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 pushforwards of \omega_Y vanish. A remarkable consequence of this result is that characteristic zero klt singularities are rational.

As one might expect, this vanishing theorem fails in positive characteristic. In this talk, we will explain how to prove a version of Grauert-Riemenchneider vanishing involving Witt differential forms, answering a question of Blickle, Esnault, Chatzistamatiou and Rülling. We will also discuss applications to singularities.