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.