Globally $F$-regular and log Fano varieties

Details Globally $F$-regular and log Fano varieties Globally $F$-regular and log Fano varieties

2009-05-04

arxiv.org/abs/0905.0404v2

Advances in Mathematics, Vol 224 Issue 3, 863--894, 2010

Mathematics

Algebraic Geometry

31 pages, minor changes throughout. The presentation of section 5 improved. To appear in Advances in Mathematics

Scientific paper

10.1016/j.aim.2009.12.020

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.

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