I am attending courses about complex manifolds and the teacher gave us the property to construct complex manifolds, that follows :
" Let X be a complex manifold and $\Gamma \subset Aut( \textbf{X})$. $\Gamma$ is a discret group and :
(i) the action is without fixed points, i.e : $\gamma.x = x$ for some x in X $\Rightarrow \gamma = id$ ;
(ii) For all $K_1, K_2$ compacts sets in X, $card(\{\gamma \in \Gamma, \gamma K_1 \cap K_2 \neq \emptyset\}) < +\infty$ ; i.e the action is properly discontinuous.
Then, X/$\Gamma$ has the structure of a complex manifold, and $\textbf{X} \rightarrow \textbf{X}/\Gamma$ is holomorphic. "
However, I cannot find any proof (I don't know if this property has a name...) and I would like to know if someone knows how to prove it or where to find any proof. Also, what does this quotient means ? What is the map ?
Thank you for your help
This is a very general construction, which works for any locally compact topological Hausdorff space $X$ and a subgroup $\Gamma$ of $G$, which is the group of homeomorphisms of $X$, acting properly discontinuously on $X$. Then we obtain an orbit space $X/\Gamma$. For example, take $X=\Bbb R^n$ and $G={\rm Isom}(\Bbb R^n)$. Then a subgroup $\Gamma$ acting properly discontinuously is discrete. A standard reference is the book Spaces of constant curvature by Joe Wolf.
A very nice note on a this topic is by Kapovich.