On Semistable Principal Bundles over a Complex Projective Manifold

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 χ of P . We prove that a holomorphic principal G–bundle EG over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class c2(ad(EG)) ∈ H (M, Q) vanishes if and only if the line bundle over EG/P defined by χ is numerically effective. Also, a principal G–bundle EG overM is semistable with c2(ad(EG)) = 0 if and only if for every pair of the form (Y , ψ), where ψ is a holomorphic map to M from a compact connected Riemann surface Y , and for every holomorphic reduction of structure group EP ⊂ ψ ∗EG to the subgroup P , the line bundle over Y associated to the principal P–bundle EP for χ is of nonnegative degree. Therefore, EG is semistable with c2(ad(EG)) = 0 if and only if for each pair (Y , ψ) of the above type the G–bundle ψ∗EG 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 Y. Miyaoka, [Mi], 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.