Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 1

Special Issue: In Honor of Eduard Looijenga, Part 2 of 3

Guest Editor: Gerard van der Geer

Birational Chow–Künneth decompositions

Pages: 105 – 140

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n1.a4


Mingmin Shen (KdV Institute for Mathematics, University of Amsterdam, The Netherlands)


We study the notion of a birational Chow–Künneth decomposition, which is essentially a decomposition of the integral birational motive of a variety. The existence of a birational Chow–Künneth decomposition is a stably birational invariant. We show that a birational Chow–Künneth decompostion exists for the following varieties: (a) Jacobian variety; (b) Hilbert scheme of points on a $K3$ surface and (c) The variety of lines on a stably rational cubic threefold or a stably rational cubic fourfold.


birational motive, Jacobian variety, hyperkähler variety, cubic threefold, cubic fourfold

2010 Mathematics Subject Classification

14C25, 14E08, 14H40

Published 15 February 2017