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.Comment: Current version: Added subsection on scheme theoretic closure and density and their relevance in the article. Updated the subsection on torsion-freeness, added a diagram to Lemma 7.11 to make the interactions of the various morphisms clea

Steenbrink vanishing extended

The notion of DB index, a measure of how far a singularity of a pair is from being Du Bois, is introduced and used to generalize vanishing theorems of Steenbrink and others with simpler and more natural proofs than the originals.Comment: To appear in the Proceedings of 12th ALGA meeting dedicated to Aron Simis and Steven Kleiman (Bulletin of the Brazilian Mathematical Society). Some typos are corrected in v2, but no essential change from v