Let $X$ be an integral scheme proper over $\mathbb{C}$. Let $L$ be a line bundle on $X$. Is there a scheme $Y$ smooth over $\mathbb{C}$, a line bundle $M$ on $Y$, and a morphism $f:X\to Y$ over $\mathbb{C}$, with $f^*M$ isomorphic to $L$?
By Lemma, p.159 of Rational equivalence on singular varieties, it is true when $X$ is quasi-projective. I would like to know the non-projective case.
