Jean-Pierre Serre, [Prolongement de faisceaux analytiques coh ́erents][1], Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 363–374. MR MR0212214 (35 #3088) p. 372 proves that there are infinitely many distinct *holomorphic* line bundles on $\mathbb{C}^2-0$.


  [1]: https://eudml.org/doc/73897