What kind of manifold is the quotient of $ \mathbb C^n $by a (non-mirror) crystallographic group A lattice of rank $k $ on $\mathbb C ^n$ is a discrete subgroup of $(\mathbb C ^n,+)$. It has the form
$$\Gamma_k = \mathbb Z e_1 + \mathbb Z e_2+ \cdots + \mathbb Z e_k$$
where $\{e_i\}_{i=1,\cdots,k}$ is an $\mathbb R$ independent family of elements of $\mathbb C ^n$.
when $k=2n$, $\Gamma_{2n}$ is called a lattice of maximal rank. 
$\Gamma_k$ acts on $\mathbb C ^n$ by translation.
The quotient of $\mathbb C ^n$ by $\Gamma_k$ leads to known manifolds. In the case of maximal rank the quotient is the torus (abelien varity). When the rank is less than 2n we get what it is called quasi-tori. and I can find many sources where they study these two cases, they study for example line bundles aver such manifolds which are related with theta functions.
Now, if I take the orientation preserving displacement group $U(n)\ltimes \mathbb C^n$, which, if I take $\gamma:=(A,b)  \in U(n)\ltimes \mathbb C^n$, acts on $z \in \mathbb C ^n$ by $\gamma\cdot z = Az+b$.  And I take $\tilde\Gamma$ a discrete subgroup (eventually a crystallographic group). I think it is legitimate to think about $\mathbb C^n/\tilde\Gamma$ as a manifold. Then my question is : Is there any special name for those kind of manifold. any special property or any particular study that focus on understanding them in an explicit manner, without heavy geometric language, like for the previous case.

I would call them "Kahler Bieberbach maniflolds" or "Kahler flat manifolds". You can find some discussion and references here. 
