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

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 \mathbb{Z}-module M is a bilinear map \psi : M \times M \to \mathbb{Z} satisfying \psi(v, v) = 0.

A symplectic form is the same thing as a homomorphism \wedge^2 M \to A, so I shall write \wedge^2 M^* for the \mathbb{Z}-module of symplectic forms on M.

We can similarly define a symplectic form V \times V \to k on a k-vector space V for any field k, or indeed on an A-module for any ring A. Any symplectic form satisfies \psi(u, v) = -\psi(v, u) 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 A be a complex abelian variety, and L_{\mathbb{Z}} = H_1(A, \mathbb{Z}). In order to explain the properties possessed by the special symplectic form on L, we will need to discuss the Hodge structure of \wedge^2 L_{\mathbb{Z}}^*.

Recall that since L is an H_1 \mathbb{Z}-Hodge structure, it comes with a complex structure h : \mathbb{C} \to \operatorname{End}(L_{\mathbb{R}}) (where L_{\mathbb{R}} = L \otimes_{\mathbb{Z}} \mathbb{R}).

This induces an action h' of \mathbb{C}^\times on \wedge^2 L_{\mathbb{R}}^*, defined by \left( h'(z)\psi \right) \left( u, \, v \right) = \psi \left( h(z)^{-1}u, \, h(z)^{-1}v \right).

As we did when studying H_1 Hodge structures, we will extend scalars to \mathbb{C} and diagonalise. Recall that we get a decomposition L_{\mathbb{C}} = L^{-1,0} \oplus L^{0,-1} \text{ with } L^{-1,0} = \overline{L^{0,-1}} where h(z) acts on L^{-1,0} as multiplication by z and on L^{0,-1} as multiplication by \bar{z}.

For the group of symplectic forms, we get \wedge^2 L_{\mathbb{C}}^* = V^{2,0} \oplus V^{1,1} \oplus V^{0,2} where h'(z) has eigenvalues

  • z^{-2} on V^{2,0} = (\wedge^2 L^{-1,0})^*,

  • (z \bar{z})^{-1} on V^{1,1} = (L^{-1,0} \wedge L^{0,-1})^*,

  • \bar{z}^{-2} on V^{0,2} = (\wedge^2 L^{0,-1})^*.

This is an example of an H^2 \mathbb{Z}-Hodge structure, defined to be

a \mathbb{Z}-module V_{\mathbb{Z}} together with a decomposition V_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{C} = V^{2,0} \oplus V^{1,1} \oplus V^{0,2} satisfying V^{2,0} = \overline{V^{0,2}} and V^{1,1} = \overline{V^{1,1}}.

Hodge symplectic forms

An H^2 Hodge structure V has a middle component V^{1,1} which is mapped to itself by complex conjugation. This has no analogue for H_1 Hodge structures, and is important for the symplectic forms which we wish to construct.

Observe that V^{2,0} \cap V_{\mathbb{Z}} = 0 because any element of V_\mathbb{Z} is fixed by conjugation, but any element of V^{2,0} is mapped into V^{0,2}. (Similarly for an H_1 Hodge structure L, we have L^{-1,0} \cap L_{\mathbb{Z}} = 0.) However, because V^{1,1} is mapped to itself by conjugation, it is possible that V^{1,1} \cap V_{\mathbb{Z}} is non-zero.

In the case V = \wedge^2 L^* for an H_1 \mathbb{Z}-Hodge structure L, an element of V^{1,1} \cap V_{\mathbb{Z}} is called a Hodge symplectic form on L. Concretely it is a symplectic form on L_{\mathbb{Z}} which vanishes on L^{-1,0} \times L^{-1,0} and on L^{0,-1} \times L^{0,-1}.

Hodge symplectic forms and abelian varieties

A generic H_1 \mathbb{Z}-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) A, there is a canonical isomorphism H^2(A, \mathbb{Z}) \cong \wedge^2 H_1(A, \mathbb{Z})^*. This is a purely topological fact: you can use the fact that A is homeomorphic to (S^1)^{2g} and the Künneth theorem to compute the cohomology of A.

Choose an embedding of A into \mathbb{P}^N_{\mathbb{C}}. Let H be a hyperplane in \mathbb{P}^N_{\mathbb{C}} which is not tangent to A at any point. Then A \cap H is a smooth subvariety of A of complex dimension g-1, so its real dimension is 2g-2. We can triangulate A \cap H and get a non-zero homology class in H_{2g-2}(A, \mathbb{Z}). This homology class is independent of the choice of hyperplane H, though it does depend on the choice of projective embedding.

By Poincaré duality, H_{2g-2}(A, \mathbb{Z}) is canonically isomorphic to H^2(A, \mathbb{Z}) so this corresponds to a cohomology class \omega \in H^2(A, \mathbb{Z}), which as explained above we can also interpret as a symplectic form on H_1(A, \mathbb{Z}).

It can be shown that a form \omega arising from a subvariety in this way is in H^{1,1}(A), so we get a non-zero Hodge symplectic form.

Next time I shall explain how we can use Hodge symplectic forms to tell which H_1 \mathbb{Z}-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.

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

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