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

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