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Let G be a simple linear algebraic group defined over the complex numbers.Fix a proper parabolic subgroup P of G and a nontrivial antidominant character\chi of P. We prove that a holomorphic principal G-bundle E over a connectedcomplex projective manifold M is semistable and the second Chern class of itsadjoint bundle vanishes in rational cohomology if and only if the line bundleover E/P defined by \chi is numerically effective. Similar results remain validfor principal bundles with a reductive linear algebraic group as the structuregroup. These generalize an earlier work of Y. Miyaoka where he gave acharacterization of semistable vector bundles over a smooth projective curve.Using these characterizations one can also produce similar criteria for thesemistability of parabolic principal bundles over a compact Riemann surface.

On semistable principal bundles over a complex projective manifold