat a_{11}\end{array}\right)$$</code> where $\hat a_{ij}$ denotes the transposed homomorphism between the dual Abelian varieties.</p> <p>In the paper: D. Orlov, Derived categories of coherent sheaves on Abelian varieties and equivalences between them. Izv. Math. 66 569 (2002) (available also on the arxiv: <a href="http://arxiv.org/abs/alg-geom/9712017" rel="nofollow">http://arxiv.org/abs/alg-geom/9712017</a>), it is claimed just after Proposition 4.12 that every symplectic isomorphism $f\colon A\times\widehat A\to B\times\widehat B$ can be factored as $f=f_1\circ f_2$ where $f_1\colon A\times \widehat A\to B\times \widehat B$, $f_2\colon A\times \widehat A\to A\times\widehat A$ are symplectic isomorphisms such that the $12$ entries of their corresponding matrices are isogenies. It is easy to prove this if $A$ and $B$ are simple Abelian varieties.</p> <p><strong>Question</strong></p> <p>Any ideas on how can it be proved in the general case? </p>