Semistability vs. nefness for (Higgs) vector bundles

Generalizing a result of Miyaoka, we prove that the semistability of a vector bundle E on a smooth projective curve over a field of characteristic zero is equivalent to the nefness of any of certain divisorial classes θs, λs in the Grassmannians Grs(E) of locally-free quotients of E and in the projective bundles PQs, respectively (here 0<s<rkE and Qs is the universal quotient bundle on Grs(E)). The result is extended to Higgs bundles. In that case a necessary and sufficient condition for semistability is that all classes λs are nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the classes λs is equivalent to the semistability of the bundle E together with the vanishing of the characteristic class .