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 θsθs, λsλs in the Grassmannians Grs(E)Grs(E) of locally-free quotients of E   and in the projective bundles PQsPQs, respectively (here 0