On Gorenstein $\mathbb{Q}_p$-rational threefolds and fourfolds

TL;DR

This paper establishes that for dimensions up to four and primes greater than five, certain classes of singularities are canonical, extending known results about rational Gorenstein singularities using dual complex analysis.

Contribution

It proves that quasi-Gorenstein F-pure and Q_p-rational singularities are canonical in dimensions up to four, under specific conditions, advancing the understanding of singularity classification.

Findings

Quasi-Gorenstein F-pure singularities are canonical for n ≤ 4 and p > 5.

Q_p-rational singularities are shown to be canonical in the same setting.

The proof involves analyzing the dual complex of dlt modifications of log canonical singularities.

Abstract

We prove that for $n \leq 4$ and $p > 5$, quasi--Gorenstein $F$--pure and $\mathbb{Q}_p$--rational $n$--fold singularities are canonical. This is analogous to the usual fact that rational Gorenstein singularities are canonical. The proof is based on a careful analysis of the dual complex of a dlt modification of a log canonical singularity. The result for $n = 4$ is contingent upon the existence of log resolutions.