Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Chow–Künneth decomposition for $3$- and $4$-folds fibred by varieties with trivial Chow group of zero-cycles


Author: Charles Vial
Journal: J. Algebraic Geom. 24 (2015), 51-80
DOI: https://doi.org/10.1090/S1056-3911-2014-00616-0
Published electronically: January 27, 2014
MathSciNet review: 3275654
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Abstract: Let $k$ be a field, and let $\Omega$ be a universal domain over $k$. Let $f:X \rightarrow S$ be a dominant morphism defined over $k$ from a smooth projective variety $X$ to a smooth projective variety $S$ of dimension $\leq 2$ such that the general fibre of $f_\Omega$ has trivial Chow group of zero-cycles. For example, $X$ could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that $X$ has dimension $\leq 4$. Then we prove that $X$ has a self-dual Murre decomposition, i.e., that $X$ has a self-dual Chow-Künneth decomposition which satisfies Murre’s conjectures (B) and (D). Moreover, we prove that the motivic Lefschetz conjecture holds for $X$ and hence so does the Lefschetz standard conjecture. We also give new examples of $3$-folds of general type which are Kimura finite dimensional, new examples of $4$-folds of general type having a self-dual Murre decomposition, as well as new examples of varieties with finite degree three unramified cohomology.


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Charles Vial
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
MR Author ID: 867800
Email: c.vial@dpmms.cam.ac.uk

Received by editor(s): September 29, 2011
Received by editor(s) in revised form: March 23, 2012
Published electronically: January 27, 2014
Additional Notes: This work was supported by a Nevile Research Fellowship at Magdalene College, Cambridge, and an EPSRC Postdoctoral Fellowship under grant EP/H028870/1.
Article copyright: © Copyright 2014 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.