Let $G$ be a connected abelian lie group such that it is diffeomorphic to $\mathbb C^n/\Gamma$ where $\Gamma$ is a discrete additive subgroup of $\mathbb C^n$. Now by Iwasawa decomposition $G$ is diffeomorphic to $K\times \mathbb R^m$ for a maximal compact subgroup $K$ of $G$.
How to show that in this case, $K$ is diffeomorphic to $V_\Gamma/\Gamma$ where $V_\Gamma:=span_{\mathbb R}(\Gamma )$?