Semistable and numerically effective principal (Higgs) bundles

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.