Globally F-regular varieties: Applications to vanishing theorems for quotients of Fano varieties
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.
scientific article; zbMATH DE number 1700888