Absolute Galois Acts Faithfully on the Components of the Moduli Space of Surfaces: A Belyi-Type Theorem in Higher Dimension

Abstract

Given an object over Q, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(Q/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 ∞, 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 (using surfaces isogenous to a product) [BCG3].