On a Grauert-Riemenschneider vanishing theorem in dimension 3
Rahul Ajit
TL;DR
This work develops a Grauert–Riemenschneider–type vanishing theorem in dimension $3$ for excellent local rings with rational singularities, under the hypothesis that the blow-up $X$ is Cohen–Macaulay, normal, and pseudorational in codimension $2$. The authors prove that higher direct images $R^{i}\pi_{*}\omega_{X}$ vanish for $i>0$, and that for any birational morphism $W\to X$ with similar hypotheses, $R^{i}\phi_{*}\omega_{W}=0$ for $i>0$, yielding that $X$ has rational singularities. This vanishing result enables a dimension-$3$ version of Lipman’s vanishing conjecture for arbitrary characteristic and leads to a Briançon–Skoda-type theory for multiplier modules. The methods integrate resolution-based arguments, the Sancho de Salas exact sequence, and duality, producing applications in adjoint ideals and singularity theory with potential further implications in positive and mixed characteristics.
Abstract
Suppose $R$ is an excellent ring of dimension $3$ and has rational singularities. Let $π:X \longrightarrow \mathrm{Spec} \ R$ be a blow-up and $φ: W \longrightarrow X$ be any projective, birational morphism such that $X$ and $W$ are both normal, Cohen-Macaulay, and have pseudorational singularities in codimension $2$. Then $R^{i}φ_{*}ω_{W}=0 \ \text{ and }R^{i}π_{*}ω_{X} = 0$ 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.