Holomorphic Picard group (${\mathrm{Pic}}_{\mathrm{hol}}(X)=H^1(X,{\mathcal{O}}_X^{\ast })$)

The holomorphic Picard group consists of isomorphism classes of holomorphic line bundles, with tensor product as its group law. The trivial bundle is the identity and dualization gives inverses. Transition functions identify it with $H^1(X,{\mathcal{O}}_X^{\ast })$. On the complex projective line every class is $\mathcal{O}(n)$, so degree gives an isomorphism with $\mathbb{Z}$.

For a rank-two holomorphic vector bundle on a nonempty complex manifold, this map from ${\mathrm{Pic}}_{\mathrm{hol}}(X)\times \mathbb{Z}$ to ${\mathrm{Pic}}_{\mathrm{hol}}(\mathbb{P}(E))$ is injective. Fibre degree detects $n$. If $p^{\ast }M$ is trivial, a nonvanishing holomorphic section is constant along every compact projective fibre in a local bundle chart, so it descends to a nonvanishing section of $M$. The proof requires no global section of $p$.