# Martin Orr's Blog

## Line bundles and morphisms to the dual variety

Posted by Martin Orr on Saturday, 28 May 2011 at 15:25

Over the complex numbers, the dual of an abelian variety is defined to have a Hodge structure dual to that of . Hence morphisms can be interpreted as bilinear forms on the Hodge structure of . Of particular importance are the morphisms corresponding to Hodge symplectic forms.

Last time we saw that can also be interpreted as a group of line bundles on . Today we will use this interpretation to define morphisms which turn out to be the same as those corresponding to Hodge symplectic forms. Then we generalise the definition of to base fields other than , which we will use next time in constructing dual abelian varieties over number fields.

## Dual abelian varieties and line bundles

Posted by Martin Orr on Monday, 09 May 2011 at 14:30

The definition I gave last time of dual abelian varieties was very much dependent on complex analytic methods. In this post I will explain how dual varieties can be interpreted geometrically: the points of correspond to a certain group of line bundles on . We construct a single line bundle on the product , the PoincarĂ© bundle, such that all line bundles on arise as restrictions of , and show that the pair satisfies a universal property.