Let $X$ be a normal variety over a field of characteristic zero with rational singularities.



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> If $\pi:Y \to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational singularities?
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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.



An invariance property of 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.


