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 $Y$?
Best Answer
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.