Martin's Blog

Hodge structures and abelian varieties

Posted by martin on Friday, 24 September 2010 at 08:48

I spend most of my time thinking about the Hodge structures attached to abelian varieties, so I decided that I should explain what these Hodge structures are. A Hodge structure is a type of algebraic structure found on the (co)homology of complex projective varieties.

Here I will discuss only the special case of the first homology of abelian varieties. This is the simplest case, but is nonetheless very important. In particular, the Hodge structures on other homology and cohomology groups for abelian varieties can be calculated from that of the first homology. Also Hodge structures on the first (but not higher) cohomology of non-abelian varieties can be reduced to the case of abelian varieties by passing to something called the Albanese variety, generalising the Jacobian of curves.

The exponential map

Let $A$ be an abelian variety over $\mathbb{C}$, of dimension $g$. Recall that $A$ is isomorphic (as a complex Lie group) to a complex torus $\mathbb{C}^g/\Lambda$, where $\Lambda$ is a full lattice in $\mathbb{C}^g$.

Definition. A full lattice in a real vector space $V$ of dimension $n$ (or complex vector space of dimension $n/2$) is a discrete subgroup of $V$ which is isomorphic as a group to $\mathbb{Z}^n$. Equivalently, it is a subgroup $\Lambda$ of $V$ such that the inclusion $\Lambda \to V$ induces an isomorphism of real vector spaces $\Lambda \otimes_{\mathbb{Z}} \mathbb{R} \to V$.

In fact the complex structure of $A$ does not really matter here: we could let $A$ be any commutative connected compact real Lie group, of dimension $n$ (this is real dimension instead of complex dimension, so $n = 2g$). Then $A$ is isomorphic (as a real Lie group) to $\mathbb{R}^n/\Lambda$, where $\Lambda$ is a full lattice in $\mathbb{R}^n$.

I shall briefly recall a proof of this fact: The tangent space to $A$ at the identity, $T_0 A$, is a real vector space of dimension $n$. For any Lie group, there is a smooth map $ \exp : T_0 A \to A. $ Since $A$ is commutative and connected, $\exp$ is a surjective group homomorphism. The kernel of $\exp$ gives us the lattice $\Lambda$. The compactness of $A$ ensures that $\Lambda$ has full rank.

All full lattices in a real vector space of dimension $n$ are equivalent. More precisely, we can choose a basis of $T_0 A$ so that $\Lambda$ is the standard full lattice consisting of vectors with integer coordinates. (This is obvious: just pick a $\mathbb{Z}$-basis of $\Lambda$. You get a $\mathbb{R}$-basis of $T_0 A$.) So $ A \cong \mathbb{R}^n/\mathbb{Z}^n \cong (\mathbb{R}/\mathbb{Z})^n \cong (S^1)^n $ and there is, up to isomorphism, only one commutative connected compact real Lie group of each dimension.

The complex structure

Forgetting the complex structure on abelian varieties, as we did above, puts the result that all abelian varieties are isomorphic to tori in a broader perspective. But we lose the ability to distinguish between different varieties of the same dimension - in other words most of the theory of abelian varieties.

Again let $A$ be a complex abelian variety of dimension $g$. To distinguish the lattices attached to different varieties of the same dimension, we use the fact that $T_0 A$ is not just a real vector space of dimension $2g$ but is a complex vector space of dimension $g$. It is not true that all full lattices in a complex vector space are equivalent - essentially because a $\mathbb{Z}$-basis of the lattice has $2g$ elements, but only $g$ of these are needed to give a $\mathbb{C}$-basis of the vector space.

You may regard the complex vector space $\mathbb{C}^g$ as fixed, and different abelian varieties as corresponding to different lattices in this space. But the Hodge-theoretic point of view is to regard the lattice $\Lambda$ (and hence the real vector space $\Lambda \otimes_{\mathbb{Z}} \mathbb{R}$) as fixed, while different abelian varieties correspond to different complex structures on $\Lambda \otimes_{\mathbb{Z}} \mathbb{R}$.

What do I mean by “complex structures” here? Well, if you have a real vector space, then in order to turn it into a complex vector space all you need to do is specify how to multiply vectors by complex numbers. Formally, this gives us the following definition:

Definition. A complex structure on a real vector space $V$ is an $\mathbb{R}$-algebra homomorphism $ \mathbb{C} \to \mathop{\mathrm{End}_{\mathbb{R}}} V $.

As an aside, to specify a complex structure on $V$, we do not need to give the action of all of $\mathbb{C}$. Indeed, by $\mathbb{R}$-linearity, it suffices to specify how $i$ acts. The action of $i$ on $V$ will be a $\mathbb{R}$-linear map $J : V \to V$ such that $J^2 = -1$, and conversely any such $J$ determines a complex structure.

Hodge structures

So, starting from a complex abelian variety $A$, we get a $\mathbb{Z}$-module $\Lambda$ (isomorphic to $\mathbb{Z}^{2g}$), plus a complex structure on $\Lambda \otimes_{\mathbb{Z}} \mathbb{R}$ coming from the isomorphism $\Lambda \otimes_{\mathbb{Z}} \mathbb{R} \cong T_0 A$. Let’s wrap this all up with a bow and give it a name.

Definition. An $H_1$ $\mathbb{Z}$-Hodge structure is a $\mathbb{Z}$-module $\Lambda$ together with a complex structure on $\Lambda \otimes_{\mathbb{Z}} \mathbb{R}$.

