Exercise 30, § 6

Let G be a compact connected complex Lie group and hence of the form \mathbf{C}^n/D, where D is a discrete subgroup of \mathbf{C}^n of rank 2n. (a) Show that every holomorphic differential 1-form $\omega$ on G is invariant. (Let $\pi : \mathbf{C}^n \to G$ be the canonical morphism. Let $\zeta_1, \ldots, \zeta_n$ be the coordinate functions on $\mathbf{C}^n$. Then $\pi^*(\omega)$ is of the form $\sum_j a_j d\zeta_j$, where the $a_j$ are holomorphic functions on $\mathbf{C}^n$ which are invariant under D and hence constant.) (b) Let $G' = \mathbf{C}^n/D'$, where D' is a discrete subgroup of $\mathbf{C}^n$ of rank 2n. Show that every analytic manifold isomorphism $u : G \to G'$ is of the form $s \mapsto v(s) + a'$, where v is a Lie group isomorphism and $a' \in G'$. (Let $\pi' : \mathbf{C}^n \to G'$ be the canonical morphism. There exists an isomorphism of analytic manifolds $\tilde{u} : \mathbf{C}^n \to \mathbf{C}^n$ such that $u \circ \pi = \pi' \circ \tilde{u}$. For every holomorphic differential 1-form $\omega'$ on $G'$, $u^*({\pi'}^*(\omega'))$ is a 1-form on $\mathbf{C}^n$ which is invariant under translations by (a). Deduce that $\tilde{u}$ is an affine mapping.)