Connections on principal bundles over curves in positive characteristics

Abstract

Let X be an irreducible smooth projective curve over an algebraically closed field k of characteristic p, with p > 5. Let G be a connected reductive algebraic group over k. Let H be a Levi factor of some parabolic subgroup of G and a character of H. Given a reduction E H of the structure group of a G-bundle EG to H, let E be the line bundle over X associated to E H for the character . If G does not contain any SL(n)/Z as a simple factor, where Z is a subgroup of the center of SL(n), we prove that a G-bundle EG over X admits a connection if and only if for every such triple (H,,E H ), the degree of the line bundle E is a multiple of p. If G has a factor of the form SL(n)/Z, then this result is valid if n is not a multiple of p. If G is a classical group but not of the form SL(n)/Z, then this criterion for the existence of connection remains valid even if p > 3.