Exercise 9, § 6

Let X be a non-empty connected compact complex manifold of dimension n. Suppose that there exist holomorphic vector fields $\xi_1, \ldots, \xi_n$ on X which are linearly independent at each point of X. Show that there exist a complex Lie group G and a discrete subgroup D of G such that X is diffeomorphic to $G/D$. (Let $c_{ijk}$ be the holomorphic functions on X such that $[ \xi_i, \xi_j ] = \sum_k c_{ijk} \xi_k$. The $c_{ijk}$ are constants because X is compact. Take G to be the simply connected complex Lie group whose Lie algebra admits the $c_{ijk}$ as constants of structure and use Theorem 5.)