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We provide a partial classification of semistable Higgs bundles over a simply connected Calabi-Yau manifolds. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces.

No abstract

We determine some classes of varieties X -that include the varieties with numerically effective tangent bundle -satisfying the following property: if E = (E, φ) is a Higgs bundle such that f * E is semistable for any morphism f : C → X, where C is a smooth projective curve, then E is slope semistable and 2rc2(E) − (r − 1)c 2 1 (E) = 0 in H 4 (X, R). We also characterize some classes of varieties such that the underlying vector bundle of a slope semistable Higgs bundle is always slope semistable.