Middle cohomology of very general hyperplane sections

Question

Let $X$ be a smooth, projective variety over $\mathbb{C}$ of dimension $n$ satisfying the property that for every $i \ge 0$, $H^{i,i}(X,\mathbb{C}) \cap H^{2i}(X,\mathbb{Q})=\mathbb{Q}c_1(\mathcal{O}_X(1))^i$. Is it then true for a very general smooth hyperplane section $Y \subset X$, we have $H^{i,i}(Y,\mathbb{C}) \cap H^{2i}(Y,\mathbb{Q})=\mathbb{Q}c_1(\mathcal{O}_Y(1))^i$? In particular, is it true that for a fixed integer $n$ and a very general complete intersection subvariety $Y$ of $\mathbb{P}^n$, we have $H^{i,i}(Y,\mathbb{C}) \cap H^{2i}(Y,\mathbb{Q})=\mathbb{Q}c_1(\mathcal{O}_Y(1))^i$? We know that this holds true for very general hypersurfaces in $\mathbb{P}^n$.

Any hint/reference will be most welcome.

Answer 1

I guess that this is not true even for hypersurfaces. Take a smooth quadric in $Y=Q \subset \mathbb{P}^{3} = X$. Then $h^{1,1}(Q) = b_{2}(Q) = 2$, so the second cohomology cannot be generated by $1$ element.