Piatetski–Shapiro–Shafarevich CM varieties conjecture

Establish that every complex algebraic variety of CM type—equivalently, a variety whose Mumford–Tate group is commutative—admits a model over a number field.

Background

The paper develops complex multiplication theory for cubic fourfolds and proves that any cubic fourfold of CM type is defined over an abelian extension of its reflex field, strengthening known results in this specific case.

In the broader context, Piatetski–Shapiro and Shafarevich formulated a conjecture asserting that varieties of CM type are defined over number fields. The authors quote this conjecture to situate their results for cubic fourfolds within the general CM framework.