On embeddings of certain spherical homogeneous spaces in prime characteristic

TL;DR

This paper investigates embeddings of certain spherical homogeneous spaces over fields of positive characteristic, demonstrating their Frobenius splitting, Cohen-Macaulayness, and normality of orbit closures, with applications to reductive monoids and determinantal varieties.

Contribution

It establishes canonical Frobenius splitting and rational resolutions for embeddings of specific homogeneous spaces, extending understanding of their geometric properties in prime characteristic.

Findings

Embeddings are Frobenius split and Cohen-Macaulay.

Orbit closures in spherical varieties are normal.

Varieties of circular complexes are Gorenstein.

Abstract

Let $\mc G$ be a reductive group over an algebraically closed field of characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic subgroup of $\mc G$. We show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen-Macaulay. Examples are the $G\times G$-orbits in normal reductive monoids with unit group $G$. Our class of homogeneous spaces also includes the open orbits of the well-known determinantal varieties and the varieties of (circular) complexes. We also show that all $G$-orbit closures in a spherical variety which is canonically Frobenius split are normal. Finally we study the Gorenstein property for the varieties of circular complexes and for a related reductive monoid.