Semistable principal G-bundles in positive characteristic

Abstract

Let $X$ be a normal projective variety defined over an algebraically closed field $k$ of positive characteristic. Let $G$ be a connected reductive group defined over $k$. We prove that some Frobenius pull back of a principal $G$-bundle admits the canonical reduction $E_P$ such that its extension by $P\to P/R_u(P)$ is strongly semistable. Then we show that there is only a small difference between semistability of a principal $G$-bundle and semistability of its Frobenius pull back. This and the boundedness of the family of semistable torsion free sheaves imply the boundedness of semistable principal $G$-bundles.