Martin Orr's Blog

Hodge symplectic forms

Posted by Martin Orr on Saturday, 18 December 2010 at 15:00

Both the Hodge structure and the Tate module of an abelian variety come with symplectic forms which are (almost) preserved by the action of the relevant group (Mumford-Tate or Galois group). The form on the Tate module, called the Weil pairing, will require some preparation. So in this post I will construct the Hodge symplectic forms (also called the Riemann forms) on the Hodge structure. Next time I will discuss some further properties of Hodge forms.

Symplectic forms

A symplectic form on an -module is a bilinear map satisfying .

A symplectic form is the same thing as a homomorphism , so I shall write for the -module of symplectic forms on .

We can similarly define a symplectic form on a -vector space for any field , or indeed on an -module for any ring . Any symplectic form satisfies and this condition is sufficient for a form to be symplectic as long as 2 is not a zero divisor in the base ring.

Hodge structures and symplectic forms

Let be a complex abelian variety, and . In order to explain the properties possessed by the special symplectic form on , we will need to discuss the Hodge structure of .

Recall that since is an -Hodge structure, it comes with a complex structure (where ).

This induces an action of on , defined by

As we did when studying Hodge structures, we will extend scalars to and diagonalise. Recall that we get a decomposition where acts on as multiplication by and on as multiplication by .

For the group of symplectic forms, we get where has eigenvalues

• on ,

• on ,

• on .

This is an example of an -Hodge structure, defined to be

a -module together with a decomposition satisfying and .

Hodge symplectic forms

An Hodge structure has a middle component which is mapped to itself by complex conjugation. This has no analogue for Hodge structures, and is important for the symplectic forms which we wish to construct.

Observe that because any element of is fixed by conjugation, but any element of is mapped into . (Similarly for an Hodge structure , we have .) However, because is mapped to itself by conjugation, it is possible that is non-zero.

In the case for an -Hodge structure , an element of is called a Hodge symplectic form on . Concretely it is a symplectic form on which vanishes on and on .

Hodge symplectic forms and abelian varieties

A generic -Hodge structure does not have a non-zero Hodge symplectic form, but a Hodge structure coming from an abelian variety does. I shall sketch the proof of this using some facts from differential geometry.

The property that distinguishes abelian varieties from arbitary complex tori is that they can be embedded in projective space. We will need to make use of this in order to construct a Hodge symplectic form.

Note that for an abelian variety (or a complex torus) , there is a canonical isomorphism . This is a purely topological fact: you can use the fact that is homeomorphic to and the Künneth theorem to compute the cohomology of .

Choose an embedding of into . Let be a hyperplane in which is not tangent to at any point. Then is a smooth subvariety of of complex dimension , so its real dimension is . We can triangulate and get a non-zero homology class in . This homology class is independent of the choice of hyperplane , though it does depend on the choice of projective embedding.

By Poincaré duality, is canonically isomorphic to so this corresponds to a cohomology class , which as explained above we can also interpret as a symplectic form on .

It can be shown that a form arising from a subvariety in this way is in , so we get a non-zero Hodge symplectic form.

Next time I shall explain how we can use Hodge symplectic forms to tell which -Hodge structures are associated with abelian varieties, and to get an isogeny between a complex abelian variety and its dual, and how this relates to the Mumford-Tate group.

Trackbacks

1. Polarisations on Hodge structures From Martin's Blog

In the last post, I discussed Hodge symplectic forms. Now I shall show that the of an abelian variety has a polarisation, which is defined to be a Hodge symplectic form satisfying a positivity condition. The importance of polarisations is that the...

2. Hodge classes on abelian varieties From Martin's Blog

In this post I will define Hodge classes and state the Hodge conjecture. I will restrict my attention to the case of abelian varieties and say the minimum amount necessary to be able to discuss the relationships between the Hodge, Tate and ...

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