Embeddings of spherical homogeneous spaces in characteristic p

TL;DR

This paper investigates embeddings of spherical homogeneous spaces over fields of characteristic p, focusing on Frobenius splittings, cohomology vanishing, and resolutions, extending results to symmetric spaces and those closed under parabolic induction.

Contribution

It introduces new properties of embeddings in characteristic p, including Frobenius splitting techniques and the existence of G-equivariant resolutions, expanding the class of spaces where these hold.

Findings

Frobenius splittings compatible with subvarieties are established.

Cohomology vanishing results are proved for these embeddings.

Existence of rational G-equivariant resolutions by toroidal embeddings is demonstrated.

Abstract

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.