## Abelian varieties

We would like to generalize one of the nice properties of Elliptic curves, namely that maps between them are determined by linear Algebra: Specifically, sublattices of the lattice $\mathbb{Z} \oplus \mathbb{Z}$. To do so we must first introduce the notion of an Abelian variety.

I would like to first address one bit of (somewhat) confusing nomenclature that necessarily arises. An Abelian variety is, first an foremost, a torus. Now, to which torus am I referring?

• Algebraic Torus: The simplest definition of an Algebraic torus is that it is the product of copies of the multiplicative group of a field; since I only work with the complex numbers, it would be isomorphic to $(\mathbb{C}^\times)^k$ for some integer $k$.
• Topological Torus: When topologists speak of a torus, they usually mean specifically $S^1 \times S^1$; that is, an elliptic curve if we forget the complex structure; the surface of a donut. Slightly more generally, an product $(S^1)^{\times k}$ can be called a torus with very little guilt. As we work over the complex numbers, we really only consider even (real) dimensional manifolds, so we would in fact only consider $(S^1)^{\times 2k}$.

Now, there is a relationship between these two, most obviously that $\mathbb{C}^\times$ is homotopic to the circle $S^1$. A little more strongly than that (which doesn’t quite agree with my definition of an Algebraic torus provided above), it turns out that there are two real forms of the torus: $\mathbb{R}^\times$ and $S^1$! So in some sense they are the same, but for my purposes we will only consider the second version of a torus.

So let’s return to the definition of an abelian variety. First and foremost, it is a complex torus. Topologically, it is simply the product $(S^1)^{\times 2k}$. However, to endow it with a complex structure we in fact consider it as the quotient

$\mathbb{C}^k/\Lambda$

where $\Lambda$ is a discrete co-compact subgroup of $\mathbb{C}^k$. It should be clear of course that a one-dimensional (complex dimension, remember) complex torus is nothing but an elliptic curve.

Now, much the same as we classify elliptic curves by looking at the upper half-plane, we would like to classify complex tori with an analogous structure. We begin to do so as follows.

A (discrete) rank $2k$ sublattice of $\mathbb{C}^k$ can be described by choosing $2k$ column vectors in $\mathbb{C}^k$ which span the whole space (as an $\mathbb{R}$-vector space). If we write these as a matrix $\Pi$ (which we call the period matrix), then this latter condition is equivalent to the matrix

$\displaystyle \begin{pmatrix} \Pi \\ \overline{\Pi}\end{pmatrix}$

being non-degenerate.

Exercise: Check this at least for the case $k = 1$, i.e. for elliptic curves. It will be easier to see this if you make a simplifying assumption that is to follow.

Now, much as the case for elliptic curves, multiplication of this basis by any matrix in $GL_k\mathbb{C}$ will yield an isomorphic complex torus (if you don’t believe me, you should check this). In particular, if we write $\Pi$ as

$\Pi = \begin{pmatrix} Z_1 & Z_2 \end{pmatrix}$

with $Z_i$ being $k \times k$ complex matrices (which are necessarily invertible), then multiplying by $Z_1^{-1}$ yeilds the simpler form

$\Pi \equiv \begin{pmatrix} Id_k & Z_1^{-1}Z_2 \end{pmatrix} \pmod {GL_k\mathbb{C}}$

more simply, we have that up to isomorphism, every complex torus has a period matrix of the form

$\Pi = \begin{pmatrix} Id_k & Z \end{pmatrix}$

where $Z$ is an invertible $k \times k$ complex matrix.

Now, as in the case of elliptic curves, maps between complex tori are particularly simple. In particular, they always factor into a linear map on the underlying vector space (which fixes the lattice; equivalently, a map of the lattices) followed by a translation. That is, if we denote by $t_y : X \to X$ the map given by $x \mapsto x + y$, then a given morphism $f : Y \to X$ can be written as $f = t_{f(0)} \circ g$ where $g$ lifts to a linear map preserving the lattices on the respective universal covers.

Now, this would be all well and good but for one important fact. As stated before, elliptic curves can always be embedded into projective space as a plane cubic. In particular, as a closed subvariety of projective space, they are in fact algebraic.

However, from the definition above, it turns out that “most” (in a suitably precise sense) complex tori are not algebraic. This is less than ideal, and so we would like to restrict ourselves to those tori which are algebraic. We call those tori Abelian Varieties.

Part of what works well with elliptic curves is also that the Weirstrass p-function exists, and satisfies a certain differential equation. Using that fact is how we are able to provide an explicit embedding of an elliptic curve (though of as $\mathbb{C}/\Lambda_\tau$) in projective space as an algebraic curve.