Vector extensions of abelian varieties

Maths > Abelian varieties > Universal vector extensions of abelian varieties

Vector extensions of abelian varieties

Posted by Martin Orr on Thursday, 17 April 2014 at 18:12

A theme of my posts on abelian varieties has been ad hoc constructions of objects which are equivalent to the (co)homology of abelian varieties together with their appropriate extra structures -- the period lattice for singular homology and the Hodge structure, the Tate module for $\ell$ -adic cohomology and its Galois representation. I want to do the same thing for de Rham cohomology. To prepare for this, I need to discuss vector extensions of abelian varieties -- that is extensions of abelian varieties by vector groups.

In this post I will define and classify extensions of an abelian variety by the additive group. We will conclude that $\operatorname{Ext}^1(A, \mathbb{G}_a)$ , the set of isomorphism classes of such extensions, forms a vector space isomorphic to the tangent space of the dual of $A$ . Most of this was discovered by Rosenlicht in the 1950s.

Definitions

Throughout this post, $k$ will be a field of characteristic zero (most of it works in positive characteristic as well, provided that you take care about separability in various places).

We define a vector group to be an algebraic group $V$ with an action of the multiplicative group $\mathbb{G}_m$ on $V$ such that there is an isomorphism of algebraic groups between $V$ and some power $\mathbb{G}_a^n$ of the additive group, under which the action of $\mathbb{G}_m$ on $V$ corresponds to its natural action on $\mathbb{G}_a^n$ . (Because we are restricting to characteristic zero, there is only one possible action of $\mathbb{G}_m$ on each $V$ , but in positive characteristic there is more than one.)

If $A$ is an abelian variety and $V$ is a vector group, then an extension of $A$ by $V$ is an algebraic group $E$ together with a short exact sequence (in positive characteristic, we also require the morphisms in the short exact sequence to be separable). Note that the short exact sequence is part of the data, so if we have an extension $0 \overset{\alpha}{\to} V \to E \to A \to 0$ and we replace $\alpha$ by $2\alpha$ then we get a different extension. Nevertheless, we will often write just $E$ as a shorthand for the extension.

The group $E$ is always commutative, even though this is not part of the definition. This follows from the fact that for any algebraic group $G$ , the quotient by the centre $G/Z(G)$ is an affine algebraic group.

Given any abelian variety $A$ and vector group $V$ , we can form the trivial extension $V \times A$ , which is just the direct product of the groups with the obvious morphisms.

Operations on vector extensions

Here are some methods of constructing new vector extensions from old.

First, the pushout. Given a vector extension $0 \to V \to E \to A \to 0$ and a morphism of vector groups $f \colon V \to V'$ , there is a unique extension $E'$ of $A$ by $V'$ such that there exists a morphism of algebraic groups $F \colon E \to E'$ making the following diagram commute: This extension $E'$ is called the pushout of $E$ by $f$ and denoted $f_* E$ .

If we fix $A$ and $V$ , the set of isomorphism classes of extensions of $A$ by $V$ is denoted $\operatorname{Ext}^1(A, V).$

We can define a $k$ -vector space structure on $\operatorname{Ext}^1(A, V)$ as follows:

Given two extensions $E$ and $E'$ of $A$ by $V$ , we take their fibre product $E \times_A E'$ , which is an extension of $A$ by $V \oplus V$ , then push out by the map $(x, y) \mapsto x + y \colon V \oplus V \to V.$ This gives the sum $E + E'$ .

To multiply an extension $E \in \operatorname{Ext}^1(A, V)$ by a scalar $x \in k^\times$ , we simply push out by the multiplication-by- $x$ map $V \to V$ . This is equivalent to replacing the morphism $\alpha \colon V \to E$ by $x^{-1} \alpha$ .

Principal bundles

In order to classify vector extensions, it is useful to consider more general objects called principal bundles.

Let $G$ be an algebraic group and $X$ a variety.

A principal $G$ -bundle over $X$ is an algebraic variety $Y$ together with a morphism $\pi \colon Y \to X$ and an action $\sigma$ of $G$ on $Y$ such that there exists an open cover of $X$ and isomorphisms $\pi^{-1}(U_i) \to U_i \times G$ which commute with the projections down to $U_i$ and under which the restriction of $\sigma$ to $\pi^{-1}(U_i)$ corresponds to the action of $G$ on $U_i \times G$ given by $g.(u, h) = (u, gh).$ Note that this implies that $\sigma$ restricts to a free and transitive action of $G(\bar{k})$ on the $\bar{k}$ -points of each fibre of $\pi$ . A principal $G$ -bundle is also called a $G$ -torsor over $X$ .

If $0 \to V \to E \to A \to 0$ is an extension of an abelian variety $A$ by a vector group $V$ , then letting $V$ act on $E$ by translations makes $E$ into a principal $V$ -bundle over $A$ . The key step in proving this is Rosenlicht's cross section theorem, which asserts that there is a rational map $A \to E$ (defined over $k$ ) such that $A \to E \to A$ is the identity wherever it is defined.

