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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn035, 28 pages, doi:10.1093/imrn/rnn035 published on April 26, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

On Semistable Principal Bundles over a Complex Projective Manifold

Indranil Biswas1 and Ugo Bruzzo2

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 {chi} 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 Formula vanishes if and only if the line bundle over EG/P defined by {chi} is numerically effective. Also, a principal G-bundle EG over M is semistable with Formula if and only if for every pair of the form (Y, {psi}), where {psi} is a holomorphic map to M from a compact connected Riemann surface Y, and for every holomorphic reduction of structure group EP sub {psi}*EG to the subgroup P, the line bundle over Y associated with the principal P-bundle EP for {chi} is of nonnegative degree. Therefore, EG is semistable with Formula if and only if for each pair (Y, {psi}) of the above type the G-bundle {psi}*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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This Article
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