Globally F -regular and Log Fano Varieties

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 Qdivisor ∆ such that −KX −∆ is ample and (X, ∆) 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-Q-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.