De Rham-Betti classes with coefficients

Abstract

Let $K$ and $L$ be algebraic extensions of the rational numbers inside the field of complex numbers. An $L$-de Rham-Betti class on a smooth projective variety $X$ over $K$ is a class in the Betti cohomology with $L$-coefficients 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 $L$-de Rham-Betti classes should be $L$-linear combinations of algebraic cycle classes. We prove that $L$-de Rham-Betti classes on products of elliptic curves are $L$-linear combinations of algebraic classes, provided $L$ contains at most one of the CM fields associated with the CM elliptic curves involved in the product. A key step consists in establishing a version of the analytic subgroup theorem with $L$-coefficients. Moreover, building on results of Deligne and André regarding the Kuga-Satake correspondence, we show that codimension-2 $L$-de Rham-Betti classes on hyper-Kähler varieties of known deformation type are $L$-linear combinations of motivated cycles, and we obtain a global de Rham-Betti Torelli theorem for K3 surfaces defined over the algebraic numbers.