This page is a translation into English of the following:
Douady, A. “Variétés abéliennes.” Séminaire Claude Chevalley 4 (1958–59), Talk no. 9.
The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.
1 Algebraic groups
An algebraic group is a pair
For every point
(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
There is an analogous statement in analytic geometry:
2.2 Consequences of Proposition 0
“By an analogous argument we can show,” or “by considering the dual group of
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
This theorem results from the combination of two lemmas.
fis defined at u; \varphiis defined at (u,u); \varphi_0is defined at (u,u).
We now show that (b)
This shows that the intersection of with the diagonal of the set
4 Functions defined on a product with values in an abelian variety
This implies that
We will successively reduce to the following particular cases:
Xis a curve; Xis a complete non-singular curve, and Yis non-singular.
Reduction to (a).
The set of points
Reduction to (b).
Proof in case (b).
This concludes the proof of the theorem.
Appendix: adjoint representations
We can also obtain Theorem 0 from the following proposition: