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.](https://eudml.org/doc/163369) 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. 



Compact complex affine Kähler manifold is a torus
If a compact Kähler manifold $M$ admits a holomorphic affine connection, its Atiyah class and therefore all its Chern classes are zero. By Yau's solution of the Calabi conjecture, this implies that a finite covering of $M$ is a complex torus.


