The generalized André–Oort conjecture for mixed Shimura varieties

A mixed Shimura variety is an algebraic variety arising from a mixed Shimura datum; its special points and special subvarieties are the distinguished points and subvarieties defined by the Shimura structure. Let $Y$ be a closed irreducible subvariety of a mixed Shimura variety. Generalized André–Oort conjecture. If $Y$ contains a Zariski-dense set of special points, then it is special. This is the mixed-Shimura analogue of the André–Oort conjecture, characterizing special subvarieties through the density of their special points. It was proved under the generalized Riemann hypothesis by Klingler, Ullmo, and Yafaev, using equidistribution results of Clozel and Ullmo.