Stability of the tangent bundle of G/P in positive characteristics

TL;DR

This paper proves that the tangent bundle of homogeneous spaces G/P associated with certain algebraic groups over fields of positive characteristic remains Frobenius stable under specified conditions, extending stability results in algebraic geometry.

Contribution

It establishes Frobenius stability of the tangent bundle for G/P in positive characteristic for various types of algebraic groups, under specific characteristic bounds.

Findings

Tangent bundle of G/P is Frobenius stable for types B, C, F4 with p > 2, and G2 with p > 3.

Stability holds with respect to the anticanonical polarization.

Results extend known stability properties to new classes of algebraic groups in positive characteristic.

Abstract

Let $G$ be an almost simple simply-connected affine algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. If $G$ has type $B_n$, $C_n$ or $F_4$, we assume that $p > 2$, and if $G$ has type $G_2$, we assume that $p > 3$. Let $P \subset G$ be a parabolic subgroup. We prove that the tangent bundle of $G/P$ is Frobenius stable with respect to the anticanonical polarization on $G/P$.