Volume 23, issue 6 (2019)

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Rationality, universal generation and the integral Hodge conjecture

Mingmin Shen

Geometry & Topology 23 (2019) 2861–2898

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.

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algebraic cycles, Hodge conjecture, cubic threefold, cubic fourfold
Mathematical Subject Classification 2010
Primary: 14C25, 14C30, 14E08
Received: 12 December 2016
Revised: 24 January 2019
Accepted: 26 February 2019
Published: 1 December 2019
Proposed: Dan Abramovich
Seconded: Richard P Thomas, Jim Bryan
Mingmin Shen
KdV Institute for Mathematics
University of Amsterdam