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 $(E,\theta )$ 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}).

International Journal of Mathematics 2014

References in corpus (4)

• Semistable Higgs bundles and representations of algebraic fundamental groups: Positive characteristic case
• Semistable Higgs bundles of small ranks are strongly Higgs semistable
• Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
• An inverse Cartier transform via exponential in positive characteristic

Cited by in corpus (2)

• Semistable modules over Lie algebroids in positive characteristic
• Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups