I had been told this once, but cannot deduce it from the standard references. Is it correct?
I'm working over $\mathbb{C}$. Let $f : X \to Y$ be a birational proper morphism where $X$ is normal with rational singularities. Assume that $f$ has connected fibers. Then $f_* \mathcal{O}_X = \mathcal{O}_Y$.
If it is not correct is it possible to repair it in a reasonable way by placing additional constraints on the fibers?