{
  "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 $[β]\\in H_1(S_g;\\mathbb{Z}/3\\mathbb{Z})^*$, 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 $X$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29351",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiyan Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8pcgaphv6bp8k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29351",
  "fulltext_url": null,
  "id": "8pcgaphv6bp8k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29351",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7204907993",
  "pdf_url": "https://arxiv.org/pdf/2608.29351",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8pcgaphv6bp8k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2608.29351",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29351"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Rigidity of the period map up to finite covers",
  "title_text": "Rigidity of the period map up to finite covers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/8pcgaphv6bp8k",
  "venue": "arXiv",
  "versions": []
}