## Globally $$F$$-regular and log Fano varieties.(English)Zbl 1193.13004

Globally $$F$$-regular varieties were introduced by K. E. Smith [Mich. Math. J. 48, Spec. Vol., 553–572 (2000; Zbl 0994.14012)] where it was shown that Fano varieties in characteristic $$0$$ reduce to globally $$F$$-regular varieties in characteristic $$p\gg 0$$. It is known that globally $$F$$-regular varieties satisfy Kodaira type vanishing results and are locally $$F$$-regular (in particular they are Cohen-Macaulay and have pseudo-rational singularities). Note that locally $$F$$-regular singularities correspond to Kawamata log terminal singularities in the context of the minimal model program.
The main result of the paper under review is to show that if $$X$$ is a globally $$F$$- regular variety of characteristic $$p\neq 0$$, then there exists an effective $$\mathbb Q$$-divisor $$\Delta$$ such that $$(X,\Delta )$$ is a log Fano variety (i.e. $$-(K_X+\Delta)$$ is ample and $$(X,\Delta )$$ is Kawamata log terminal). The authors also show that log Fano varieties in characteristic $$0$$ reduce to globally $$F$$-regular varieties in characteristic $$p\gg 0$$.

### MSC:

 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 14J45 Fano varieties 14B05 Singularities in algebraic geometry

### Keywords:

Fano varieties; globally $$F$$-regular

Zbl 0994.14012
Full Text:

### References:

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