I've got two isomorphic holomorphic vector bundle $E,E' \to X$, where $X=X'-\{x_0\}$ where $X'$ is a complex variety. I know that $E'$ is the restriction of an holomorphic bundle over $X'$. Can I extend $E$ as a bundle over $X$? It is obvious I've got a bundle which restriction is isomorphic to $X$, but I would like to have exactly $X$. I explain why:I've got a vector bundle with connection over a Riemann surface minus some (finite ) points. So locally this is isomorphic to a bundle over the pointed disk $\mathbb{D}^*$ with connection ,which is to say a trivial vector bundle with connection. So the trivial bundle can be extended over $\mathbb{D}$ , and I would like to extend my original bundle to the whole Riemann surface using the isomorphisms I pointed out.
I tried to write things as transition maps, but I've got problems unifying them all and now I'm stuck.