Rigidity of spreadings and fields of definition

Abstract

Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely˘ı curves and Shimura varieties. Rigidity is related to maximal Higgs fields which come from variations of Hodge structure. Basic properties for these due to P. Griffiths, W. Schmid, C. Simpson and, on the arithmetic side, to Y. André and I. Satake all play a role. This note tries to give a largely self-contained exposition of these manifold ideas and techniques, presenting, where possible, short new proofs for key results.