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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.

Abstract. A decorated vector bundle on a smooth projective curve X is a pair (E, ϕ) consisting of a vector bundle and a morphism ϕ : (E ⊗a ) ⊕b → (det E) ⊗c ⊗ N, where N ∈ Pic(X). There is a suitable semistability condition for such objects which has to be checked for any weighted filtration of E. We prove, at least when a = 2, that it is enough to consider filtrations of length ≤ 2. In this case decorated bundles are very close to quadric bundles and to check semistability condition one can just consider the former. A similar result for L-twisted bundles and quadric bundles was already proved ([1], [6]). Our proof provides an explicit algorithm which requires a destabilizing filtration and ensures a destabilizing subfiltration of length at most two. Quadric bundles can be thought as a generalization of orthogonal bundles. We show that the simplified semistability condition for decorated bundles coincides with the usual semistability condition for orthogonal bundles. Finally we note that our proof can be easily generalized to decorated vector bundles on nodal curves.

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