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

TL;DR

This paper proves a conjecture about semistable Higgs bundles of rank 2 and establishes new inequalities for H-nef Higgs bundles, extending known results for vector bundles.

Contribution

It confirms the conjecture for rank 2 Higgs bundles and introduces generalized inequalities for H-nef Higgs bundles.

Findings

Proof of the conjecture for rank 2 Higgs bundles

New inequalities for H-nef Higgs bundles

Extension of Fulton-Lazarsfeld 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.