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

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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 LanShengZuo12ii[1]. In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with p satisfying some bounds (Theorem TensorTheorem). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan-Sheng-Zuo conjecture holds (Corollary TensorStableBundle).