Hi,

this is a very vague question, but I'm also glad about vague answers...

One knows that for an abelian variety over the complex numbers, one has a canonical exact sequence

$0\rightarrow \Omega^1(X) \rightarrow H^1_{DR}(X) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0$.

On the other hand there is a canonical group isomorphism

$Ext(X,G_a) \simeq H^1(X,\mathcal O_X)$,

which I will call the iso (+). Here on the left we have the group of extensions of $X$ by the additive group $G_a$, i.e. in our case just the complex numbers considered as algebraic group.

Know I would like to have a group $H$ which has the following properties:

1) It induces an iso $H \simeq H^1_{DR}(X)$

2) It projects on $Ext(X,G_a)$ naturally

3) The properties in 1) and 2) are compatible with the iso (+).

Does such a group exist? How do you get it?

Does this have to do with biextensions?

Thanks