Restricting Higgs bundles to curves

We determine some classes of varieties $X$ --- that include the varieties with numerically effective tangent bundle --- satisfying the followingproperty: if $E=(E,phi)$ is a Higgs bundle such that $f^astE$ is semistable for any morphism $fcolon C o X$, where $C$ is a smooth projective curve, then $E$ is slope semistable and $2r c_2(E)-(r-1) c_1^2(E)=0$ 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.