 Solution

A rank-$r$ holomorphic vector bundle is a complex manifold $E$ with a holomorphic projection to $X$ and local holomorphic trivializations $E{|}_{U_i}\cong U_i\times {\mathbb{C}}^r$ that are complex-linear on each fibre. Its transition maps have the form $(x,v)\mapsto (x,g_{ij}(x)v)$ with holomorphic $g_{ij}:U_i\cap U_j\to G{L}_r(\mathbb{C})$ satisfying the cocycle identities.

The holomorphic Picard group consists of isomorphism classes of holomorphic line bundles, with tensor product as multiplication, the trivial line bundle as identity and the dual bundle as inverse. The classification on the complex projective line givesThe inverse is degree; $\mathcal{O}(1)$ is the positive generator.