K3 surfaces and cubic fourfolds with Abelian motive

TL;DR

This paper proves that certain cubic fourfolds and related hyperKahler varieties have abelian and finite dimensional Chow motives, revealing deep connections between their geometric structures and motives.

Contribution

It establishes the abelian and finite dimensionality of motives for specific classes of cubic fourfolds and hyperKahler varieties, extending the understanding of their motive structures.

Findings

Cubic fourfolds with high-rank algebraic 2-cycle lattice have abelian motives.

HyperKahler varieties related to these fourfolds also have abelian and finite dimensional motives.

Constructs families of Fano varieties with finite dimensional Chow motives.

Abstract

We show that cubic fourfolds with lattice of algebraic 2-cycles of rank greater than 19 have abelian and finite dimensional (in the sense of Kimura) Chow motive. This also implies Abelianity and finite dimensionality of the motive of related hyperKahler varieties, such as the Fano variety of lines and the LLSvS 8fold. A similar remark allows us to show the Abelianity of the motive of an infinity of LSV 10folds, and of other hyperKahler 10folds associated to the twisted intermediate Jacobian fibration of cubic fourfolds with an associated K3 surface. After that, starting from certain 4-dimensional families of K3 surfaces, we construct two families of Fano varieties whose Chow motive is finite dimensional. Varieties from the first family are some quadric surface fibrations, and contain the finite dimensional transcendental motive of a K3 surface. Varieties from the second family are singular cubic fourfolds, and their motives are Schur-finite and Abelian in Voevodsky's triangulated category of motives.