  According to Miyaoka, a vector bundle E on a smooth projective curve is
semistable if and only if a certain numerical class in the projectivized bundle
PE is nef. We establish a similar criterion for the semistability of Higgs
bundles: namely, such a bundle is semistable if and only if for every integer s
between 0 and the rank of E, a suitable numerical class in the scheme
parametrizing the rank s locally-free Higgs quotients of E is nef. We also
extend this result to higher-dimensional complex projective varieties by
showing that the nefness of the above mentioned classes is equivalent to the
semistability of the Higgs bundle E together with the vanishing of the
discriminant of E.
