Semistability vs. nefness for (Higgs) vector bundles

Details Semistability vs. nefness for (Higgs) vector bundles Semistability vs. nefness for (Higgs) vector bundles

2003-10-03

arxiv.org/abs/math/0310040v3

Diff. Geom. Appl. 24 (2006) 403-416

Mathematics

Algebraic Geometry

Comments: 20 pages, Latex2e, no figures. v2 includes a generalization to complex projective manifolds of any dimension. To app

Scientific paper

10.1016/j.difgeo.2005.12.007

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.

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