Let $E,F$ be two holomorphic vector bundles on a compact Kahler manifold $X$. Denote by $\mathbb{P}(E), \mathbb{P}(F)$ the associated projective bundles and $L\_E=\mathcal{O}\_E(-1), L\_F=\mathcal{O}\_F(-1)$ the tautological line bundles. Let $f:E\to F$ a bundle map.


Is there a "canonical" induced map
$$\tilde{f}:\tilde{L}\_E\to \tilde{L}\_F$$
where $\tilde{L}\_E,\tilde{L}\_F\to Y$ are some pull back of $L\_E,L\_F$ to some space $Y$?



Induced homomorphism on tautological line bundles $\mathcal{O}_E(1),\mathcal{O}_F(1)$
Let $Z = \mathbb{P}(E) \times\_X \mathbb{P}(F)$. Consider the composition
$$
p\_1^\*L\_E \to p^\*E \to p^\*F \to p^\*F/p\_2^\*L\_F,
$$
where $p\_1 \colon Z \to \mathbb{P}(E)$, $p\_2 \colon Z \to \mathbb{P}(F)$, and $p \colon Z \to X$ are the natural projections,
and the middle arrow above is the pullback of $f$. Let
$$
Y \subset Z
$$
be the zero locus of the composition. Then on $Y$ the composition vanishes, hence the composition of the first two arrows factors (in a unique way) through a morphism
$$
(p\_1^\*L\_E)\vert\_Y \to (p\_2^\*L\_F)\vert\_Y
$$
as required.


