To my knowledge, there is no characteristic-free proof of Grauert–Riemenschneider vanishing theorem: Let $\pi \colon \widetilde{X} \to X$ a desingularization of $X$, $\mathcal{F}$ an ample locally free sheaf on $X$ such that $\pi^{\ast}(\mathcal{F})$ is also locally free, then $R^p\pi_*(\pi^{\ast}(\mathcal{F}) \otimes \omega_{\widetilde{X}}) = 0$ for all $p \geq 1$, where $\omega_{\widetilde{X}}$ denotes the canonical sheaf on $\widetilde{X}$.
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