# Martin's Blog

## 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 Poincaré bundle, such that all line bundles on arise as restrictions of , and show that the pair satisfies a universal property.

### Homologically equivalent line bundles

Let be a complex abelian variety and . Recall that denotes the group of isomorphism classes of line bundles on . We shall define an equivalence relation, called homological equivalence, on .

Recall that line bundles on are classified by the Appell-Humbert theorem. This will play a vital role throughout this post.

Appell-Humbert Theorem. Every line bundle on a complex abelian variety is isomorphic to for exactly one pair of a Hermitian form on and a semi-character .

We say that two line bundles are homologically equivalent if their Hermitian forms are the same.

In order to explain the terminology, we should interpret this via homology: because abelian varieties are non-singular varieties, line bundles biject with linear equivalence classes of Weil divisors. A Weil divisor is a linear combination of subvarieties of codimension 1. Now a subvariety of complex codimension 1 is a real manifold of dimension which gives us a homology class in .

This gives a homomorphism . If you compose this with Poincaré duality then you get the cohomology class which corresponds to the Hermitian form . So two line bundles are homologically equivalent iff they have the same homology class in .

### The dual abelian variety and

Write for the group of isomorphism classes of line bundles on homologically equivalent to . By using Appell-Humbert again, we shall show that is isomorphic to the group of complex points of .

By Appell-Humbert, line bundles homologically equivalent to correspond to semi-characters with respect to the zero Hermitian form. The latter are just group homomorphisms so we get

We defined the dual abelian variety to be the complex torus coming from the -dual Hodge structure .

Since is a free -module, the functor is exact. Applying it to the short exact sequence gives a short exact sequence

So we get an isomorphism of groups with a point corresponding to the line bundle .

### The Poincaré bundle

We would like to show that the above group isomorphism is geometrically well-behaved. The right way to interpret "geometrically well-behaved" turns out to be that the line bundles on fit together (as we vary ) into a single line bundle on .

We define a Poincaré bundle for to be a line bundle on satisfying

1. for all ;
2. is trivial.

Condition 1 is the important condition; condition 2 is just a normalisation to ensure that the Poincaré bundle is unique.

We construct by using the Appell-Humbert theorem again: we need to give a Hermitian form on (or equivalently a Hodge symplectic form on ) and a semi-character and then we take .

We know that . Subject to this condition, we can do a calculation which shows that

Hence in order to satisfy condition 1, we must have Combined with condition 2, this forces

This shows that there is indeed a unique Poincaré bundle on .

### Universal property of the Poincaré bundle

We finish by proving that the Poincaré bundle satisfies a universal property. In this proposition, you should regard as being "a family of line bundles on parameterised by ".

This is not just a universal property of , but of the pair : this pair is universal among "families of line bundles on (homologically equivalent to ) parameterised by normal varieties and trivial at ".

As always, the universal pair is unique up to unique isomorphism, and so we could use this property to define the dual abelian variety. This is exactly how we will soon define the dual abelian variety over fields other than .

Proposition. Let be a normal variety over and a line bundle on such that

1. for all ;
2. is trivial.

Then there is a unique morphism of varieties such that .

Sketch proof. Set theoretically, must be defined by: is the point in corresponding to . We need to show that this is a morphism of varieties.

Let be the graph of in .

Let be the line bundle on . Then and hence is closed in .

The projection is bijective, so by Zariski's Main Theorem, it is an isomorphism of varieties. Hence is a morphism of varieties.

Tags abelian-varieties, alg-geom, hodge, maths

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