On the Chow ring of Fano varieties on the Fatighenti-Mongardi list

TL;DR

This paper proves that many Fano varieties of K3 type constructed by Fatighenti-Mongardi admit a multiplicative Chow-K"unneth decomposition, impacting the understanding of their Chow rings.

Contribution

It establishes the existence of a multiplicative Chow-K"unneth decomposition for numerous Fano varieties of K3 type from the Fatighenti-Mongardi list, confirming a conjecture.

Findings

Many Fano varieties of K3 type admit a multiplicative Chow-K"unneth decomposition.

This result has significant implications for the structure of their Chow rings.

The work confirms a conjecture for a broad class of Fano varieties.

Abstract

Conjecturally, Fano varieties of K3 type admit a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. We prove this for many of the families of Fano varieties of K3 type constructed by Fatighenti-Mongardi. This has interesting consequences for the Chow ring of these varieties.