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
Email: c.vial@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/S1056-3911-2014-00616-0
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.

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