= Picard injection for holomorphic projective bundles
{c}
{title2=$(M,n)\mapsto p^*M\otimes L^{\otimes n}$}
For a rank-two <holomorphic vector bundle> on a nonempty <complex manifold>, this map from $\operatorname{Pic}_{\rm hol}(X)\times\mathbb Z$ to $\operatorname{Pic}_{\rm hol}(\mathbb P(E))$ is injective. Fibre degree detects $n$. If $p^*M$ 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 $p$.
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