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

Smith, Karen E.

Mich. Math. J. 48, Spec. Vol., 553-572 (2000).

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.

Reviewer: Irena Swanson (Las Cruces)

MSC:

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

Keywords:

vanishing theorems; globally \(F\)-regular varieties; character \(p\); Fano varieties; stable Frobenius splitting