A family of special cubic fourfolds with motive of abelian type

TL;DR

This paper demonstrates the existence of one-dimensional families of special cubic fourfolds whose Chow motives are of abelian type and finite dimensional, with implications for related Hyperkähler varieties.

Contribution

It establishes the abelian type and finite dimensionality of motives for specific families of cubic fourfolds and related Hyperkähler varieties, expanding understanding of their motive structures.

Findings

Existence of 1D families of cubic fourfolds with abelian motives

Implication of abelianity for Fano varieties of lines

Finite dimensionality of motives for related Hyperkähler varieties

Abstract

In this short note, we show that there exist one dimensional families of cubic fourfolds with Chow motive of abelian type and finite dimensional inside every Hassett divisor of special cubic fourfolds. This also implies abelianity and finite dimensionality of the motive of related Hyperk\"ahler varieties, such as the Fano variety of lines and the LLSvS 8fold.