On a Conjecture of Lan-Sheng-Zuo on Semistable Higgs Bundles: Rank 3 Case

TL;DR

This paper proves that rank 3 nilpotent semistable Higgs bundles on smooth projective curves over fields of characteristic p>2 are strongly semistable, partially confirming a conjecture and establishing tensor product properties for such bundles.

Contribution

It establishes the strong semistability of rank 3 nilpotent semistable Higgs bundles and proves a tensor product theorem under certain conditions, advancing understanding of Higgs bundle stability.

Findings

Rank 3 nilpotent semistable Higgs bundles are strongly semistable.

A tensor product theorem for strongly semistable Higgs bundles is proved.

Reproves a tensor theorem for semistable Higgs bundles assuming the conjecture.

Abstract

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$ of characteristic $p>2$. We prove that any rank $3$ nilpotent semistable Higgs bundle $(E,\theta)$ on $X$ is a strongly semistable Higgs bundle. This gives a partially affirmative answer to a conjecture of Lan-Sheng-Zuo \cite{LanShengZuo12ii}\footnotemark[1]. In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with $p$ satisfying some bounds (Theorem \ref{TensorTheorem}). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan-Sheng-Zuo conjecture holds (Corollary \ref{TensorStableBundle}).