Let $j: U \to X$ be an open dense map of quasiprojective varieties. Is there an ample line bundle $\mathcal L$ on $U$ such that $j_\ast \mathcal L$ is also an ample line bundle?
As pointed out by Saal Hardali in the comments below, it's not immediately clear that there's even a line bundle $\mathcal L$ on $U$ such that $j_\ast \mathcal L$ is a line bundle, never mind ample. I suppose that makes me a bit greedy, but so be it!
I'm being stupid. If $j: U \to X$ is an open immersion and $\mathcal F$ is a quasicoherent sheaf on $U$, then the stalk of $j_\ast \mathcal F$ vanishes at any point $P \in X \setminus U$ such that locally at $P$, $X \setminus U$ is a hypersurface. So in such a case (which is pretty generic), $j_\ast \mathcal F$ is not a line bundle or locally free.