I am reading Deligne: Hodge III, and am puzzled by a certain statement in section 10. If anyone could give a reference or a hint for how to prove this, I would be grateful. Maybe it is obvious and I just don't see why.


We consider an extension $G$ of an abelian variety by a torus. Then Deligne claims that the kernel of the exponential map $Lie(G) \to G$ can be identified with $H\_1(G, \mathbb{Z} )$. Why is this true?



Describing the kernel of the exponential map as a homology group
The exponential map realizes Lie(G) as the universal covering space of G, and the kernel is group of covering transformations. Thus the kernel is $\pi\_1(G)$, which, being commutative in this case, equals $H\_1(G,Z)$. (I assume we are over the complex numbers.)


