The arithmetic and geometry of a generic hypersurface section

Abstract

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