Rationality, universal generation and the integral Hodge conjecture

TL;DR

This paper explores the connection between rationality and the integral Hodge conjecture using algebraic cycles, with applications to cubic hypersurfaces and their associated geometric structures.

Contribution

It establishes universal generation of 1-cycles on cubic hypersurfaces and links stable rationality to algebraic properties of Hodge classes in specific dimensions.

Findings

Chow group of 1-cycles on cubic hypersurfaces is universally generated by lines.

If a generic cubic fourfold is stably rational, certain Hodge classes are algebraic.

Stable rationality relates to the geometry of intermediate Jacobians in dimensions 3 and 5.

Abstract

We use the universal generation of algebraic cycles to relate (stable) rationality to the integral Hodge conjecture. We show that the Chow group of 1-cycles on a cubic hypersurface is universally generated by lines. Applications are mainly in cubic hypersurfaces of low dimensions. For example, we show that if a generic cubic fourfold is stably rational then the Beauville--Bogomolov form on its variety of lines, viewed as an integral Hodge class on the self product of its variety of lines, is algebraic. In dimension $3$ and $5$, we relate stable rationality with the geometry of the associated intermediate Jacobian.