Let $Y$ be a smooth complex projective variety of dimension $n+1$ and $X \subset Y$ be an ample smooth projective hypersurface. Does there exist an example of such $X$ and $Y$ such that the pullback map in cohomology fails to be injective in some degree at most $2n$.
By Lefschetz hyperplane theorem there is no counterexample in degree at most $n$. Moreover the map on homology the other way is surjective, which by poincare duality tells you something about injectivity of pullback on cohomology.