Embeddings of spherical homogeneous spaces in characteristic p

Let G be a reductive group over an algebraically closed field of characteristic p>0. We study properties of embeddings of spherical homogeneous G-spaces. We look at Frobenius splittings, canonical or by a (p−1)-th power, compatible with certain subvarieties. We show the existence of rational G-equivariant resolutions by toroidal embeddings, and give results about cohomology vanishing and surjectivity of restriction maps of global sections of line bundles. We show that the class of homogeneous spaces for which our results hold contains the symmetric homogeneous spaces in characteristic ≠2 and is closed under parabolic induction.