Restricting Higgs bundles to curves

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.