The Frobenius–stable version of the Grauert–Riemenschneider vanishing theorem fails

We show the failure of the Frobenius–stable version of the Grauert–Riemenschneider vanishing theorem for threefolds in any positive characteristic, and for terminal 3-folds in characteristic p ∈ { 2 , 3 , 5 } p \in \{2, 3, 5\} . To prove this, we introduce the notion of F p \mathbb {F}_p -rationality for singularities in positive characteristic and we show that klt singularities in dimension at most 4 are F p \mathbb {F}_p -rational. We apply this to prove a Frobenius–stable version of the Kawamata–Viehweg vanishing theorem on K K -trivial 3-folds.