Abstract: A useful vanishing theorem for understanding characteristic zero singularities is Grauert-Riemenschneider vanishing, which asserts that if f: Y -> X is a projective birational morphism and Y is smooth, then higher pushfowards of the sheaf of top forms on Y vanish. A remarkable consequence of this result is that characteristic zero klt singularities are rational.

As one might expect, this vanishing theorem fails over fields of positive characteristic. In this talk, we will explain how to prove a Witt vector version of Grauert-Riemenchneider vanishing, and consequences to the rationality of certain singularities in positive characteristic.