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The Hodge filtration and universal vector extensions

Posted by Martin Orr on Friday, 13 June 2014 at 20:10

We will begin this post by looking at the isomorphism between the Hodge filtration H^{0,-1}(A) \subset H_1(A, \mathbb{C}) of a complex abelian variety A and the natural filtration T_0(A^\vee)^\vee \subset T_0(E_A) on the tangent space to the universal vector extension of A.

The significance of this isomorphism is that the Hodge filtration, as we defined it before, is constructed by transcendental methods, valid only over \mathbb{C}, but the universal vector extension is an object of algebraic geometry. So this gives us an analogue for the Hodge filtration for abelian varieties over any base field. Furthermore, in the usual way of algebraic geometry, the construction of the universal vector extension can be carried out uniformly in families of abelian varieties.

We will use the construction of the universal vector extension in families to show that “the Hodge filtration varies algebraically in families.” We will first have to explain what this statement means. We will also mention briefly why H^{-1,0}(A) does not vary algebraically.

A note on the general philosophy of this post: the usual construction of an algebraic-geometric object isomorphic to the Hodge filtration uses de Rham cohomology, which works for H^n of an arbitrary smooth projective variety. My aim in using universal vector extensions is to give an ad hoc construction of de Rham (co)homology, valid only for H_1 of an abelian variety, requiring less sophisticated technology than the general construction. This fits with previous discussion on this blog of the Hodge structure on H_1, constructed via the exponential map from the tangent space of A, and of the \ell-adic H_1, constructed as the Tate module.

The universal vector extension and the Hodge filtration

Let A be a complex abelian variety and let 0 \to V_A \to E_A \to A \to 0 \tag{*} be its universal vector extension. Recall that we showed last time that V_A \cong T_0(A^\vee)^\vee \cong \bar{V} and E_A(\mathbb{C}) \cong V \oplus \bar{V} / \{ (\lambda, \lambda) \mid \lambda \in \Lamba \} where V = T_0(A) and \Lambda = \ker \exp \colon V \to A(\mathbb{C}).

Now there are canonical isomorphisms V \oplus \bar{V} \cong V \otimes_\mathbb{R} \mathbb{C} \cong H_1(A, \mathbb{C}) which map V to H^{-1,0}(A) (the subspace of V \otimes_\mathbb{R} \mathbb{C} on which the two actions of \mathbb{C} agree) and \bar{V} to H^{0,-1}(A) (the subspace on which these two actions are complex conjugate).

Hence, looking at the tangent spaces of (*), we get canonically isomorphic short exact sequences: \usepackage{xypic} \xymatrix{ 0 \ar[r] & T_0(A^\vee)^\vee \ar[r] \ar[d] & T_0(E_A) \ar[r] \ar[d] & T_0(A) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & H^{0,-1}(A) \ar[r] & H_1(A, \mathbb{C}) \ar[r] & H^{-1,0}(A) \ar[r] & 0 }

In particular, we get a purely algebraic-geometric construction of a filtered complex vector space T_0(A^\vee)^\vee \subset T_0(E_A) which is canonically isomorphic to the Hodge filtration H^{0,-1}(A) \subset H_1(A, \mathbb{C}). Note that there is no natural construction of the subspace (rather than quotient) of T_0(E_A) which corresponds to H^{-1,0}(A) under this isomorphism.

How does the Hodge filtration vary in families?

Let \pi \colon \mathcal{A} \to S be an algebraic family of complex abelian varieties – that is, \pi is a smooth proper morphism of complex algebraic varieties with irreducible fibres, and we are implicitly given morphisms of S-varieties m \colon \mathcal{A} \times_S \mathcal{A} \to \mathcal{A}, \iota \colon \mathcal{A} \to \mathcal{A} and e \colon S \to \mathcal{A} which satisfy the axioms for a group law. It follows that each fibre \mathcal{A}_s = \pi^{-1}(s) is an abelian variety.

What does it mean to say “the Hodge filtration of \mathcal{A}_s varies algebraically”?

The idea of a family of vector spaces varying algebraically over S is captured by the notion of an (algebraic) vector bundle (these are defined as in the Wikipedia article, except that we are working in the category of complex algebraic varieties instead of topological spaces, and that our vector spaces are over \mathbb{C} instead of over \mathbb{R}).

We can break down the statement that the Hodge filtration varies algebraically into two parts, looking at the two parts of the filtration:

Theorem 1. There are sensible choices of a vector bundle \mathcal{V} on S and of isomorphisms of vector spaces \iota_s \colon \mathcal{V}_s \to H_1(\mathcal{A}_s, \mathbb{C}) for each point s \in S(\mathbb{C}).

Theorem 2. There is a vector subbundle \mathcal{V}^{0,-1} \subset \mathcal{V} such that \iota_s(\mathcal{V}^{0,-1}_s) = H^{0,-1}(\mathcal{A}_s) for each point s \in S(\mathbb{C}).

