We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E 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 real coefficients is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.Comment: v1: 25 pages. This submission supersedes arXiv:0809.3936. v2: 28 pages, includes changes suggested by the referees. v3: Final version to appear in Advances in Mathematic

Abstract. After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.

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Abstract. Let X be a compact connected Kähler-Einstein manifold with c 1 (T X) ≥ 0. If there is a semistable Higgs vector bundle (E , θ) on X with θ = 0, then we show that c 1 (T X) = 0; any X satisfying this condition is called a Calabi-Yau manifold, and it admits a Ricci-flat Kähler form [Ya]. Let (E , θ) be a polystable Higgs vector bundle on a compact Ricci-flat Kähler manifold X. Let h be an Hermitian structure on E satisfying the Yang-Mills-Higgs equation for (E , θ). We prove that h also satisfies the Yang-Mills-Higgs equation for (E , 0). A similar result is proved for Hermitian structures on principal Higgs bundles on X satisfying the Yang-Mills-Higgs equation.