  Given an object over the algebraic closure Qbar of Q, there is often no
reason for invariants of the corresponding holomorphic object to be preserved
by the absolute Galois group Gal(Qbar/Q), and in general this is not true,
although it is sometimes surprising to observe in practice. The case of covers
of the projective line branched only over the points 0, 1, and infinity,
through Belyi's theorem, leads to Grothendieck's dessins d'enfants program for
understanding the absolute Galois group through its faithful action on such
covers. This note is motivated by Catanese's question about a
higher-dimensional analogue: does the absolute Galois group act faithfully on
the deformation equivalence classes of smooth surfaces? (These equivalence
classes are of course by definition the strongest deformation invariants.) We
give a short proof of a weaker result: the absolute Galois group acts
faithfully on the irreducible components of the moduli space of smooth surfaces
(of general type, canonically polarized). Bauer, Catanese, and Grunewald have
recently answered Catanese's original question using a different construction.
