Rational Singularities

A resolution-free definition of rational singularities is introduced, and it is proved that for a variety admitting a resolution of singularities, this is equivalent to the usual definition. It is also demonstrated that rational singularities are equivalent to pseudo-rational singularities. As applications, several open questions about the higher direct images of structure sheaves and dualizing sheaves are answered and it is proved that Cohen-Macaulay klt singularities are rational in arbitrary characteristic.