  Varieties without deformations are defined over a number field. Several old
and new examples of this phenomenon are discussed such as Bely\u \i\ 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. Andr\'e 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.
