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Given an object over the algebraic closure Qbar of Q, there is often noreason for invariants of the corresponding holomorphic object to be preservedby 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 coversof the projective line branched only over the points 0, 1, and infinity,through Belyi's theorem, leads to Grothendieck's dessins d'enfants program forunderstanding the absolute Galois group through its faithful action on suchcovers. This note is motivated by Catanese's question about ahigher-dimensional analogue: does the absolute Galois group act faithfully onthe deformation equivalence classes of smooth surfaces? (These equivalenceclasses are of course by definition the strongest deformation invariants.) Wegive a short proof of a weaker result: the absolute Galois group actsfaithfully on the irreducible components of the moduli space of smooth surfaces(of general type, canonically polarized). Bauer, Catanese, and Grunewald haverecently answered Catanese's original question using a different construction.

Absolute Galois acts faithfully on the components of the moduli space of surfaces: A Belyi-type theorem in higher dimension