关于二阶实局部系的非阿贝尔Noether-Lefschetz轨迹的代数性
On the algebraicity of non-abelian Noether-Lefschetz loci of rank-two real local systems
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中文总结 AI 辅助
本文在曲线模空间上研究非阿贝尔Noether-Lefschetz轨迹的代数性，证明其等价于表示像的离散性，并通过狭缝手术构造两个显式有理族，其中离散单值族回答了Baldi-Lam的问题。
中文摘要 AI 辅助
作为Hodge轨迹的非阿贝尔类比，Simpson引入了非阿贝尔Noether-Lefschetz轨迹，并猜想其对$\mathbb Z$PVHS具有代数性。本文在曲线模空间$\mathcal M_g$上研究此问题。对于容许权一$\mathbb R$PVHS的非酉表示$\rho:\pi_1(\Sigma_g)\longrightarrow \mathrm{SL}_2(\mathbb R)$，我们证明了正维非阿贝尔Noether-Lefschetz分量的代数性等价于$\operatorname{im}\rho$的离散性，且此时该分量恰为一个带标记的固定靶orbifold Hurwitz分量。作为应用，我们通过狭缝手术构造了两个显式有理族：一个具有非离散单值且Noether-Lefschetz像非代数，另一个具有离散单值但其周期映射仍非均匀化。后者对Baldi-Lam关于$\mathbb Q$PVHS的问题给出了肯定回答。
英文摘要
As a non-abelian analogue of the Hodge locus, Simpson introduced the non-abelian Noether--Lefschetz locus and conjectured its algebraicity for $\mathbb Z$PVHS. In this paper, we study this question on the moduli space of curves $\mathcal M_g$. For a non-unitary representation \[ ρ:π_1(Σ_g)\longrightarrow \mathrm{SL}_2(\mathbb R) \] which admits a $\mathbb R$PVHS of weight one, we prove that algebraicity of a positive-dimensional non-abelian Noether--Lefschetz component is equivalent to discreteness of $\operatorname{im}ρ$, and in this case the component is precisely a marked fixed-target orbifold Hurwitz component. As applications, we construct two explicit rational families by slit surgery: one with non-discrete monodromy and non-algebraic Noether--Lefschetz image, and another with discrete monodromy whose period map is nevertheless non-uniformizing. The latter gives an affirmative answer to a question of Baldi--Lam concerning $\mathbb Q$PVHS.
发表机构
- Wuhan University（武汉大学）
- Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）
- Johannes Gutenberg-Universität Mainz（美因茨约翰内斯古腾堡大学）
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