Restriction Theorems for Principal Bundles and Some Consequences

TL;DR

This paper proves restriction theorems for principal G-bundles over smooth projective varieties, showing that semistability or stability is preserved when restricting to general high-degree complete intersection curves, in arbitrary characteristic.

Contribution

It provides a proof of restriction theorems for principal bundles with reductive algebraic groups in any characteristic, extending known results to broader settings.

Findings

Restriction of semistable principal G-bundles remains semistable on general high-degree curves.

Restriction of stable principal G-bundles remains stable on general high-degree curves.

Results hold over arbitrary characteristic fields.

Abstract

The aim of this paper is to give a proof of the restriction theorems for principal bundles with a reductive algebraic group as structure group in arbitrary characteristic. Let $G$ be a reductive algebraic group over any field $k=\bar{k}$, let $X$ be a smooth projective variety over $k$, let $H$ be a very ample line bundle on $X$ and let $E$ be a semistable (resp. stable) principal $G$-bundle on $X$ w.r.t. $H$. The main result of this paper is that the restriction of $E$ to a general smooth curve which is a complete intersection of ample hypersurfaces of sufficiently high degree's is again semistable (resp. stable).