Algebraic cycles and intersections of 2 quadrics

TL;DR

This paper proves that smooth intersections of two quadrics in projective space have a special Chow-K"unneth decomposition, ensuring certain algebraic cycles behave well with respect to cohomology.

Contribution

It establishes the existence of a multiplicative Chow-K"unneth decomposition for these varieties, linking algebraic cycles to their cohomological properties.

Findings

Varieties have Hodge level 1

Existence of multiplicative Chow-K"unneth decomposition

Tautological subring injects into cohomology

Abstract

A smooth intersection $Y$ of two quadrics in $\mathbb{P}^{2g+1}$ has Hodge level 1. We show that such varieties $Y$ 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.