Given a Cohen-Macaulay scheme of klt type X and a resolution π Y X, we show that R¹π_*ωY=0. We deduce that if dim (X) =3, then X satisfies Grauert-Riemenschneider vanishing and therefore has rational singularities. We also obtain that in arbitrary dimension, if X is of finite type over a perfect field of characteristic p>0, then X has Qₚ-rational singularities.
Baudin et al. (Thu,) studied this question.