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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.

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    You can call it a complex Bieberbach manifold. However, I am not sure why are you using $SO(n)$: Maybe you mean $SO(2n)$ (since you are acting on $R^{2n}$) or $U(n)$ (in case you want the quotient to be a complex manifold). – 2017-02-13
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    @MoisheCohen Thank you, I fixed it. Would you mind giving me a reference. I want to know for example if sections of line bundles are studied in similar way as in the lattice case using (automorphic)-theta functions. – 2017-02-13

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

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Cohen answer gave me a lot of insight to look for Beiberbach manifolds, which allowed me to learn many stuff. Here I give some of the information I gathered on this topic.

In short, the $\mathbb C ^n / \Gamma$ is not necessarily a manifold. It is one if $\Gamma$ is torsion free (i.e. no element is of finite order besides the neutral element.) and discrete subgroup of SE(n). In general $\mathbb C ^n / \Gamma$ has a stricture called orbifold, it is introduced by Satake to describe the smooth structure of spaces that look like manifolds, except on a few subsets, where they look like quotients of linear domains by a finite group of linear transformations.

If $\Gamma$ is crystallographic (discrete and co-compact) and torsion-free, then $\mathbb C ^n / \Gamma$ is a manifold and is called a Bieberbach maniflold.