On Gorenstein ${\mathbb{Q}}_p$-rational threefold and fourfold singularities

Abstract

We prove that for $n\le 4$ 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.

Final version. To appear in Épijournal de Géométrie Algébrique