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arXiv 1805.04076math.AG
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ruled surfaces上稳定的Higgs束

Stable Higgs Bundles on ruled surfaces

Snehajit Misra

备注：
Final Version
发表信息：
Indian Journal of Pure and Applied Mathematics, 51, 735-747 (2020)
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摘要：

设$π: X = \mathbb{P}_C(E) \longrightarrow C$是特征为0的代数闭域$k$上的直纹曲面，$X$上有固定的极化$L$。本文证明了在$π$下，$C$上的（半）稳定Higgs丛的拉回是一个$L$ -（半）稳定Higgs丛。相反，如果$(V,θ)$是$X$上的一个$L$ -（半）稳定的Higgs丛，其中对于$C$和$c_2(V)=0$上的某个度为$d$的因子$\bf d \rm $，有$c_1(V)= π^*(\bf d \rm)$，那么在$C$上存在一个度为$d$的（半）稳定的Higgs丛$(W,ψ)$，它在$π$下的拉回同构于$(V,θ)$。由此，我们得到了（半）稳定Higgs丛的模空间之间的同构。当基场为$\mathbb{C}$且基场为$g(C)\geq 2$时，证明了$X$上存在非平凡稳定的Higgs丛.

英文摘要：

Let $π: X = \mathbb{P}_C(E) \longrightarrow C$ be a ruled surface over an algebraically closed field $k$ of characteristic 0, with a fixed polarization $L$ on $X$. In this paper, we show that pullback of a (semi)stable Higgs bundle on $C$ under $π$ is a $L$-(semi)stable Higgs bundle. Conversely, if $(V,θ)$ is a $L$-(semi)stable Higgs bundle on $X$ with $c_1(V)= π^*(\bf d \rm)$ for some divisor $\bf d \rm $ of degree $d$ on $C$ and $c_2(V)=0$, then there exists a (semi)stable Higgs bundle $(W,ψ)$ of degree $d$ on $C$ whose pullback under $π$ is isomorphic to $(V,θ)$. As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on $X$ whenever $g(C)\geq 2$ and the base field is $\mathbb{C}$.

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