Rigidity of the period map up to finite covers
Abstract
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]∈ H1(Sg;ℤ/3ℤ)*, and let S̃→ Sg be the corresponding triple cover with deck transformation σ. For h≤ g, every non-abelian homomorphism from either Mod(Sg,[β]), the stabilizer of [β] in Mod(Sg), or Mod(S̃,σ), the centralizer of σ in Mod(S̃), to Sp2h(ℤ) is, up to conjugation, the standard symplectic representation on H1(Sg;ℤ). As an application, we obtain a rigidity theorem for holomorphic maps from the moduli space Rg(3) of genus-g curves equipped with a $3$-sheeted (unbranched) normal covering to the moduli space 𝒜h of h-dimensional principally polarized abelian varieties. We prove that, for g≥ 6 and h≤ g, the unique nonconstant holomorphic map from Rg(3), equipped with either of its two natural complex-orbifold structures, to 𝒜h is the period map sending a cover Y→ X to the Jacobian of the base curve X.
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BibTeX
@misc{zhong2026rigidity,
title = {{Rigidity of the period map up to finite covers}},
author = {Xiyan Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29351},
eprint = {2608.29351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29351},
}
RIS
TY - UNPB
TI - Rigidity of the period map up to finite covers
AU - Xiyan Zhong
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.29351
UR - https://arxiv.org/abs/2608.29351
AB - 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]∈ H_1(S_g;ℤ/3ℤ)^*, and let S̃→ S_g be the corresponding triple cover with deck transformation σ. For h≤ g, every non-abelian homomorphism from either Mod(S_g,[β]), the stabilizer of [β] in Mod(S_g), or Mod(S̃,σ), the centralizer of σ in Mod(S̃), to Sp_2h(ℤ) is, up to conjugation, the standard symplectic representation on H_1(S_g;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 𝒜_h of h-dimensional principally polarized abelian varieties. We prove that, for g≥ 6 and h≤ g, the unique nonconstant holomorphic map from R_g^(3), equipped with either of its two natural complex-orbifold structures, to 𝒜_h is the period map sending a cover Y→ X to the Jacobian of the base curve X.
ER -