Semistability vs. nefness for (Higgs) vector bundles

Abstract

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 θ s , λ s in the Grassmannians Gr s ( E ) of locally-free quotients of E and in the projective bundles P Q s , respectively (here 0 < s < rk E and Q s is the universal quotient bundle on Gr s ( E ) ). The result is extended to Higgs bundles. In that case a necessary and sufficient condition for semistability is that all classes λ s are nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the classes λ s is equivalent to the semistability of the bundle E together with the vanishing of the characteristic class Δ ( E ) = c 2 ( E ) − r − 1 2 r c 1 ( E ) 2 .