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 also look at cohomology vanishing and show the existence of rational G-equivariant resolutions by toroidal embeddings. We show that the class of homogeneous spaces for which our results hold contains the symmetric homogeneous spaces in characteristic not 2 and is closed under parabolic induction.