关于半稳定Higgs丛的Lan-Sheng-Zuo猜想：秩3情形
On a Conjecture of Lan-Sheng-Zuo on Semistable Higgs Bundles: Rank 3 Case
摘要：
设$X$是特征为$p>2$的代数闭域$k$上亏格为$g$的光滑投影曲线.证明了$X$上的任意秩$3$幂零半稳定Higgs丛$(E,θ)$是强半稳定Higgs丛.这部分地肯定了兰生左\cite{LanShengZuo12ii} \footnotemark [1]的一个猜想.此外，我们证明了强半稳定Higgs丛的张量积定理，其中$p$满足一定的界（定理\ref{TensorTheorem}）。在此基础上，我们在兰生左猜想成立的条件下，重新证明了半稳定Higgs丛的张量定理（推论\ref{TensorStableBundle}）.
英文摘要：
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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