If the Hodge conjecture (respectively the Tate conjecture or the Mumford-Tate conjecture) holds for a smooth projective variety X over a field k of characteristic zero, then it holds for a generic member X_t of a k-rational Lefschetz pencil of hypersurface sections of X of sufficiently high degree. The Mumford-Tate conjecture is true for the Hodge Q-structure associated with vanishing cycles on X_t. If the transcendental part of the second cohomology of a K3 surface S over a number field is an absolutely irreducible module under the action of the Hodge group Hg(S) then the punctual Hilbert scheme Hilb~2 (S) is a hyperkahler fourfold satisfying the conjectures of Hodge, Tate and Mumford-Tate.
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