Exercise 1, § 1

Let G be a finite dimensional, connected, commutative complex Lie group, and let V be its Lie algebra.

a) The map exp$_G: V\rightarrow G$ is a surjective homomorphism; its kernel $\Gamma$ is a discrete subgroup of V.

$b) G$ is compact if and only if $\Gamma$ is a lattice in V; then G is said to be a complex torus.

c) Let $\Gamma$ be the discrete subgroup of$\surd \mathbf{C}^2_2$ generated by the elements $e_1=$ $(1,0), e_2= (0,1), e_3= (2, i)$; put $G =\mathbf{C}/\Gamma , H = (\Gamma +\mathbf{C}e_1)/\Gamma$. Show that H is isomorphic to $\mathbf{C}^*$, that $G/H$ is a complex torus of dimension 1, but that G contains no non-zero complex torus.

d) Every finite dimensional connected compact complex Lie group is a complex torus (cf. Chap. III, §6, no. 3, Prop. 6).