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 Q-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-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. (C) 2009 Elsevier Inc. All rights reserved.