On a Grauert-Riemenschneider vanishing theorem in dimension 3

Abstract

Suppose $R$ is an excellent ring of dimension $3$ and has rational singularities. Let $\pi :X⟶\mathrm{Spec}R$ be a blow-up and $\varphi :W⟶X$ be any projective, birational morphism such that $X$ and $W$ are both normal, Cohen-Macaulay, and have pseudorational singularities in codimension $2$. Then $R^i{\varphi }_{\ast }{\omega }_W=0\text{ and }{R}^i{\pi }_{\ast }{\omega }_X=0$ for all $i>0$ and $X$ has rational singularities. We use this result to prove Lipman's vanishing conjecture in dimension $3$ for arbitrary characteristics and provide a few applications.

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