Picard injection for holomorphic projective bundles ($(M,n)\mapsto p^{\ast }M\otimes L^{\otimes n}$)

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 $p$.