A remark on the motive of the Fano variety of lines of a cubic

TL;DR

This paper explores the relationship between the motives of a smooth cubic hypersurface and its Fano variety of lines, showing finite-dimensionality transfers from the hypersurface to the Fano variety in certain cases.

Contribution

It establishes a new relation between the Chow motives of cubic hypersurfaces and their Fano varieties, proving finite-dimensionality in specific dimensions.

Findings

Finite-dimensional motive of $X$ implies finite-dimensional motive of $F$.

Finite-dimensionality proven for Fano varieties of cubics in dimensions 3 and 5.

Results extend to certain cubics in other dimensions.

Abstract

Let $X$ be a smooth cubic hypersurface, and let $F$ be the Fano variety of lines on $X$. We establish a relation between the Chow motives of $X$ and $F$. This relation implies in particular that if $X$ has finite-dimensional motive (in the sense of Kimura), then $F$ also has finite-dimensional motive. This proves finite-dimensionality for motives of Fano varieties of cubics of dimension $3$ and $5$, and of certain cubics in other dimensions.