On a Grauert-Riemenschneider vanishing theorem in dimension 3

TL;DR

This paper proves a vanishing theorem for rational singularities in dimension 3, confirming Lipman's conjecture across all characteristics and offering new applications in algebraic geometry.

Contribution

It establishes a Grauert-Riemenschneider type vanishing theorem for three-dimensional rational singularities, extending known results to arbitrary characteristic.

Findings

Vanishing of higher direct images of the canonical sheaf under certain morphisms.

Validation of Lipman's vanishing conjecture in dimension 3 for all characteristics.

New applications in the study of rational singularities and algebraic geometry.

Abstract

Suppose $R$ is an excellent ring of dimension $3$ and has rational singularities. Let $\pi:X \longrightarrow \mathrm{Spec} \ R$ be a blow-up and $\phi: W \longrightarrow X$ be any projective, birational morphism such that $X$ and $W$ are both normal, Cohen-Macaulay, and have pseudorational singularities in codimension $2$. Then $R^{i}\phi_{*}\omega_{W}=0 \ \text{ and }R^{i}\pi_{*}\omega_{X} = 0$ 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.