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 |
References |
Additional Information
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.
References
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References
- S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), no. 5, 1235-x-1253. MR 714776 (85i:14002), DOI https://doi.org/10.2307/2374341
- J.-L. Colliot-Thélène and B. Kahn, Cycles de codimension $2$ et $H^3$ non ramifié pour les variétés sur les corps finis, preprint, arXiv:1104.3350v2.
- Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Christophe Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), no. 3, 763–801 (French). MR 714830 (85d:14010), DOI https://doi.org/10.1215/S0012-7094-83-05038-X
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- Sergey Gorchinskiy and Vladimir Guletskiĭ, Motives and representability of algebraic cycles on threefolds over a field, J. Algebraic Geom. 21 (2012), no. 2, 347–373. MR 2877438, DOI https://doi.org/10.1090/S1056-3911-2011-00548-1
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- B. Brent Gordon, Masaki Hanamura, and Jacob P. Murre, Absolute Chow–Künneth projectors for modular varieties, J. Reine Angew. Math. 580 (2005), 139–155. MR 2130589 (2006a:14006), DOI https://doi.org/10.1515/crll.2005.2005.580.139
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- János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180 (98c:14001)
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- J. P. Murre, On the motive of an algebraic surface, J. Reine Angew. Math. 409 (1990), 190–204. MR 1061525 (91g:14003), DOI https://doi.org/10.1515/crll.1990.409.190
- J. P. Murre, On a conjectural filtration on the Chow groups of an algebraic variety. I. The general conjectures and some examples, Indag. Math. (N.S.) 4 (1993), no. 2, 177–188. MR 1225267 (94j:14006a), DOI https://doi.org/10.1016/0019-3577%2893%2990038-Z
- A. J. Scholl, Classical motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 163–187. MR 1265529 (95b:11060)
- Charles Vial, Niveau and coniveau filtrations on cohomology groups and Chow groups, Proc. Lond. Math. Soc. 106 (2013), no. 2, 1191–1195. MR 3021467
- Charles Vial, Projectors on the intermediate algebraic Jacobians, New York J. Math. 19 (2013), 793–822.
- Charles Vial, Pure motives with representable Chow groups, C. R. Math. Acad. Sci. Paris 348 (2010), no. 21-22, 1191–1195 (English, with English and French summaries). MR 2738925 (2012c:14010), DOI https://doi.org/10.1016/j.crma.2010.10.017
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329–353. MR 715656 (85b:14041)
- Claire Voisin, Sur les zéro-cycles de certaines hypersurfaces munies d’un automorphisme, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 4, 473–492 (French). MR 1205880 (93m:14005)
Additional Information
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.