The transcendental motive of a cubic fourfold

TL;DR

This paper introduces and studies the transcendental motive of a cubic fourfold, establishing its relations to Fano varieties, Prym motives, and K3 surfaces, with implications for rationality conjectures.

Contribution

It defines the transcendental motive of cubic fourfolds and links it to Fano varieties, Prym motives, and K3 surfaces, providing new insights into their motive structures.

Findings

The transcendental motive of a cubic fourfold is isomorphic to that of its Fano variety.

For special cubic fourfolds, the motive relates to the transcendental motive of a K3 surface.

The motive is finite dimensional if and only if the associated K3 surface has finite dimensional motive.

Abstract

In this note we introduce the transcendental part $t(X)$ of the motive of a cubic fourfold $X$ and prove that it is isomorphic to the (twisted) transcendental part $h_2^{tr}(F(X))$ in a suitable Chow-K\"unneth decomposition for the motive of the Fano variety of lines $F(X)$. Then we prove that $t(X)$ is isomorphic to the Prym motive associated to the surface $S_l \subset F(X)$ of lines meeting a general line $l$. If $X$ is a special cubic fourfold in the sense of Hodge theory, and $F(X)\cong S^{[2]}$, with $S$ a $K3$, then we show that $t(X) \cong t_2(S)(1)$, where $t_2(S)$ is the transcendental motive. Therefore the motive $h(X)$ is finite dimensional if and only if $S$ has a finite dimensional motive. If $X$ is very general then $t(X)$ cannot be isomorphic to the (twisted) transcendental motive of a surface. We relate the existence of an isomorphism $t(X) \cong t_2(S)(1)$ to conjectures by Hassett and Kuznetsov on the rationality of a special cubic fourfold. Finally we consider the case of cubic fourfolds X admitting a fibration over $\mathbf{P}^2$, whose fibers are either quadrics or del Pezzo surfaces of degree 6, and prove the isomorphism $t_2(S)(1) \cong t(X)$, with $S$ a K3 surface.