De Rham-Betti classes with coefficients

TL;DR

This paper investigates the nature of de Rham-Betti classes with coefficients on algebraic varieties, proving they are algebraic or motivated under certain conditions, and establishes related Torelli theorems.

Contribution

It proves that certain de Rham-Betti classes are algebraic or motivated, extending the understanding of their structure on products of elliptic curves and hyper-K"ahler varieties.

Findings

De Rham-Betti classes on products of elliptic curves are algebraic if L contains at most one CM field.

Codimension-2 classes on hyper-K"ahler varieties are motivated cycles.

A global Torelli theorem for K3 surfaces over algebraic numbers is established.

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 Andr\'e regarding the Kuga-Satake correspondence, we show that codimension-2 $L$-de Rham-Betti classes on hyper-K\"ahler 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.