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 (MumfordTate 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 nonzero.
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 nonzero 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 nonzero 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 nonzero 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 MumfordTate group.