The bialgebraic characterization of dR-absolutely special subvarieties

Let $S$ be defined over $\bar{\mathbb{Q}}$ and let $(\mathbb{V}^{\sigma})_{\sigma}$ be a $\bar{\mathbb{Q}}$-absolute variation on $S$. A closed irreducible subvariety is $\bar{\mathbb{Q}}$-bialgebraic if it is defined over $\bar{\mathbb{Q}}$ and an analytic irreducible component of its inverse image under the universal-cover map is an analytic irreducible component of the inverse image of a $\bar{\mathbb{Q}}$-subvariety of the compact dual. A subvariety is dR-absolutely special when it is maximal with respect to its generic dR-absolute Mumford–Tate group. The bialgebraic characterization conjecture. The $\bar{\mathbb{Q}}$-bialgebraic subvarieties of $S$ are exactly the dR-absolutely special subvarieties. The paper proves that every dR-absolutely special subvariety is $\bar{\mathbb{Q}}$-bialgebraic; the converse is conjectural.