{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33537",
  "author_total": 2,
  "authors": [
    {
      "name": "Tianzhi Hu"
    },
    {
      "name": "Kang Zuo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vrgjv97mfzbh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33537",
  "fulltext_url": null,
  "id": "2vrgjv97mfzbh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33537",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214729314",
  "pdf_url": "https://arxiv.org/pdf/2609.33537",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vrgjv97mfzbh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33537",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33537"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the algebraicity of non-abelian Noether-Lefschetz loci of rank-two real local systems",
  "title_text": "On the algebraicity of non-abelian Noether-Lefschetz loci of rank-two real local systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/2vrgjv97mfzbh",
  "venue": "arXiv",
  "versions": []
}