# Holomorphic line bundles on a punctured disc

Is every holomorphic line bundle on the - say - punctured unit disc $\dot{\Delta} \subseteq \mathbf{C}$ trivial? Griffiths-Harris (p. 39) prove that $H^{p,q}_{\overline{\partial}}(\Delta) = 0$ (for $q \geq 1$), and mention that by replacing discs by annuli that one could prove also $H^{p,q}_{\overline{\partial}}(\dot{\Delta}^k \times \Delta^\ell) = 0$. They seem to imply that in particular $H^{p,q}_{\overline{\partial}}(\dot{\Delta}) = 0$. If this were true, using Dolbeault's theorem and the Kummer sequence one could conclude $H^1(\dot{\Delta},\mathcal{O}^\times_{\dot{\Delta}}) = 0$, hence that every holomorphic line bundle on $\dot{\Delta}$ is trivial.

-

Yes, every holomorphic vector bundle of any rank is trivial on the punctured disk $\dot{\Delta}$ . Indeed, since $\dot{\Delta}$ is a Stein manifold ( like any non-compact Riemann surface ! ) the Oka meta-principle (here a Theorem of Grauert ) says that the classification of holomorphic vector bundles on that manifold is the same as that of topological vector bundles. But since the punctured disk is homotopically equivalent to a circle , all topological complex vector bundles are trivial, hence the triviality of all holomorphic vector bundles. (Do not confuse with the Möbius vector bundle, which is a non-trivial real vector bundle!)
On page 229, Theorem 30.3 states that every holomorphic line bundle on a non-compact Riemann surface $X$ is trivial, and immediately below (on the same page) Theorem 30.4 proves that holomorphic vector bundles of any rank on $X$ are also trivial.