# Abelian varieties

*1958–59*

#### Translator’s note

*This page is a translation into English of the following:*

Douady, A. “Variétés abéliennes.” *Séminaire Claude Chevalley* **4** (1958–59), Talk no. 9. `http://www.numdam.org/item/SCC_1958-1959__4__A9_0`

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `f36a214`

# 1 Algebraic groups

An *algebraic group* is a pair

For every point

Every point

If

(There would be no problem with asking for

An *abelian variety* is an algebraic group whose variety is connected (and thus irreducible) and complete.

We will show that this implies that the group is commutative.

# 2 A property of complete varieties

Recall that a variety

## 2.1 Proposition 0

Let

*Proof*. If

Since

There is an analogous statement in analytic geometry:
let

## 2.2 Consequences of Proposition 0

If

*Proof*. Let

“By an analogous argument we can show,” or “by considering the dual group of

If

Let

*Proof*. Consider

In particular:

The underlying group of an abelian variety is abelian.

(For another proof of this result, see the Appendix).

# 3 Functions with values in an abelian variety

Every function

This theorem results from the combination of two lemmas.

If

*Proof*. Let

f is defined atu ;\varphi is defined at(u,u) ;\varphi_0 is defined at(u,u) .

Firstly, (a)

We now show that (b)

This shows that the intersection of with the diagonal of the set

Let

If

*Proof*. Since

# 4 Functions defined on a product with values in an abelian variety

Let

This implies that

*Proof*. Let

We will successively reduce to the following particular cases:

X is a curve;X is a complete non-singular curve, andY is non-singular.

*Reduction to (a).*
The set of points

*Reduction to (b).*
If

*Proof in case (b).*
The variety

This concludes the proof of the theorem.

Every function

*Proof*. Set

Every function

*Proof*. By Theorem 1,

# Appendix: adjoint representations

We can also obtain Theorem 0 from the following proposition:

Let

*Proof*. Let

In characteristic

We deduce Proposition 2 and Theorem 0 from Proposition 3 by noting that

# Bibliography

*Fondements de la Géométrie algébrique*. Paris, Secrétariat mathématique, 1958.