Compact complex affine Kähler manifold is a torus 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.  
 A: 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.
A: Consider $S^2$  it is a projective manifold ( a manifold endowed with an atlas $(U_i), f_i:U_i\rightarrow P\mathbb{R}^n$ such that $f_i\circ f_j^{-1}$ is a restriction of an element of $PGl(n,\mathbb{R})$. Benzecri has shown in its thesis that if $M$ is a projective real manifold, so is $M\times S^1$. $S^2$ is endowed with a real projective structure, thus $S^2\times S^1$ is a real affine manifold, $S^2\times S^1\times S^1$ is also a real affine manifold. The holonomy of this manifold can be defined by homothetic maps so preserves a complex structure on $\mathbb{R}^4$. This manifold is endowed with a Kahler structure, the complex structure here is not necessarily associated with a symplectic structure such that $(M,J,\omega)$ is Kahler.
http://www.numdam.org/article/BSMF_1960__88__229_0.pdf
