= 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 $g_{ij}:U_i\cap U_j\to GL_r(\mathbb C)$ 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> gives
$$
\boxed{\operatorname{Pic}(\mathbb P^1)\cong\mathbb Z,\qquad
n\longmapsto[\mathcal O_{\mathbb P^1}(n)].}
$$
The inverse is degree; $\mathcal O(1)$ is the positive generator.
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