I will be vague about what “sensible choices” means in Theorem 1. It basically means that \mathcal{V} and \iota_s should be functorial with respect to morphisms of families over S and to changes of the base S.

Let us quickly note one non-sensible choice: we could take the constant vector bundle \mathbb{C}^{2g} \times S. There exist isomorphisms \mathcal{V}_s \to H_1(\mathcal{A}_s, \mathbb{C}) for each s, simply because these are vector spaces of the same dimension, but there are no natural (functorial) choices of such isomorphisms.

The Hodge decomposition does not vary algebraically

Let us note that Theorem 2 does not apply to H^{-1,0}. This can be stated informally as “the Hodge decomposition does not vary algebraically”, where the Hodge decomposition means the pair of subspaces H^{0,-1}(A) and H^{-1,0}(A) of H_1(A, \mathbb{C}).

Theorem 3. There is NOT a vector subbundle \mathcal{V}^{-1,0} \subset \mathcal{V} such that \iota_s(\mathcal{V}^{-1,0}_s) = H^{-1,0}(\mathcal{A}_s) for each point s \in S(\mathbb{C}).

I will not prove Theorem 3, but here is an indication of why it is true. The key point is that H^{-1,0}(\mathcal{A}_s) is the complex conjugate of H^{0,-1}(\mathcal{A}_s).

The complex conjugate of an algebraic vector subbundle cannot itself be an algebraic vector subbundle unless it is a locally constant subbundle. This ultimately reduces to the fact that a function f \colon X \to \mathbb{C} (for X an algebraic variety) and its complex conjugate x \mapsto \overline{f(x)} cannot both be algebraic, unless f is constant.

And the Hodge filtration is not locally constant unless the family \mathcal{A} \to S is locally trivial.

The universal vector extension in families

In order to prove Theorems 1 and 2 by using the comparison between the tangent space of the universal vector extension and the Hodge filtration, we need to be able to construct the universal vector extension of a family of abelian varieties.

Given a family of abelian varieties \pi \colon \mathcal{A} \to S, we define an extension of \mathcal{A} by a vector bundle \mathbb{V} over S to be a short exact sequence of group varieties over S 0 \to \mathcal{V} \to \mathcal{E} \to \mathcal{A} \to 0. A universal vector extension of \mathcal{A} can be defined in the same way as before.

Let us suppose first that S is affine and that the relative tangent sheaf T_{\mathcal{A}^\vee/S} of the dual famiily of abelian varieties is free. In this case we can generalise the approach of the previous two posts to construct the universal vector extension.

We need affineness of S to obtain canonical isomorphisms \operatorname{Ext}^1(\mathcal{A}, \mathbb{G}_a \times S) \to H^1(\mathcal{A}, \mathcal{O}_{\mathcal{A}}) \to H^0(S, e^* T_{\mathcal{A}^\vee/S}) via the classification of principal \mathbb{G}_a-bundles on \mathcal{A}. More precisely, affineness ensures that H^1(\mathcal{A}_{|U}, \mathcal{O}_\mathcal{A|U}) (for open subsets U \subset S) form a sheaf, and that the zero section e \colon S \to \mathcal{A} can be lifted to every principal \mathbb{G}_a-bundle on \mathcal{A}.

We need the condition that T_{\mathcal{A}^\vee/S} is free in order to choose a basis of global sections for e^* T_{\mathcal{A}^\vee/S}.

Now consider an arbitary base variety S. The smooothness of \mathcal{A}^\vee \to S implies that the T_{\mathcal{A}^\vee/S} is locally free, and so we can cover S by open sets on which the above argument applies. We can use the uniqueness properties of vector extensions to glue together the universal vector extensions obtained on each set in the cover, and obtain a universal vector extension of \mathcal{A}: 0 \to (e^* T_{\mathcal{A}^\vee/S})^\vee \to \mathcal{E}_\mathcal{A} \to \mathcal{A} \to 0.

The Hodge filtration in families

Taking the relative tangent bundles of the above sequence, pulled back along the zero sections, gives a filtered complex vector bundle on S (e^* T_{\mathcal{A}^\vee/S})^\vee \subset e^* T_{\mathcal{E}_{\mathcal{A}}/S}.

The fibres of this filtration at s \in S(\mathbb{C}) are T_0(\mathcal{A}_s^\vee)^\vee \subset T_0(E_{\mathcal{A}_s}) which we already know is canonically isomorphic to the Hodge filtration H^{0,-1}(\mathcal{A}_s) \subset H_1(\mathcal{A}_s, \mathbb{C}).

Hence we have proved Theorems 1 and 2: the Hodge filtration of \mathcal{A}_s varies algebraically.

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

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