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