On Principal Bundles with Vanishing Chern Classes

Semistable principal bundles with vanishing Chern classes over abelian varieties and complete homogeneous spaces in arbitrary characteristics are studied. The results of [14] on vector bundles are generalized to the context of principal bundles. If E_G is a semistable principal G–bundle over a complete homogeneous space with vanishing characteristic classes, then E_G is trivial. A semistable principal G–bundle over an abelian variety admits a reduction of structure group to a Borel subgroup of G. In [20] it was proved that a semistable vector bundle with vanishing Chern classes over a projective manifold X with vanishing Chern classes admits a flat connection. We prove that a principal G–bundle over X, where G is a complex reductive group, with vanishing characteristic classes admits a flat connection.