Globally F-regular varieties: Applications to vanishing theorems for quotients of Fano varieties. (English) Zbl 0994.14012

The main results of this paper are on the vanishing of higher cohomology modules of certain line bundles. The author introduces globally F-regular varieties as follows: a projective variety over a Frobenius-finite (prime characteristic) field is globally F-regular if it admits some section ring that is F-regular. A scheme (in characteristic 0) is of strongly F-regular type if in some standard “reduction to characteristic \(p\)” a dense set of closed fibers (of positive prime characteristic) is globally F-regular. The author proves that these varieties satisfy strong vanishing properties. Examples of varieties of globally F-regular type are Fano varieties with rational singularities, projective toric varieties, and geometric invariant quotients. Smith proves equivalent formulations of global F-regularity in terms of stable Frobenius splitting. All the theory and notions are carefully and elegantly explained.


14F17 Vanishing theorems in algebraic geometry
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14J45 Fano varieties
14G20 Local ground fields in algebraic geometry
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