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 E-G over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class c(2)(ad(E-G)) is an element of H-4(M, Q) vanishes if and only if the line bundle over E-G/P defined by chi is numerically effective. Also, a principal G-bundle E-G over M is semistable with c(2)(ad(E-G)) = 0 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 E-P subset of psi*E-G to the subgroup P, the line bundle over Y associated with the principal P-bundle E-P for chi is of nonnegative degree. Therefore, E-G is semistable with c(2)(ad(E-G)) = 0 if and only if for each pair (Y,psi) of the above type the G-bundle psi*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.

On semistable principal bundles on complex projective manifolds / Biswas, I.; Bruzzo, U.. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - 2008:1(2008), pp. 1-28. [10.1093/imrn/rnn035]

On semistable principal bundles on complex projective manifolds

Bruzzo, U.
2008-01-01

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 chi 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 c(2)(ad(E-G)) is an element of H-4(M, Q) vanishes if and only if the line bundle over E-G/P defined by chi is numerically effective. Also, a principal G-bundle E-G over M is semistable with c(2)(ad(E-G)) = 0 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 E-P subset of psi*E-G to the subgroup P, the line bundle over Y associated with the principal P-bundle E-P for chi is of nonnegative degree. Therefore, E-G is semistable with c(2)(ad(E-G)) = 0 if and only if for each pair (Y,psi) of the above type the G-bundle psi*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.
2008
2008
1
1
28
rnn035
https://arxiv.org/abs/0803.4042
Biswas, I.; Bruzzo, U.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/12564
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