TITLE: Generalizations of Belyi's theorem
QUESTION [13 upvotes]: Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic function $\phi: X\to\mathbb{P}^1\mathbb{C} $ ramified at most at $0,1,$ and $\infty$;
3) $X$ is isomorphic to $\Gamma \backslash \mathbb{H}$ (compactified at cusps) for a finite index subgroup $[\mathrm{PSL}_2(\mathbb{Z}), \Gamma]<\infty.$
The remaining question is:
$$\boxed{\text{
Is there a way to treat singularities in this or a similar framework? }}$$
The following of my original questions have been answered:
Can this be generalized to arbitrary projective nonsingular varities of higher dimensions? (I discussed this with one professor here  in Goettingen. That seems to be ongoing research. Please see also comment of David Roberts.)
What compactification do they mean here? $\Gamma \backslash \mathbb{H} \cup \mathbb{Q}$! (see the answer of Robin Chapman)
What is a nice reference for the proof of Belyi's theorem? (see answer of YBL and Koeck + the comments of Emerton)
How does the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ enter the picture? (see comment of Ariyan and answer of YBL)
Where can I find nice examples where these computations have been done explicitly?
(see answer of Andy Putnam and JSE)

REPLY [5 votes]: Recently I was wondering about generalizations of Beyli's theorem to higher dimensions and did some googling. As this issue is only discussed briefly in David Roberts' comment, I thought I contribute what references I found hoping someone might find it useful: 
There is one direction of research which looks for actions of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on algebraic objects other than curves. A general criterion for an algebraic variety to be defined over a number field has been provided in 
this paper of Gabino González-Diez. In the special case of surfaces, there was a question of Catanese on the Galois action on moduli surfaces. This question has been answered in papers of Easton and Vakil (published in International Math. Res. Notices 20, 2007) and of Bauer, Catanese and Grunewald. 
There is an interesting 
survey of Goldring  (published in the Serge Lang memorial proceedings "Number theory, analysis and geometry"), which also discusses higher-dimensional generalizations of Belyi's theorem. Apparently one way of generalizing Belyi's theorem is 

Braungardt's question: Is every connected quasi-projective variety $X$
  that is defined over $\overline{\mathbb{Q}}$ birational to a finite
  étale cover of some moduli space of curves $\mathcal{M}_{g,n}$?

It was formulated in the paper: V. Braungardt: Covers of moduli surfaces. Compositio Math. 140 (2004) 1033-1036. In this paper, there are also some partial results on this question. There is also a paper of Paranjape on realization of surfaces defined over $\overline{\mathbb{Q}}$ as branched covers of $\mathbb{P}^2$.