Motives and representability of algebraic cycles on threefolds over a field

Authors:
Sergey Gorchinskiy and Vladimir Guletskiĭ

Journal:
J. Algebraic Geom. **21** (2012), 347-373

DOI:
https://doi.org/10.1090/S1056-3911-2011-00548-1

Published electronically:
May 31, 2011

Corrigendum:
J. Algebraic Geom. **22** (2013), 795-796.

MathSciNet review:
2877438

Full-text PDF

Abstract |
References |
Additional Information

Abstract: We study algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in $\mathbb Q$. We decompose the motive of a non-singular projective threefold $X$ with representable algebraic part of $CH_0(X)$ into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the Griffiths’ intermediate Jacobian $J^2(X)$ when the ground field is $\mathbb C$. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibred by surfaces with algebraic $H^2$. This gives new examples of three-dimensional varieties whose motives are finite-dimensional.

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*Motivic decomposition of abelian schemes and the Fourier transform*, J. Reine Angew. Math. **422** (1991), 201–219. MR **1133323 (92m:14055)**
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Additional Information

**Sergey Gorchinskiy**

Affiliation:
Steklov Mathematical Institute, Gubkina str. 8, 119991, Moscow, Russia

MR Author ID:
786536

Email:
gorchins@mi.ras.ru

**Vladimir Guletskiĭ**

Affiliation:
Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, England, United Kingdom

Email:
vladimir.guletskii@liverpool.ac.uk

Received by editor(s):
July 3, 2009

Received by editor(s) in revised form:
September 21, 2009

Published electronically:
May 31, 2011

Additional Notes:
The first author was partially supported by the grants RFBR 08-01-00095, NSh-1987.2008.1 and MK-297.2009.1.