The absolute Galois group acts faithfully on the connected components of the moduli space of surfaces of general type
Abstract
We show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $\sigma \in Gal(\bar{\Q} /\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^{\sigma}$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting preprint by Robert Easton and Ravi Vakil (\cite{ev}), where it is proven, with a completely different type of examples, that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of irreducible components of the moduli spaces of surfaces of general type. We also give other simpler examples of surfaces with nonisomorphic fundamental groups which are Galois conjugate, hence have isomorphic algebraic fundamental groups.
 Publication:

arXiv eprints
 Pub Date:
 June 2007
 DOI:
 10.48550/arXiv.0706.1466
 arXiv:
 arXiv:0706.1466
 Bibcode:
 2007arXiv0706.1466B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 11R32;
 14J10;
 14J29;
 14 M99
 EPrint:
 13 pages and 2 figures