Semistable principal G-bundles in positive characteristic

Original · EN

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 Eₚ such that its extension by P→ P/Rᵤ(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.