Semistable Higgs bundles of small ranks are strongly Higgs semistable
We give a proof of the conjecture that a semistable Higgs bundle is strongly Higgs semistable in the case of small ranks, based upon the fact that there exists a gr-semistable Griffiths-transverse filtration on a $\nabla$-invariant semistable vector bundle. The latter is a generalization of a result of Carlos Simpson.