# Frobenius splitting and geometry of $G$-Schubert varieties

## TL;DR

This paper demonstrates Frobenius splitting properties of $G$-Schubert varieties in equivariant embeddings, revealing their singularity characteristics and extending results to $ ext{R}$-Schubert varieties, with implications for algebraic geometry in positive characteristic.

## Contribution

It proves Frobenius splitting compatibility for all $G$-Schubert varieties in equivariant embeddings and extends these results to $ ext{R}$-Schubert varieties, highlighting their geometric properties.

## Key findings

- $X$ admits a Frobenius splitting compatible with all $G$-Schubert varieties.
- Smooth, projective, toroidal $X$ have $G$-Schubert varieties with stable Frobenius splitting.
- Existence of a non-normal $G$-Schubert variety in the wonderful compactification of type $G_2$.

## Abstract

Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, projective and toroidal, then any $G$-Schubert variety in $X$ admits a stable Frobenius splitting along an ample divisors. Although this indicates that $G$-Schubert varieties have nice singularities we present an example of a non-normal $G$-Schubert variety in the wonderful compactification of a group of type $G_2$. Finally we also extend the Frobenius splitting results to the more general class of $\mathcal R$-Schubert varieties.