Let $X$ be a normal variety over a field of characteristic zero with rational singularities.
If $\pi:Y \to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational singularities?
It is easy to see that this is true if $\dim(X) = 2$, but the higher dimensional case seems more difficult and perhaps it is even false. If true, I would also be interested in analogous results in positive or mixed characteristic e.g., for pseudo-rational singularities.
No, $Y$ need not have rational singularities. See Section III of the paper for an example in dimension three: Cutkosky, A characterization of rational surface singularities, Inventiones Mathematicae, 1990. In that example, $Y$ ($Z$ in the notation of the paper) is normal but not Cohen-Macaulay, so it is not a rational singularity.
For an explicit example (in dimension three) with the same property, see Theorem 3.11 in Huckaba and Huneke, Normal ideals in regular rings, Journal fur die Reine und Angewande Mathematik, 1999.
