  If there is a topologically locally constant family of smooth algebraic
varieties together with an admissible normal function on the total space, then
the latter is constant on any fiber if this holds on some fiber. Combined with
spreading out, it implies for instance that an irreducible component of the
zero locus of an admissible normal function is defined over k if it has a
k-rational point where k is an algebraically closed subfield of the complex
number field with finite transcendence degree. This generalizes a result of F.
Charles that was shown in case the normal function is associated with an
algebraic cycle defined over k.
