Algebraic cycles and Fano threefolds of genus 8

TL;DR

This paper proves that prime Fano threefolds of genus 8 possess a special Chow-K"unneth decomposition, leading to injectivity results for certain subrings of their Chow rings into cohomology.

Contribution

It establishes the existence of a multiplicative Chow-K"unneth decomposition for prime Fano threefolds of genus 8, a new structural property in algebraic geometry.

Findings

Prime Fano threefolds of genus 8 have a multiplicative Chow-K"unneth decomposition.

A tautological subring of the Chow ring injects into cohomology.

The result advances understanding of the Chow ring structure of Fano threefolds.

Abstract

We show that prime Fano threefolds $Y$ of genus 8 have a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.