On Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type

TL;DR

This paper proves Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type, establishing rational singularities in dimension three and $Q_p$-rational singularities in positive characteristic.

Contribution

It extends Grauert-Riemenschneider vanishing to Cohen-Macaulay schemes of klt type, a significant class in algebraic geometry, in arbitrary dimensions and positive characteristic.

Findings

Proves $R^1 o_*\omega_Y=0$ for resolutions of klt type schemes.

Shows three-dimensional klt schemes have rational singularities.

Establishes $Q_p$-rational singularities for schemes over perfect fields of characteristic $p>0$.

Abstract

Given a Cohen-Macaulay scheme of klt type $X$ and a resolution $\pi\colon Y\to X$, we show that $R^1\pi_*\omega_Y=0$. We deduce that if $\mathrm{dim}(X)=3$, then $X$ satisfies Grauert-Riemenschneider vanishing and therefore has rational singularities. We also obtain that in arbitrary dimension, if $X$ is of finite type over a perfect field of characteristic $p>0$, then $X$ has $\mathbb{Q}_p$-rational singularities.