The arithmetic and geometry of a generic hypersurface section

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 $\mathbb{Q}$-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 $\operatorname{Hg}(S)$, then the punctual Hilbert scheme $\operatorname{Hilb}^2(S)$ is a hyperkähler fourfold satisfying the conjectures of Hodge, Tate and Mumford–Tate.