  For any subfield K of the complex numbers which is not contained in an
imaginary quadratic number field, we construct conjugate varieties whose
algebras of K-rational (p,p)-classes are not isomorphic. This compares to the
Hodge conjecture which predicts isomorphisms when K is contained in an
imaginary quadratic number field; additionally, it shows that the complex Hodge
structure on the complex cohomology algebra is not invariant under the
Aut(\C)-action on varieties. In our proofs, we find simply connected conjugate
varieties whose multilinear intersection forms on their second real cohomology
groups are not (weakly) isomorphic. Using these, we detect non-homeomorphic
conjugate varieties for any fundamental group and in any birational equivalence
class of dimension at least 10.
