Before giving a motivation let me ask the precise question firstly.

By a complex affine manifold I mean a complex manifold $M$ with the property that there exists an holomorphic atlas for which transition functions are restrictions of functions belonging to $Aff(\mathbb{C}^{\dim_\mathbb{C}M})$, the group of complex affine motions of $\mathbb{C}^{\dim_\mathbb{C}M}$.

**Question:** Suppose $M$ is a compact complex affine manifold admitting Kähler metric. Does it imply that $M$ has a complex torus as a finite covering? What restriction does it imply on a Kähler metric? Does it have to be a flat metric induced from the torus?

The reason for such a question is *Remark 2* at the end of Ma. Kato's paper Compact Differentiable 4-Folds with Quaternionic Structures. Apparently Calabi-Yau theorem seems to be of use here. Since I do not understand the explanation given there, the clarification of an argument in that case would be appreciated as well.