De Rham-Betti classes with coefficients

TL;DR

This work develops a systematic framework for de Rham--Betti cohomology with coefficients and investigates their algebraicity by extending Grothendieck's period conjecture to $L$-coefficients. It establishes that $L$-de Rham--Betti classes on products of elliptic curves are $L$-algebraic under mild CM-field restrictions and proves analogous motivic-status results for codimension-2 classes on known-deformation-type hyper-Kähler varieties via the Kuga–Satake correspondence. The authors extend Wüstholz's analytic subgroup theorem to the coefficient setting and derive comprehensive descent, base-change, and motivic-torsor properties within André motives with coefficients. As a consequence, they obtain a global de Rham--Betti Torelli theorem for K3 surfaces over $ar{oldsymbol{Q}}$ and show that many de Rham--Betti classes on hyper-Kähler varieties are motivated or algebraic, especially in cases of large Picard rank. Overall, the results advance the understanding of period conjectures with coefficients, connect CM Shimura-period relations to de Rham--Betti classes, and illuminate the interplay between motives, Hodge theory, and hyper-Kähler geometry.

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.

Theorems & Definitions (108)

• Theorem 1: Corollary \ref{['cor:LdRB-abelian']}
• Theorem 2: Theorem \ref{['T:dRB-elliptic']}
• Theorem 3: special instance of Theorem \ref{['P:GdRBisometry']}
• Theorem 4: Global de Rham--Betti Torelli theorem for K3 surfaces over $\overline \mathds{Q}$ ; see Theorem \ref{['T2:dRB-Torelli']}
• Theorem 5: Proposition \ref{['P:dRB-tensor2']} and Theorem \ref{['T:dRB-cod2']}
• Theorem 6
• Definition 2.1: De Rham--Betti objects
• Definition 2.2: Base-change of de Rham--Betti objects
• Definition 2.3: $L'$-de Rham--Betti classes
• Definition 2.4: The de Rham--Betti group and the de Rham--Betti torsor of periods