Martin Orr's Blog

Dual abelian varieties over the complex numbers

Posted by Martin Orr on Tuesday, 26 April 2011 at 12:35

In this post I will define dual abelian varieties over the complex numbers. The motivation is that polarisations can be interpreted as isogenies from an abelian variety to its dual. For the moment, all this is tied to Hodge structures so only works over the complex numbers, but this is the view of polarisations which will we will generalise later to other fields.

Chow's Theorem

I left out a bit in the previous post - I sketched a proof that a polarisable complex torus is projective in the sense that it has a holomorphic embedding in projective space, but in order to show that it is an abelian variety we need to know that the image of this embedding is algebraic. Fortunately, this is automatically true due to the following theorem of complex geometry.

Chow's Theorem. A closed analytic submanifold of complex projective space is an algebraic variety.

A related theorem says that holomorphic maps between complex projective varieties are automatically algebraic morphisms. Furthermore morphisms of complex tori biject with morphisms of their Hodge structures. So we conclude that the following categories are equivalent:

Another theorem in the same vein says that holomorphic invertible sheaves on a complex projective variety are algebraic.

-dual Hodge structures

Recall that we defined a polarisation of an Hodge structure to be a certain kind of bilinear form. A bilinear form on a vector space is equivalent to a linear map from to its dual vector space. We would like to similarly define a notion of dual for Hodge structures, so that a polarisation can be viewed as a morphism from a Hodge structure to its dual.

What we are about to define is not what is usually called the dual of a Hodge structure. In the usual terminology, the dual of an Hodge structure is an Hodge structure. But to stay inside the above equivalence of categories, we need to work only with Hodge structures. For want of a better name, I shall call the object we are about to define the -dual of a Hodge structure.

Let be an -Hodge structure, with defining the complex structure on .

The -module underlying the -dual Hodge structure is of course

The complex structure is a little less obvious. Recall that one of the conditions for a symplectic form on to be a polarisation is that or equivalently Because of this conjugation which happens when we move between the left and right arguments of , we define to be the conjugate of the most obvious thing:

Alternatively, in terms of the decomposition , we are defining and likewise for .

With this definition, bilinear forms on satisfying biject with morphisms of Hodge structures via

As with any duality, is a contravariant functor from the category of -Hodge structures to itself, and is naturally isomorphic to , by the map sending to "evaluate at ". (However when we interpret the dual variety geometrically, it turns out to be better to use the isomorphism sending to "evaluate at ". This is at least partially justified by the fact that we are working with symplectic pairings.)

Dual abelian varieties

Let be an abelian variety over and its Hodge structure. The dual abelian variety of is defined to be the abelian variety associated to the -dual Hodge structure . In order for this definition to make sense, we need to check that is polarisable.

Choose a polarisation of , and let be the associated morphism . Because the Hermitian form of is positive definite, is a nondegenerate symplectic form. Hence is an isomorphism of vector spaces. However is injective but not necessarily surjective -- it has a finite cokernel.

(A morphism of Hodge structures which becomes an isomorphism when restricted to -Hodge structures is called an isogeny of Hodge structures, because if and are polarisable then these are precisely the morphisms which correspond to isogenies of the associated abelian varieties.)

Let be the index of in . For any , there is a unique such that . We define a symplectic form by

Because was a polarisation, so is . So is polarisable, and the complex torus is isomorphic to an abelian variety .

The above proof tells us that an abelian variety is always isogenous to its dual -- each polarisation of gives rise to an isogeny . However and are not always isomorphic. A polarisation whose associated isogeny is an isomorphism is called a principal polarisation.

You can show that, given a free -module with a nondegenerate symplectic form , there is a submodule of finite index such that the restriction of to induces an isomorphism . Hence every complex abelian variety is isogenous to a principally polarisable abelian variety.

In the case of elliptic curves, every elliptic curve has a unique principal polarisation. This is because the group of symplectic forms on a free -module of rank 2 is isomorphic to . There are two possible isomorphisms, one of which makes positive integers correspond to polarisations. The symplectic form which corresponds to under this isomorphism is the unique principal polarisation.

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