Chow motives of genus one fibrations

TL;DR

This paper proves that the Chow motives of a genus 1 fibration and its Jacobian are isomorphic, and applies this to establish Kimura finite-dimensionality for certain surfaces, extending previous results to arbitrary characteristic.

Contribution

It establishes the isomorphism of Chow motives for genus 1 fibrations and their Jacobians, and extends Kimura finite-dimensionality results to broader classes of surfaces.

Findings

Chow motives of genus 1 fibrations and their Jacobians are isomorphic.

Kimura finite-dimensionality holds for certain non-general type surfaces with geometric genus 0.

Generalizes Bloch-Kas-Lieberman's result to arbitrary characteristic.

Abstract

Let $f: X \rightarrow C$ be a genus 1 fibration from a smooth projective surface, i.e. its generic fiber is a regular genus 1 curve. Let $j: J \rightarrow C$ be the Jacobian fibration of $f$. In this paper, we prove that the Chow motives of $X$ and $J$ are isomorphic. As an application, combined with our concomitant work on motives of quasi-elliptic fibrations, we prove Kimura finite-dimensionality for smooth projective surfaces not of general type with geometric genus 0. This generalizes Bloch-Kas-Lieberman's result to arbitrary characteristic.