Holomorphic bundles on O(−k) are algebraic

We show that holomorphic bundles on O(−k) for k ≥ 0 are algebraic. We also show that such bundles are trivial outside the zero section. 1 Preliminaires The line bundle on P given by transition function z is usually denoted O(−k). Since we will be studying bundles over this space, we will denote O(−k) by Mk when we want to view this space as the base of a bundle. We give Mk the following charts Mk = U ∪ V , for U = C = {(z, u)} V = C = {(ξ, v)} U ∩ V = (C − {0})×C with change of coordinates (ξ, v) = (z, zu). Since Mk = O(−k) is a vector bundle on P 1 it follows that Pic(O(−k)) = Pic(P) = Z, hence holomorphic line bundles on Mk are classified by their Chern classes. Therefore it is clear that holomorphic line bundles over Mk are algebraic. We will denote by O(j) the line bundle on Mk given by transition funcion z . If E is a rank n bundle over Mk, then over the zero section (which is a P ) E splits as a sum of line bundles by Grothendieck’s theorem. Denoting the zero section by l it follows that for some integers ji uniquely determined up to order El ≃ ⊕ n i=1O(ji). We will show that such E is an algebraic extension of the line bundles O(ji).