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.

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

References in corpus (2)

• Introduction to the log minimal model program for log canonical pairs
• Generalized test ideals, sharp F-purity, and sharp test elements

Cited by in corpus (61)

• Normal singularities with torus actions
• A survey of test ideals
• Fundamental groups of $F$-regular singularities via $F$-signature
• K-Stability for Fano Manifolds with Torus Action of Complexity One
• Derived splinters in positive characteristic
• Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras
• Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic
• Depth of $F$-singularities and base change of relative canonical sheaves
• Finite torsors over strongly $F$-regular singularities
• On log del Pezzo surfaces in large characteristic
• Test ideals in rings with finitely generated anti-canonical algebras
• Bigness of the tangent bundle of del Pezzo surfaces and $D$-simplicity
• Valuations and Frobenius
• On the three dimensional minimal model program in positive characteristic
• The Minimal Model Program for threefolds in characteristic five
• Nef anti-canonical divisors and rationally connected fibrations
• On canonical bundle formulae and subadjunctions
• An analog of adjoint ideals and PLT singularities in mixed characteristic
• Frobenius Splitting in Commutative Algebra
• On base point freeness in positive characteristic
• Singularities of locally acyclic cluster algebras
• F-signature of pairs and the asymptotic behavior of Frobenius splittings
• Rational points on log Fano threefolds over a finite field
• When is the Albanese morphism an algebraic fiber space in positive characteristic?
• $F$-signature under birational morphisms
• Adjoint ideals and a correspondence between log canonicity and F-purity
• The weak Beauville--Bogomolov decomposition in characteristic $p\ge 0$
• On images of weak Fano manifolds
• On the F-purity of isolated log canonical singularities
• Singularities of Cox Rings of Fano Varieties
• Schubert varieties are log Fano over the integers
• Test ideals in non-Q-Gorenstein rings
• Arithmetic and geometric deformations of $F$-pure and $F$-regular singularities
• Reductive quotients of klt singularities
• Local cohomology bounds and test ideals
• Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field
• On images of Mori dream spaces
• Bogomolov--Sommese type vanishing for globally $F$-regular threefolds
• $F$-singularities: applications of characteristic $p$ methods to singularity theory
• Global $F$-splitting of surfaces admitting an int-amplified endomorphim
• General hyperplane sections of threefolds in positive characteristic
• Classification of globally F-regular $F$-sandwiches of Hirzebruch surfaces
• A Gorenstein criterion for strongly $F$-regular and log terminal singularities
• Weak Akizuki-Nakano vanishing theorem for globally $F$-split 3-folds
• Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic
• The F-signature Function on the Ample Cone
• On the semicontinuity problem of fibers and global $F$-regularity
• A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf II
• Characterization of log del Pezzo pairs via anticanonical models
• On the moduli spaces of parabolic symplectic orthogonal bundles on curves
• Singularities of non-$\mathbb{Q}$-Gorenstein varieties admitting a polarized endomorphism
• On proper splinters in positive characteristic
• Surfaces of globally $F$-regular type are of Fano type
• Ample Weil divisors
• Globally F-regular $F$-sandwiches of degree $p$ of a projective space
• Kollár's injectivity theorem for globally $F$-regular varieties
• Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers
• On Rational Connectedness of Globally F-Regular Threefolds
• Elliptic singularities and threefold flops in positive characteristic
• On the superadditivity of anticanonical Iitaka dimension
• Remarks on log Calabi--Yau structure of varieties admitting polarized endomorphism