Supose $L$ and $L'$ are holomorphic line bundles over $\mathbb{CP}^n$ such that $L|_{H} \simeq L'|_{H}$ for every hyperplane $H \subset \mathbb{CP}^n$. Does it follow that $L \simeq L'$?
Using the fact that every $x \in \mathbb{CP}^n$ is contained in a hyperplane one gets that $L_x \simeq L'_x$ for every $x$ but I don't know how to prove that the isomorphisms glue to a global one.