On the Chow Ring of Fano Fourfolds of K3 type

TL;DR

This paper proves that many Fano fourfolds of K3 type have a Chow ring structure similar to K3 surfaces, using a multiplicative Chow-Künneth decomposition, and explores implications for the Franchetta property.

Contribution

It establishes the existence of a multiplicative Chow-Künneth decomposition for a broad class of Fano fourfolds of K3 type, linking their Chow ring structure to that of K3 surfaces.

Findings

Fano fourfolds of K3 type have a multiplicative Chow-Künneth decomposition.

The Chow ring of these Fano varieties behaves like that of K3 surfaces.

Provides criteria for the Franchetta property in blown-up projective varieties.

Abstract

We show that a wide range of Fano varieties of K3 type, recently constructed by Bernardara, Fatighenti, Manivel and Tanturri, have a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. It follows that the Chow ring of these Fano varieties behaves like that of K3 surfaces. As a side result, we obtain some criteria for the Franchetta property of blown-up projective varieties.