  We combine Deligne's global invariant cycle theorem, and the algebraicity
theorem of Cattani, Deligne and Kaplan, for the connected components of the
locus of Hodge classes, to conclude that under simple assumptions these
components are defined over number fields (assuming the initial family is), as
expected from the Hodge conjecture. We also show that the Hodge conjecture for
(weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly)
absolute Hodge classes on varieties defined over number fields.
