Let $f\colon X \to Y$ be a dominant morphism between normal projective algebraic varieties; assume $f_*\mathcal {O}_X=\mathcal{O}_Y $; let’s also assume that it is defined over the complex numbers, but I do not think this is relevant for the question. I do NOT assume that $f$ is flat.

Let $L$ be a line bundle on $X$ which is trivial along all fibres of $f$. Is $L$ the pull-back of a line bundle from $Y$?

(I would expect that $f_*L$ is a line bundle, and the natural map $$ f^*f_*L\to L $$ is an isomorphism. When $f$ is flat, this is the seesaw principle.)

I am very interested in the case where $f$ is a birational map.

Thanks!