On a conjecture about Higgs bundles for rank 2 and some inequalities

Abstract

We briefly review an open conjecture about Higgs bundles that are semistable with after pulling back to any curve, and prove it in the rank 2 case. We also prove a set of inequalities holding for H-nef Higgs bundles that generalize some of the Fulton-Lazarsfeld inequalities for numerically effective vector bundles.Comment: 13 pages. v2: 14 pages. Some results have been strengthened and the exposition has been reorganized. v3: minor changes, final version to appear in Mediterranean J. Mat