A 1-motive over a field $k$ is an algebraic torus $T$, an abelian variety $A$, a group scheme $G$ that's an extension of $A$ by $T$, a finitely generated free abelian group $L$, and a group homomorphism $L \longrightarrow G(k)$.

I'm currently reading a paper of Carlson, and I want to use his construction to identify something that came up in a problem that I'm working on. However, on the first page of that paper he defines a complex group scheme, but appears to leave out the requirement that $T$ and $G$ be group schemes. Later on (in section 4), he constructs the trace motive, and consistent with his definition, appears to only define the $\mathbb{C}$-points. I'm missing something -- but I don't really know what.

Does the fact that the groups in the trace motive come from group schemes somehow follow from some general nonsense about $\mathbb{C}$? Is it long and unenlightening to write down? Or am I just completely misunderstanding the paper?

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    $\begingroup$ As far as I can tell, he assumes that $A$ is an abelian variety. Given this, the only thing you need to know is that any analytic extension of $A$ by $\mathbb{C}^\times$ is algebraic. For this, note that any such extension determines and is determined by a point of the dual abelian variety (the Picard variety of $A$, in other words). $\endgroup$ – Keerthi Madapusi Pera Feb 1 '12 at 1:23
  • $\begingroup$ Hmm... I'm an algebraic geometer, so I don't know that much complex geometry. Why is that extension analytic again? A lot of the 1-motive he defines directly from the divisor groups. $\endgroup$ – mediocrates Feb 4 '12 at 2:56

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