Some new Fano varieties with a multiplicative Chow-K\"unneth decomposition

TL;DR

This paper proves that certain Fano varieties, formed as intersections of Grassmannians with hyperplanes, have a special Chow-K"unneth decomposition, leading to injectivity results for their Chow rings.

Contribution

It establishes the existence of a multiplicative Chow-K"unneth decomposition for these Fano varieties, a new result in the study of algebraic cycles.

Findings

Existence of multiplicative Chow-K"unneth decomposition for Y

Injectivity of a tautological subring into cohomology

Advancement in understanding Chow rings of Fano varieties

Abstract

Let $Y$ be a smooth dimensionally transverse intersection of the Grassmannian $\hbox{Gr}(2,n)$ with 3 Pl\"ucker hyperplanes. We show that $Y$ admits 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.