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Maths > Abelian varieties > Polarisations, dual abelian varieties and the Weil pairing

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.

H_1-dual Hodge structures

Recall that we defined a polarisation of an H_1 Hodge structure to be a certain kind of bilinear form. A bilinear form on a vector space V is equivalent to a linear map from V 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 H_1 Hodge structure is an H^1 Hodge structure. But to stay inside the above equivalence of categories, we need to work only with H_1 Hodge structures. For want of a better name, I shall call the object we are about to define the H_1-dual of a Hodge structure.

Let \Lambda be an H_1 \mathbb{Z}-Hodge structure, with h : \mathbb{C} \to \operatorname{End} \Lambda_\mathbb{R} defining the complex structure on \Lambda_\mathbb{R}.

The \mathbb{Z}-module underlying the H_1-dual Hodge structure \Lambda^\vee is of course \Lambda_\mathbb{Z}^\vee = \operatorname{Hom}_\mathbb{Z}(\Lambda_\mathbb{Z}, \mathbb{Z}).

The complex structure h^\vee : \mathbb{C} \to \operatorname{End} \Lambda_\mathbb{R}^\vee is a little less obvious. Recall that one of the conditions for a symplectic form on \Lambda_\mathbb{Z} to be a polarisation is that \psi(h(z)u, h(z)v) = z\bar{z} \, \psi(u, v) \text{ for } u, v \in \Lambda_\mathbb{R} or equivalently \psi(h(z)u, v) = \psi(u, h(\bar{z})v). Because of this conjugation which happens when we move h(z) between the left and right arguments of \psi, we define h^\vee to be the conjugate of the most obvious thing:

h^\vee(z) \in \operatorname{End} \Lambda_\mathbb{R}^\vee \text{ is the dual map of } h(\bar{z}) \in \operatorname{End} \Lambda_\mathbb{R}.

Alternatively, in terms of the decomposition \Lambda_\mathbb{C} = \Lambda^{-1,0} \oplus \Lambda^{0,-1}, we are defining (\Lambda^\vee)^{-1,0} = \{ f \in \Lambda_\mathbb{C}^\vee \mid f(\Lambda^{-1,0}) = 0 \} and likewise for (\Lambda^\vee)^{0,-1}.

With this definition, bilinear forms \psi on \Lambda_\mathbb{Z} satisfying \psi(h(z)u, v) = \psi(u, h(\bar{z})v) biject with morphisms of Hodge structures \phi : \Lambda \to \Lambda^\vee via \phi(u)(v) = \psi(u, v).

As with any duality, \Lambda \mapsto \Lambda^\vee is a contravariant functor from the category of H_1 \mathbb{Z}-Hodge structures to itself, and \Lambda^{\vee\vee} is naturally isomorphic to \Lambda, by the map sending v \in \Lambda to "evaluate at v". (However when we interpret the dual variety geometrically, it turns out to be better to use the isomorphism \Lambda \to \Lambda^{\vee\vee} sending v to "evaluate at -v". This is at least partially justified by the fact that we are working with symplectic pairings.)

Dual abelian varieties

Let A be an abelian variety over \mathbb{C} and \Lambda its H_1 Hodge structure. The dual abelian variety of A is defined to be the abelian variety associated to the H_1-dual Hodge structure \Lambda^\vee. In order for this definition to make sense, we need to check that \Lambda^\vee is polarisable.

Choose a polarisation \psi of \Lambda, and let \phi be the associated morphism \Lambda \to \Lambda^\vee. Because the Hermitian form of \psi is positive definite, \psi is a nondegenerate symplectic form. Hence \phi_\mathbb{Q} is an isomorphism of vector spaces. However \phi_\mathbb{Z} is injective but not necessarily surjective -- it has a finite cokernel.

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

Let n be the index of \phi(\Lambda_\mathbb{Z}) in \Lambda_\mathbb{Z}^\vee. For any f \in \Lambda_\mathbb{Z}^\vee, there is a unique \lambda_f \in \Lambda such that nf = \phi(\lambda_f). We define a symplectic form \psi' : \Lambda_\mathbb{Z}^\vee \times \Lambda_\mathbb{Z}^\vee \to \mathbb{Z} by \psi'(f, g) = f(\lambda_g).

Because \psi was a polarisation, so is \psi'. So \Lambda^\vee is polarisable, and the complex torus \Lambda^\vee_\mathbb{R}/\Lambda^\vee_\mathbb{Z} is isomorphic to an abelian variety A^\vee.

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

You can show that, given a free \mathbb{Z}-module \Lambda with a nondegenerate symplectic form \psi, there is a submodule of finite index \Lambda' such that the restriction of \psi to \Lambda' induces an isomorphism \Lambda' \to \Lambda'^\vee. 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 \mathbb{Z}-module of rank 2 is isomorphic to \mathbb{Z}. There are two possible isomorphisms, one of which makes positive integers correspond to polarisations. The symplectic form which corresponds to 1 under this isomorphism is the unique principal polarisation.

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

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