Let $X$ be a complex manifold and $E$ a holomorphic vector bundle over $X$ of rank $2$. Let $\mathbb{P}(E)$ denote the projectivization of $E$, with the natural map $p: \mathbb{P}(E) \rightarrow X$. It is apparently true that if $M$ is a nontrivial (holomorphic) line bundle on $X$, then $p^*M$ is a nontrivial (holomorphic) line bundle on $\mathbb{P}(E)$, but I am not sure how to prove it.
The result would be immediate if $p$ had a section, but I don't believe this should be the case in general? I would especially appreciate an answer that is essentially elementary (e.g. no spectral sequences), since this question was asked on an exam for an essentially elementary class, but all answers are welcome.
The pullback, $p^*$, is an injective map on integral cohomology. Thus if $c_1(M)\neq 0$, $c_1(p^*M)=p^*c_1(M)\neq 0$. Then you just need to show that a complex line bundle is non-trivial if and only if its first Chern class is non-zero. While that works for $C^\infty$ bundles, I think Pic(X) might give you holomorphically non-trivial line bundles with vanishing first Chern class.