Let $X$ be a complex manifold and $E\rightarrow X$ a holomorphic vector bundle of rank $n+1$. Consider the projectivization $\pi: \mathbb{P}(E) \rightarrow X$ of this bundle, where our "projectization" is a $\mathbb{CP}^{n}$-bundle over $X$ whose fiber over $x\in X$ is the projectivization of the vector space $E_{x}$.
In papers, the "canonical" map $\pi^{*}E \rightarrow \mathcal{O}_{\mathbb{P}(E)}(1)$ is defined and used. Here, $\mathcal{O}_{\mathbb{P}(E)}(1)$ is the hyperplane bundle over $\mathbb{P}(E)$. My question is: How can one define this map canonically? I wonder why one does not need a metric on $\pi^{*}E$. With such a metric, although it depends on its choice, one can define a map $\pi^{*}E \rightarrow \mathcal{O}_{\mathbb{P}(E)}(1)$. Can you define the map without it?
I appreciate your answer, and any comments are welcome. Thank you in advance.
There are two conventions about $\mathbb{P}(E)$ in the literature: in the first the fiber over $x$ is the space of lines in $E_x$, and in the second (so-called Grothendieck's convention) the fiber is the space of hyperplanes in $E_x$. It is important don't to confuse between the two conventions.
In the Grothendieck's convention, one has $\pi_*(\mathcal{O}(1)) \cong E$, hence by adjunction there is a canonical morphism $\pi^*E \to \mathcal{O}(1)$.
In the other convention, one has $\pi_*(\mathcal{O}(1)) \cong E^\vee$, hence by adjunction there is a canonical morphism $\pi^*E^\vee \to \mathcal{O}(1)$.