On Semistable Principal Bundles over a Complex Projective Manifold
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2–4, 34013, Trieste, Italy
Correspondence: Correspondence to be sent to: indranil{at}math.tifr.res.in
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 
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 
vanishes if and only if the line bundle over EG/P defined by is numerically effective. Also, a principal G-bundle EG over M is semistable with 
if and only if for every pair of the form (Y, 
), where is a holomorphic map to M from a compact connected Riemann surface Y, and for every holomorphic reduction of structure group EP 
*EG to the subgroup P, the line bundle over Y associated with the principal P-bundle EP for is of nonnegative degree. Therefore, EG is semistable with 
if and only if for each pair (Y, ) of the above type the G-bundle *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.