Let V be a motivic variation of Hodge structure on a K-variety
S, let H be the associated K-algebraic Hodge bundle, and let
ΟβAut(C/K) be an automorphism. The absolute Hodge
conjecture predicts that given a Hodge vector vβHC,sβ above sβS(C) which lies inside Vsβ, the
conjugate vector vΟββHC,sΟββ is Hodge
and lies inside VsΟββ. We study this problem in the
situation where we have an algebraic subvariety ZβSCβ
containing s whose algebraic monodromy group HZβ fixes v. Using
relationships between HZβ and HZΟββ coming from
the theories of complex and β-adic local systems, we establish a criterion
that implies the absolute Hodge conjecture for v subject to a group-theoretic
condition on HZβ. We then use our criterion to establish new cases
of the absolute Hodge conjecture.Comment: Comments welcome