#### Volume 23, issue 6 (2019)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Rationality, universal generation and the integral Hodge conjecture

### Mingmin Shen

Geometry & Topology 23 (2019) 2861–2898
##### 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 dimensions $3$ and $5$, we relate stable rationality with the geometry of the associated intermediate Jacobian.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/gt

We have not been able to recognize your IP address 18.204.56.185 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.

##### Keywords
algebraic cycles, Hodge conjecture, cubic threefold, cubic fourfold
##### Mathematical Subject Classification 2010
Primary: 14C25, 14C30, 14E08