Globally $F$-regular and log Fano varieties

Abstract

We prove that every globally $F$-regular variety is log Fano. In other words, if a prime characteristic variety $X$ is globally $F$-regular, then it admits an effective $\bQ$-divisor $\Delta$ such that $-K_X - \Delta$ is ample and $(X, \Delta)$ has controlled (Kawamata log terminal, in fact globally $F$-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non-$\bQ$-Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally $F$-regular type. Our techniques apply also to $F$-split varieties, which we show to satisfy a "log Calabi-Yau" condition. We also prove a Kawamata-Viehweg vanishing theorem for globally $F$-regular pairs.