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This article is available for purchase or by subscription. See below.
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.
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Keywords
algebraic cycles, Hodge conjecture, cubic threefold, cubic
fourfold
Mathematical Subject Classification 2010
Primary: 14C25, 14C30, 14E08
Publication
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