## Complex abelian varieties and Riemann forms

Posted by Martin Orr on Wednesday, 30 December 2009 at 21:48

The theory of abelian varieties is very beautiful, both in its arithmetic and geometrical aspects, and also looking just over ```
where there are nice applications of complex analysis.
In this post I will work over
```

```
, and sketch a proof that a complex torus is isomorphic to an abelian variety if and only if it admits a Riemann form.
This will assume some knowledge of the theory of complex manifolds.
```

### Introduction

An *abelian variety* is a projective variety ` with a multiplication morphism `

` and an inverse morphism `

```
, satisfying the usual axioms for a group.
It is a surprising fact that the condition that
```

` is projective is enough to guarantee that the multiplication is commutative.`

Any abelian variety over ` is isomorphic (as a complex Lie group) to a `

*complex torus* i.e. ` where `

` is a `

`-lattice of rank `

` in `

```
.
The isomorphism is induced by the exponential map
```

```
(this is a homomorphism because
```

` is commutative).`

In the case `, every complex torus `

```
is isomorphic to an elliptic curve.
But in higher dimensions, not every complex torus is isomorphic to an abelian variety,
because not very complex torus can be embedded in projective space.
```

### Riemann forms

A *Riemann form* is an alternating form ` such that`

`for all`

`;`

- the corresponding Hermitian form
`is positive-definite; and`

`takes integer values on`

`.`

Here I am defining a Hermitian form to be a function ` which is linear in the `

*first* variable, and such that `.`

If ` is a Hermitian form, then the imaginary part `

```
is a real alternating form satisfying condition (1) above.
Conversely any real alternating form satisfying this identity is the imaginary part of a unique Hermitian form, given by the formula in (2).
```

(Similarly the real part of a Hermitian form is a symmetric form satisfying ` and any such symmetric form is the real part of a unique Hermitian form.)`

A Hermitian form ` is `

*positive-definite* if ` for all `

`.`

In terms of the corresponding alternating form, this is ` for all `

`.`

### Hodge metrics

Given a complex torus `, we can view `

```
as the tangent space at the identity.
Hence a non-degenerate alternating form
```

` on `

` gives rise to a 2-form `

` on `

```
(which is by definition an alternating form on the tangent space at each point of
```

```
;
we define
```

` at `

` by translating `

` from the identity to `

`).`

Since ` has constant coefficients in the natural coordinate charts on `

```
,
it is a closed 2-form, i.e. a symplectic form.
```

Conversely, given a translation-invariant symplectic form you can evaluate it at the identity to get a non-degenerate alternating form on ```
.
So there is a bijection between non-degenerate alternating forms on
```

` and translation-invariant symplectic forms on `

`.`

Restricting this to alternating forms which come from a positive-definite Hermitian form,
we get a bijection between alternating forms satisfying (1) and (2) above and translation-invariant Kähler metrics on `.`

Condition (3), namely that ` takes integer values on `

```
,
is equivalent to the corresponding translation-invariant 2-form being in the integral cohomology
```

`.`

A Kähler metric where the corresponding 2-form is in an integral cohomology class is called a *Hodge metric*.
Hence Riemann forms for ` biject with translation-invariant Hodge metrics on `

`.`

### Kodaira embedding theorem

Now the fact that any complex torus with a Riemann form is an abelian variety follows immediately from the Kodaira embedding theorem:

Theorem.Any compact complex manifold with a Hodge metric can be embedded in projective space.

The general proof of the Kodaira embedding theorem is hard. Even the dimension 1 case, that any compact Riemann surface is projective, requires some deep analysis.

In the case of complex tori, I believe that it is possible to give an easier proof by explicitly constructing meromorphic functions, using theta functions.

### Other direction

To prove the other direction, that any abelian variety corresponds to a complex torus admitting a Riemann form, is easier.

Given any projective complex manifold `, just restrict the Fubini-Study metric on `

` to get a Hodge metric on `

`.`

For a complex torus, average this metric over the manifold to get a translation-invariant Hodge metric. We have seen that this corresponds to a Riemann form.

Alex Youcissaid on Wednesday, 01 October 2014 at 07:44 :I'm sure you know this, but there is another nice application of complex analysis lurking about. You said "it's a surprising fact that being projective is enough to guarantee commutativity." That said, if A is an abelian variety over C, as in your definition, then A is, in particular, a complex Lie group. Looking at its adjoint representation, this is a map A to GL_n(C) which is holomorphic. Since A is compact, and GL_n(C) affine, this implies that the adjoint representation is trivial, and so A is abelian.

This also works for A any compact complex Lie group, and so it suffices to take A to be proper. But, as I'm sure you know, this was equivalent to projective anyways.

Nice blog by the way. :)