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 $\mathbb Z$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.