We study Miyaoka-type semistability criteria for principal Higgs $G$-bundles $\fE$ on complex projective manifolds of any dimension. We prove that $\fE$ has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of $G$, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian-Yang-Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in $\R$ is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.

Semistable and numerically effective principal (Higgs) bundles / Bruzzo, Ugo; Graña Otero, B.. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 226:4(2011), pp. 3655-3676. [10.1016/j.aim.2010.10.026]

Semistable and numerically effective principal (Higgs) bundles

Bruzzo, Ugo;
2011-01-01

Abstract

We study Miyaoka-type semistability criteria for principal Higgs $G$-bundles $\fE$ on complex projective manifolds of any dimension. We prove that $\fE$ has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of $G$, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian-Yang-Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in $\R$ is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.
2011
226
4
3655
3676
https://arxiv.org/abs/0905.2870
Bruzzo, Ugo; Graña Otero, B.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/16261
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