The above paragraph works for any algebraic group $A$ and any solvable algebraic group $V$ but the converse which we will discuss later works only for abelian varieties $A$ and vector groups $V$ .

Cohomological classification

It is a standard fact that isomorphism classes of principal $G$ -bundles on a variety $X$ are in canonical bijection with the sheaf cohomology group $H^1(X, \underline{G})$ where $\underline{G}$ is the sheaf $\underline{G}(U) = \operatorname{Mor}(U, G)$ . This is because, if we take isomorphisms $\alpha_i \colon \pi^{-1}(U_i) \to U_i \times G$ as in the definition of a principal $G$ -bundle, then the functions $p_2 \alpha_i - p_2 \alpha_j \colon \pi^{-1}(U_i \times U_j) \to G$ are constant on the fibres of $\pi$ and form a Čech cocycle on $X$ with values in $\underline{G}$ .

Combining this with the previous section, we get that there is a canonical map $u \colon \operatorname{Ext}^1(A, \mathbb{G}_a) \to H^1(A, \mathcal{O}_A)$ for any abelian variety $A$ . (The sheaf $\underline{\mathbb{G}}_a$ is the same as sheaf of regular functions $\mathcal{O}_A$ .) One can show that this map is $k$ -linear.

This map $u$ is an isomorphism by the following proposition.

Proposition. Every principal $\mathbb{G}_a$ -bundle over an abelian variety $A$ can uniquely be given the structure of an extension of $A$ by $\mathbb{G}_a$ .

We observe that $H^1(A, \mathcal{O}_A)$ is canonically isomorphic to the tangent space $T_0(A^\vee)$ of the dual abelian variety $A^\vee$ at the origin, by a nice argument using dual numbers (see Chapter III, Proposition 2.1 in Milne's notes on abelian varieties -- note that $A^\vee = \operatorname{Pic}^0 A$ ).

Hence we get a canonical isomorphism of $k$ -vector spaces $\operatorname{Ext}^1(A, \mathbb{G}_a) \to T_0(A^\vee).$

We conclude with a sketch proof of the proposition.

Sketch proof of proposition. The extra ingredient which we must give to a principal $\mathbb{G}_a$ -bundle $(E, \pi, \sigma)$ in order to make it an extension of $A$ by $\mathbb{G}_a$ is an algebraic group law $E \times E \to E$ , compatible with the action of $\mathbb{G}_a$ and with the projection to $A$ .

The uniqueness is easy: suppose that we had two group laws $+_1$ and $+_2$ on $E$ , and given $x \in E$ consider the function $f \colon E \to E$ defined by $f(y) = (x +_1 y) -_1 (x +_2 y).$ Because both group laws are compatible with the action of $\mathbb{G}_a$ and the projection to $A$ , $f$ factors as $E \to A \to \mathbb{G}_a \to E$ and the morphism $A \to \mathbb{G}_a$ must be constant because $A$ is complete and $\mathbb{G}_a$ is affine.

In order to prove existence, define a translation of the principal $\mathbb{G}_a$ -bundle $E$ to be a $k$ -morphism $\tau \colon E \to E$ such that

$\pi\tau - \tau \colon E \to A$ is constant (i.e. $\tau$ lifts a translation of $A$ ); and

$\tau$ commutes with $\sigma$ .

The key step is to prove that every translation of $A$ can be lifted to a translation of $E$ . We can deduce from this that there is a bijection between $E(k)$ and the set of translations of $E$ ; since the latter forms a group, we get a group law on $E(k)$ if we choose a point to be the identity element. This group law is compatible with $\pi$ and $\sigma$ if we choose this point in $\pi^{-1}(0)$ (Rosenlicht's cross section theorem implies that there is a $k$ -point in $\pi^{-1}(0)$ ). This construction is functorial in the base ring, so in fact we get an algebraic group law.

Now we prove that every translation of $A$ can be lifted to a translation of $E$ . Let $x \in A(k)$ , and let $t_x \colon A \to A$ be the corresponding translation. We get a new principal $\mathbb{G}_a$ -bundle $(E, t_x \circ \pi, \sigma)$ , and a translation of $E$ as defined above is the same as an isomorphism of principal $\mathbb{G}_a$ -bundles $(E, t_x \circ \pi, \sigma) \to (E, \pi, \sigma).$ Hence to prove the proposition, it suffices to prove that these bundles are isomorphic.

The translation $t_x \colon A \to A$ induces a linear map If the bundle $(E, \pi, \sigma)$ corresponds to the class $\xi \in H^1(A, \mathcal{O}_A)$ , then $(E, t_x \circ \pi, \sigma)$ corresponds to . So we are reduced to showing that is the identity.

The map defines a group homomorphism $\rho \colon A(k) \to \operatorname{GL}(H^1(A, \mathcal{O}_A)).$ Since $H^1(A, \mathcal{O}_A)$ is a finite-dimensional vector space, $\operatorname{GL}(H^1(A, \mathcal{O}_A))$ is an affine algebraic group. The homomorphism $\rho$ is functorial in the base ring, so it is a morphism of algebraic groups from an abelian variety to an affine group, and so it is trivial.

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