On semistable principal bundles on complex projective manifolds

Let G be a simple linear algebraic group defined over the field of complex numbers. Fix a proper parabolic subgroup P of G, and also fix a nontrivial antidominant character chi of P. We prove that a holomorphic principal G-bundle E-G over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class c(2)(ad(E-G)) is an element of H-4(M, Q) vanishes if and only if the line bundle over E-G/P defined by chi is numerically effective. Also, a principal G-bundle E-G over M is semistable with c(2)(ad(E-G)) = 0 if and only if for every pair of the form (Y, psi), where psi is a holomorphic map to M from a compact connected Riemann surface Y, and for every holomorphic reduction of structure group E-P subset of psi*E-G to the subgroup P, the line bundle over Y associated with the principal P-bundle E-P for chi is of nonnegative degree. Therefore, E-G is semistable with c(2)(ad(E-G)) = 0 if and only if for each pair (Y,psi) of the above type the G-bundle psi*E-G over Y is semistable. Similar results remain valid for principal bundles over M with a reductive linear algebraic group as the structure group. These generalize an earlier work of Miyaoka [12], where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations, one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.