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.][1] 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][2] (published in International Math. Res. Notices 20, 2007) and of [Bauer, Catanese and Grunewald][3]. 

There is an interesting 
[survey of Goldring][4]  (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][5] on realization of surfaces defined over $\overline{\mathbb{Q}}$ as branched covers of $\mathbb{P}^2$.  

  [1]: http://www.math.technion.ac.il/Office_adm/mathedu/all/messages/1332141089.pdf
  [2]: http://arxiv.org/abs/0704.3231
  [3]: http://arxiv.org/abs/1303.2248
  [4]: http://www.math.technion.ac.il/Office_adm/mathedu/all/messages/1332141089.pdf
  [5]: http://www.ias.ac.in/mathsci/vol112/aug2002/Pm2009.pdf