I'm having a difficult time solving some exercice. I should prove the following :
Show that any holomorphic line bundle on a disc $\Delta\subset\mathbb{C}$ is trivial. Deduce that any holomorphic line bundle on $\mathbb{P}^1$ is of the form $\mathcal{O}(n)$, for some integer $n$ (Actually the same is true for $\mathbb{P}^m$ as well).
The part about the disk is not too hard using the fact that $H^{(p,q)}(\Delta)=0$ when $p\geq 0,q>0$.
This said, I don't seem to able to solve the second part or even see the link between the two. I have found solutions in books but they all take a different approach of this one and I am quite curious of the link between the disk and the projective line manifests iteself in that regard.