  A de Rham-Betti class on a smooth projective variety $X$ over an algebraic
extension $K$ of the rational numbers is a rational class in the Betti
cohomology of the analytification of$X$ that descends to a class in the
algebraic de Rham cohomology of $X$ via the period comparison isomorphism. The
period conjecture of Grothendieck implies that de Rham-Betti classes should be
algebraic. We prove that any de Rham-Betti class on a product of elliptic
curves is algebraic. This is achieved by showing that the Tannakian torsor
associated to a de Rham-Betti object is connected, and by exploiting the
analytic subgroup theorem of W\"ustholz. In the case of products of non-CM
elliptic curves, we prove the stronger result that $\overline{\mathds{Q}}$-de
Rham-Betti classes are $\overline \mathds{Q}$-linear combinations of algebraic
classes by showing that the period comparison isomorphism generates the torsor
of motivic periods. A key step consists in establishing a version of the
analytic subgroup theorem with $\overline \mathds{Q}$-coefficients. Finally,
building on results of Deligne and Andr\'e regarding the Kuga-Satake
correspondence, we further show that any de Rham-Betti isometry between the
second cohomology groups of hyper-K\"ahler varieties, with second Betti number
not 3, is Hodge. As two applications we show that codimension-2 de Rham-Betti
classes on hyper-K\"ahler varieties of known deformation type are Hodge and we
obtain a global de Rham-Betti Torelli theorem for K3 surfaces over $\overline
\mathds{Q}$.
