# Proper actions in complex geometry

Complex geometry books often treat quotients by 'properly discontinuous' actions, such as the action of a lattice $$L \subseteq \mathbb{C}$$ on the complex line. The resulting canonical projection $$p \colon \mathbb{C} \twoheadrightarrow \mathbb{C}/L$$ satisfies the universal property of topological quotients and is a covering map.

Using this, it is very easy to show that there exists a (essentially unique) Riemann surface structure on the quotient such that the canonical projection is a holomorphic map (uniqueness follows from the fact that, in this case, the canonical projection is a quotient in the category of Riemann surfaces).

Indeed, the structure can be defined by taking as charts the local inverses of the canonical projection.

Since the map $$p$$ is

• a quotient
• a covering map (hence we have lifting of maps)

it is very easy to study the quotient spaces $$\mathbb{C}/L$$.

This is interesting, but in complex geometry we also have to deal with quotients by 'geometric groups' and projections such as $$p \colon \mathbb{C}^{n+1} \setminus \{0\} \twoheadrightarrow \mathbb{P}^n.$$ The right generalization seems to be that of 'proper action'.

The only book I found treating this subject was Lee's book on smooth manifolds. I checked many complex geometry books but I found nothing.

Can someone point me to other books / resources (preferably in the context of complex geometry) for quotients by complex Lie groups acting properly, and to an analogue theory of covering maps in this context? The endgoal would be to have tools in order to study quotients without continuously using charts.

I'm also interested in any kind of information we can find about the quotient using our knowledge of the étalé space.

For example, it would be nice to deduce informations about the holomorphic vector bundles over $$\mathbb{P}^n$$ from those over $$\mathbb{C}^{n+1} \setminus \{0\}$$.