ON A CONJECTURE OF LAN–SHENG–ZUO ON SEMISTABLE HIGGS BUNDLES: RANK 3 CASE

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 [Semistable Higgs bundles and representations of algebraic fundamental groups: positive characteristic case, preprint (2012), arXiv:1210.8280][(Very recently, A. Langer [Semistable modules over Lie algebroids in positive characteristic, preprint (2013), arXiv:1311.2794] and independently Lan–Sheng–Yang–Zuo [Semistable Higgs bundles of small ranks are strongly Higgs semistable, preprint (2013), arXiv:1311.2405] have proven the conjecture for ranks less than or equal to p case.)] In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with p satisfying some bounds (Theorem 4.3). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan–Sheng–Zuo conjecture holds (Corollary 4.4).