Abstract

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 forumla vanishes if and only if the line bundle over EG/P defined by χ is numerically effective. Also, a principal G-bundle EG over M is semistable with forumla 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 with the principal P-bundle EP for χ is of nonnegative degree. Therefore, EG is semistable with forumla 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 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.

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