On Semistable Principal Bundles over a Complex Projective Manifold

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 E G over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class vanishes if and only if the line bundle over E G /P defined by χ is numerically effective. Also, a principal G-bundle E G over M is semistable with 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 E P ⊂ ψ*E G to the subgroup P, the line bundle over Y associated with the principal P-bundle E P for χ is of nonnegative degree. Therefore, E G is semistable with if and only if for each pair (Y, ψ) of the above type the G-bundle ψ*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.