On the algebraicity of non-abelian Noether-Lefschetz loci of rank-two real local systems

Abstract

As a non-abelian analogue of the Hodge locus, Simpson introduced the non-abelian Noether--Lefschetz locus and conjectured its algebraicity for ℤPVHS. In this paper, we study this question on the moduli space of curves ℳ_g. For a non-unitary representation ρ:π₁(Σ_g)⟶ SL₂(ℝ) which admits a ℝPVHS of weight one, we prove that algebraicity of a positive-dimensional non-abelian Noether--Lefschetz component is equivalent to discreteness of imρ, 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 ℚPVHS.

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BibTeX RIS

BibTeX

@misc{hu2026algebraicity,
  title = {{On the algebraicity of non-abelian Noether-Lefschetz loci of rank-two real local systems}},
  author = {Tianzhi Hu and Kang Zuo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33537},
  eprint = {2609.33537},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33537},
}

RIS

TY  - UNPB
TI  - On the algebraicity of non-abelian Noether-Lefschetz loci of rank-two real local systems
AU  - Tianzhi Hu
AU  - Kang Zuo
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33537
UR  - https://arxiv.org/abs/2609.33537
AB  - As a non-abelian analogue of the Hodge locus, Simpson introduced the non-abelian Noether--Lefschetz locus and conjectured its algebraicity for ℤPVHS. In this paper, we study this question on the moduli space of curves ℳ_g. For a non-unitary representation ρ:π_1(Σ_g)⟶ SL_2(ℝ) which admits a ℝPVHS of weight one, we prove that algebraicity of a positive-dimensional non-abelian Noether--Lefschetz component is equivalent to discreteness of imρ, 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 ℚPVHS.
ER  -

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