If $E\to X$ is a holomorphic vector bundle, it is well known that the tautological line bundle $\mathcal{O}_E(1)$ over the projectivization $\pi:\mathbb{P}(E^*)\to X$ satisfies $$\pi_*\mathcal{O}_E(1)=E$$ as a derived pushforward.
Question: Now given a complex of holomorphic vector bundle $E^{\bullet}=0\to E^0\to E^1\to E^2\to ...\to E^l\to 0$ over $X$, is there a space $p:Y\to X$ and a line bundle $L\to Y$ such that $$p_*L=E^{\bullet}$$ as a derived pushforward?