A Gorenstein criterion for strongly $F$-regular and log terminal singularities

TL;DR

This paper proves a conjecture linking Gorenstein properties of strongly F-regular rings to F-pure thresholds, under conditions on the anti-canonical cover, and extends criteria to log canonical singularities.

Contribution

It confirms a conjecture relating Gorenstein conditions to F-pure thresholds and provides new criteria for quasi-Gorenstein and log canonical singularities.

Findings

Proves the conjecture under the assumption of Noetherian anti-canonical cover.

Establishes a criterion for quasi-Gorenstein singularities based on F-pure thresholds.

Provides a similar criterion for log canonical singularities.

Abstract

A conjecture of Hirose, Watanabe, and Yoshida offers a characterization of when a standard graded strongly $F$-regular ring is Gorenstein, in terms of an $F$-pure threshold. We prove this conjecture under the additional hypothesis that the anti-canonical cover of the ring is Noetherian. Moreover, under this hypothesis on the anti-canonical cover, we give a similar criterion for when a normal $F$-pure (resp. log canonical) singularity is quasi-Gorenstein, in terms of an $F$-pure (resp. log canonical) threshold.