The aim of this paper is to prove two basic restriction theorem for principal bundles on smooth projective varieties in arbitrary characteristic generalizing the analogues theorems of Mehta-Ramanathan for vector bundles.
More precisely, let G be a reductive algebraic group over an algebraically closed field k and let X be a smooth, projective variety over k together with a very ample line bundle O(1). The main result of the paper is that if E is a semistable (resp. stable) principal G-bundle on X w.r.t O(1), then the restriction of E to a general, high multi-degree, complete-intersection curve is again semistable (resp. stable).
As easy applications of these theorems we show the openness of semistability (resp. stability) for a family of principal bundles parametrized by a finite-type k-scheme.