Vectorial extensions of abelian varities

Hi,

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

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There must be more to your conditions. I can just define the 'natural' projection $H^1_{DR}(X)\to Ext(X,G_a)$ to be the composition of the canonical projection onto $H^1(X,\mathcal{O}_X)$ followed by the canonical isomorphism with $Ext(X,G_a)$. What's not natural about this? –  Keerthi Madapusi Pera Sep 29 '11 at 12:48