This is not a standard definition: usually one defines a more general object called a $\mathbb{Z}$-Hodge structure. I have called this an $H_1$ Hodge structure because this definition only applies to the special case of Hodge structures on the first homology of a variety – which is more concrete than the general case. (I have not mentioned homology anywhere before. It comes in because we can identify $\Lambda$ with $H_1(A, \mathbb{Z})$.)

The $\mathbb{Z}$ in the name comes from the fact that we start with a $\mathbb{Z}$-module; sometimes this is replaced by another subring of $\mathbb{R}$, usually $\mathbb{Q}$.

Hodge structures are useful because, being algebraic objects, we can employ algebraic techniques to study them, complementing the geometric techniques one uses to study abelian varieties. Importantly the Hodge structure contains all the information of the original abelian variety: two abelian varieties are isomorphic iff their $H_1$ $\mathbb{Z}$-Hodge structures are isomorphic.

In particular, we can define an algebraic group called the Mumford-Tate group, such that the Hodge structure becomes a representation of the Mumford-Tate group. Then we can use tools from representation theory to learn about abelian varieties. I will talk about the Mumford-Tate group in my next post.

Diagonalising

If you have seen Hodge structures before, you might wonder in what way my definition above is a special case of the usual one: I have no subspaces $V^{p,q}$. The relationship comes by diagonalising the complex structure. Even if you haven’t seen Hodge structures before, any time you have an endomorphism of a vector space, diagonalising is a nice thing to do.

Let $h : \mathbb{C} \to \mathop{\mathrm{End}_{\mathbb{R}}} V$ be a complex structure and $J = h(i)$.

We know that $J$ is diagonalisable because its minimal polynomial $X^2 + 1$ has distinct roots. Since the eigenvalues $\pm i$ of $J$ are complex, we need to work with $V_{\mathbb{C}} = V \otimes_{\mathbb{R}} \mathbb{C}$ in order to carry out the diagonalisation.

Write $V^{-1,0}$ for the $+i$-eigenspace of $J$ in $V_\mathbb{C}$, and $V^{0,-1}$ for the $-i$-eigenspace (this funny notation is to fit with the notation for general Hodge structures). Then $V_{\mathbb{C}} = V^{-1,0} \oplus V^{0,-1}$ and the spaces $V^{-1,0}$ and $V^{0,-1}$ are complex conjugates of each other.

Note that we have diagonalised not just $J$, but all of $h(\mathbb{C})$: if $z \in \mathbb{C}$, then $h(z)$ acts on $V^{-1,0}$ as multiplication by $z$ and on $V^{0,-1}$ as multiplication by $\bar{z}$.

So a more standard-looking definition of an $H_1$ $\mathbb{Z}$-Hodge structure would be:

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

You get the general definition of a $\mathbb{Z}$-Hodge structure by also having pieces $V^{p,q}$ indexed by all pairs of integers $(p,q)$.

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  1. Mumford-Tate groups From Martin's Blog
    In this post I will define the Mumford-Tate group of an abelian variety. This is a -algebraic group, such that the Hodge structure is a representation of this group. The Mumford-Tate group is important in the study of Hodge theory, and surprisingl...
  2. Hodge symplectic forms From Martin's Blog
    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, wi...

Comments

  1. Barinder Banwait said 2 days later:

    What is a $H_n$ $\mathbb{Z}$-Hodge structure? Is it a $\mathbb{Z}$-module $\Lambda$ with a complex structure on $H_n(\mathbb{C}^g/\Lambda,\mathbb{R})$, where g is the “dimension” of $\Lambda$? Or, by your very last line, is it a direct sum of pieces $V^{p,q}$ where p+q = n, satisfying $V^{p,q} = \overline{V^{q,p}}$ for all pairs?

    In “The Exponential map” section, you start a paragraph with “All free lattices…”. Should that be “All full lattices…”?

    Looking forward to your next post on MT group!

  2. Martin Orr said 4 days later:

    Well the term $H_1$ $\mathbb{Z}$-Hodge structure is something I made up, but the corresponding definition of $H_n$ $\mathbb{Z}$-Hodge structure would be:

    a $\mathbb{Z}$-module $\Lambda$ together with a direct sum decomposition $\Lambda \otimes_{\mathbb{Z}} \mathbb{C} = \bigoplus_{p,q} V^{p,q}$ satisfying $V^{p,q} = \overline{V^{q,p}}$ for all pairs, where the sum is over pairs of nonpositive integers with $p+q=-n$.

    This convention for the signs of $p$ and $q$ is chosen because Hodge theorists usually work with cohomology. Continuing my nonstandard terminology, the definition of $H^n$ $\mathbb{Z}$-Hodge structure would be the same, but with the “nonpositive” replaced by “nonnegative” and $p+q=-n$ by $p+q=n$. This preference for cohomology is reflected in the notation: $V^{p,q}$ has the indices superscripted, the same as in $H^n$ for cohomology.

    The equivalence with complex structures is something that only comes up for $H_1$ (and $H^1$).

    The cohomology $H^n(A, \mathbb{Z})$ of an abelian variety, or any projective complex manifold, is the $\mathbb{Z}$-module part of a $H^n$ $\mathbb{Z}$-Hodge structure for every $n$ (I am not sure if this works for homology when it is for non-abelian varieties). I didn’t talk about this because it would be harder work to motivate. In the case of abelian varieties, all the information is already in the $H_1$.

    You are right about “All full lattices”.

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