THE YANG-MILLS FUNCTIONAL AND BOGOMOLOV INEQUALITY FOR ARBITRARY PRINCIPAL BUNDLES OVER KAHLER MANIFOLDS

We study the Yang-Mills functional for principal fiber bundles with structure group a compact Lie group K over a Kahler manifold. In particular, we analyze the absolute minimizers for this functional and prove that they are exactly the Einstein K-connections. By means of the structure of the Yang-Mills functional at an absolute minimum, we prove that the characteristic classes of a principal K-bundle which admits an Einstein connection satisfy two inequalities. One of them is a generalization of the Bogomolov inequality whereas the other is an inequality related to the center of the structure group. Therefore, this way we offer a new and natural proof of the Bogomolov inequality that helps understanding its origin. Finally, in view of the Hitchin-Kobayashi correspondence we prove that every (poly-)stable principal K-C-bundle has to satisfy this generalized Bogomolov type inequality.