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arXiv · 1306.3666

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

Lingguang Li
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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 $(E,θ)$ 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}).

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Lingguang Li. 2014-03-22. On a Conjecture of Lan-Sheng-Zuo on Semistable Higgs Bundles: Rank 3 Case. https://doi.org/10.1142/s0129167x1450013x

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