2007/06/11 by Ingrid Bauer, Fabrizio Catanese, Bauer, Ingrid +3 · 1 citation
Mathematics · #11R32 #14 M99 #14J10 #14J29 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.NT #msc:11R32 #msc:14 #msc:14J10 #msc:14J29 #msc:M99
paper · pdf · doi:10.48550/arxiv.0706.1466
13 pages and 2 figures
arxiv created 2007/06/11 · openalex publication_date 2007/06/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the Galois group Gal(\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 σ∈ Gal(\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 Xσ 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 (\citee-v), where it is proven, with a completely different type of examples, that the Galois group Gal(\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.