Budur–Wang special varieties package conjecture for absolute sets

Let $X$ be a smooth complex algebraic variety and let $r>0$ be an integer. Write $M_B(X,r)$ for the Betti moduli space of semisimple complex rank-$r$ local systems on $X$, and call a subset absolute $\overline{\mathbb{Q}}$-constructible or absolute $\overline{\mathbb{Q}}$-closed when it has the corresponding property under every $\sigma\in\operatorname{Gal}(\mathbb{C}/\mathbb{Q})$. An absolute $\overline{\mathbb{Q}}$-point is a local system of geometric origin, and $\overline{\mathbb{Q}}$-pseudo-isomorphisms are as defined for these moduli spaces.

Budur–Wang's special varieties package conjecture. The following conditions hold in $M_B(X,r)$: (A1) absolute $\overline{\mathbb{Q}}$-constructible subsets are generated from absolute $\overline{\mathbb{Q}}$-closed subsets by finite unions, intersections, and complements; their Euclidean, equivalently Zariski, closures are absolute $\overline{\mathbb{Q}}$-closed; and irreducible components of absolute $\overline{\mathbb{Q}}$-closed subsets are absolute $\overline{\mathbb{Q}}$-closed. (A2) Every nonempty absolute $\overline{\mathbb{Q}}$-constructible subset contains a Zariski-dense subset of absolute $\overline{\mathbb{Q}}$-points. (A3) A point is absolute $\overline{\mathbb{Q}}$ if and only if it is a local system of geometric origin. (A4) The Zariski closure of a set of absolute $\overline{\mathbb{Q}}$-points is $\overline{\mathbb{Q}}$-pseudo-isomorphic to an absolute $\overline{\mathbb{Q}}$-constructible subset of some $M_B(X',r')$, for a possibly different smooth complex algebraic variety $X'$ and rank $r'$, with the pseudo-isomorphism restricting to a bijection on absolute $\overline{\mathbb{Q}}$-points.

These conditions generalize the special-subvariety philosophy for Shimura varieties and seek to characterize local systems of geometric origin. The paper states that the conjecture is known in rank one and for complex affine tori or abelian varieties, while A1 is known when $X$ is projective; the general case remains open.