Strong semistability of Higgs bundles over curves

TL;DR

This paper investigates the strong semistability of Higgs bundles over algebraic curves, proving it for genus 0 and 1, and providing explicit counterexamples for higher genus.

Contribution

It completes the proof of the Lan-Sheng-Zuo conjecture for curves, showing semistability implies strong semistability in low genus, and constructing counterexamples in higher genus.

Findings

Semistable Higgs bundles are strongly semistable for genus g ≤ 1.

Explicit examples of semistable but not strongly semistable Higgs bundles are constructed for genus g ≥ 2.

Results complement existing theorems on small rank Higgs bundles.

Abstract

In this paper we complete the study of the Lan-Sheng-Zuo conjecture proposed in arXiv:1210.8280 for the curve case. Precisely, we prove that every semistable Higgs bundle is strongly semistable for curves of genus $g\leq 1$, and over any curves of genus $g\ge2$ construct explicit examples of semistable Higgs bundles of arbitrary big rank (the first example is $p=2,r=3$) which are not strongly semistable. These results are complementary to the strongly semistability theorem of Lan-Sheng-Yang-Zuo and Langer for semistable Higgs bundles of small rank.