有限覆盖下周期映射的刚性
Rigidity of the period map up to finite covers
AI总结：
该研究对映射类群的双仿射表示、低维辛表示分类，得到R_g^{(3)}到A_h的周期映射的刚性定理，确定了唯一非常全纯映射的形式。
AI中文摘要：
我们首先对具有有限多个边界分支或穿孔的曲面的映射类群的双仿射表示进行完整分类。我们还证明，亏格为g、带有两个边界分支的曲面的映射类群的每个维数不超过3g-3的线性表示都是双仿射的。然后我们对与三重覆盖相关的映射类群的低维辛表示进行分类。设[β]∈H₁(S_g;ℤ/3ℤ)∗，且S̃→S_g是对应的三重覆盖，带有甲板变换σ。对于h≤g，从Mod(S_g,[β])（即Mod(S_g)中[β]的稳定子群）或Mod(S̃,σ)（即Mod(S̃)中σ的中心化子）到Sp_{2h}(ℤ)的每个非阿贝尔同态，在共轭意义下，都是H₁(S_g;ℤ)上的标准辛表示。作为应用，我们得到了从亏格为g、带有3叶（无分支）正则覆盖的曲线的模空间R_g^{(3)}到h维主极化阿贝尔簇的模空间A_h的全纯映射的刚性定理。我们证明，当g≥6且h≤g时，配备其两种自然复-orbifold结构之一的R_g^{(3)}到A_h的唯一非常全纯映射，是将覆盖Y→X映射到基曲线X的雅可比的周期映射。
英文摘要：
We first give a complete classification of bi-affine representations of mapping class groups of surfaces with finitely many boundary components or punctures. We also show that every linear representation of the mapping class group of a genus-$g$ surface with two boundary components of dimension at most $3g-3$ is bi-affine. We then classify low-dimensional symplectic representations of the mapping class group associated to triple covers. Let $[β]\in H_1(S_g;\mathbb{Z}/3\mathbb{Z})^*$, and let $\widetilde{S}\to S_g$ be the corresponding triple cover with deck transformation $σ$. For $h\le g$, every non-abelian homomorphism from either $\mathrm{Mod}(S_g,[β])$, the stabilizer of $[β]$ in $\mathrm{Mod}(S_g)$, or $\mathrm{Mod}(\widetilde{S},σ)$, the centralizer of $σ$ in $\mathrm{Mod}(\widetilde{S})$, to $\mathrm{Sp}_{2h}(\mathbb{Z})$ is, up to conjugation, the standard symplectic representation on $H_1(S_g;\mathbb{Z})$. As an application, we obtain a rigidity theorem for holomorphic maps from the moduli space $R_g^{(3)}$ of genus-$g$ curves equipped with a $3$-sheeted (unbranched) normal covering to the moduli space $\mathcal{A}_h$ of $h$-dimensional principally polarized abelian varieties. We prove that, for $g\ge 6$ and $h\le g$, the unique nonconstant holomorphic map from $R_g^{(3)}$, equipped with either of its two natural complex-orbifold structures, to $\mathcal{A}_h$ is the period map sending a cover $Y\to X$ to the Jacobian of the base curve $X$.
