  We show that an irreducible component of the Hodge locus of a polarizable
variation of Hodge structure of weight 0 on a smooth complex variety X is
defined over an algebraically closed subfield k of finite transcendence degree
if X is defined over k and the component contains a k-rational point. We also
prove a similar assertion for the Hodge locus inside the Hodge bundle if the
Hodge bundle together with the connection is defined over k. This is closely
related with the theory of absolute Hodge classes. The proof uses the spread of
the Hodge locus, and is quite similar to the case of the zero locus of an
admissible normal function.
