The motive of the Fano surface of lines

TL;DR

This paper proves that the Chow motive of the Fano surface of lines on a smooth cubic threefold is finite-dimensional, providing an example of a variety with an Abelian type motive not dominated by a product of curves.

Contribution

It establishes the finite-dimensionality of the Chow motive for a specific class of surfaces, expanding understanding of motives beyond dominated varieties.

Findings

Chow motive of the Fano surface is finite-dimensional

The Fano surface's motive is of Abelian type

Provides an example of a non-dominated variety with finite-dimensional motive

Abstract

The purpose of this note is to prove that the Chow motive of the Fano surface of lines on the smooth cubic threefold is finite-dimensional in the sense of Kimura. This gives an example of a smooth projective variety that is not dominated by a product of curves but whose Chow motive is of Abelian type.