Semistable principal Higgs bundles

We give a Miyaoka-type semistability criterion for principal Higgs G-bundles E on complex projective manifolds of any dimension, i.e., we prove that E is semistable and the second Chern class of its adjoint bundle vanishes if and only if certain line bundles, obtained from the characters of the parabolic subgroups of G, are numerically effective. We also give alternative characterizations in terms of a notion of numerical effectiveness of Higgs vector bundles we have recently introduced.