Tohoku Mathematical Journal

Remarks on motives of abelian type

Charles Vial

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A motive over a field $k$ is of abelian type if it belongs to the thick and rigid subcategory of Chow motives spanned by the motives of abelian varieties over $k$. This paper contains three sections of independent interest. First, we show that a motive which becomes of abelian type after a base field extension of algebraically closed fields is of abelian type. Given a field extension $K/k$ and a motive $M$ over $k$, we also show that $M$ is finite-dimensional if and only if $M_K$ is finite-dimensional. As a corollary, we obtain Chow–Künneth decompositions for varieties that become isomorphic to an abelian variety after some field extension. Second, let $\varOmega$ be a universal domain containing $k$. We show that Murre's conjectures for motives of abelian type over $k$ reduce to Murre's conjecture (D) for products of curves over $\varOmega$. In particular, we show that Murre's conjecture (D) for products of curves over $\varOmega$ implies Beauville's vanishing conjecture on abelian varieties over $k$. Finally, we give criteria on Chow groups for a motive to be of abelian type. For instance, we show that $M$ is of abelian type if and only if the total Chow group of algebraically trivial cycles $\mathrm{CH}_*(M_\varOmega)_\mathrm{alg}$ is spanned, via the action of correspondences, by the Chow groups of products of curves. We also show that a morphism of motives $f: N \rightarrow M$, with $N$ finite-dimensional, which induces a surjection $f_* : \mathrm{CH}_*(N_\varOmega)_\mathrm{alg} \rightarrow \mathrm{CH}_*(M_\varOmega)_\mathrm{alg}$ also induces a surjection $f_* : \mathrm{CH}_*(N_\varOmega)_\mathrm{hom} \rightarrow \mathrm{CH}_*(M_\varOmega)_\mathrm{hom}$ on homologically trivial cycles.

Article information

Source
Tohoku Math. J. (2), Volume 69, Number 2 (2017), 195-220.

Dates
First available in Project Euclid: 24 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1498269623

Digital Object Identifier
doi:10.2748/tmj/1498269623

Mathematical Reviews number (MathSciNet)
MR3682163

Zentralblatt MATH identifier
06775252

Subjects
Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14C25: Algebraic cycles 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]

Keywords
Algebraic cycles Chow groups motives abelian varieties finite-dimensionality

Citation

Vial, Charles. Remarks on motives of abelian type. Tohoku Math. J. (2) 69 (2017), no. 2, 195--220. doi:10.2748/tmj/1498269623. https://projecteuclid.org/euclid.tmj/1498269623


Export citation

References

  • Yves André, Pour une théorie inconditionnelle des motifs, Inst. Hautes études Sci. Publ. Math. no. 83 (1996), 5–49.
  • Yves André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), volume 17 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris, 2004.
  • Donu Arapura, Motivation for Hodge cycles, Adv. Math. 207 (2006), no. 2, 762–781.
  • Arnaud Beauville, Sur l'anneau de Chow d'une variété abélienne, Math. Ann. 273 (1986), no. 4, 647–651.
  • S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), no. 5, 1235–1253.
  • Spencer Bloch, Lectures on algebraic cycles, Duke University Mathematics Series, IV. Duke University Mathematics Department, Durham, N.C., 1980.
  • Jean-Pierre Demailly, Thomas Peternell and Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345.
  • Christopher Deninger and Jacob Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991), 201–219.
  • William Fulton, Intersection theory, volume 2 of 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], Springer-Verlag, Berlin, second edition, 1998.
  • Sergey Gorchinskiy and Vladimir Guletskiǐ, Motives and representability of algebraic cycles on threefolds over a field, J. Algebraic Geom. 21 (2012), no. 2, 347–373.
  • Jaya N. N. Iyer, Murre's conjectures and explicit Chow-Künneth projections for varieties with a NEF tangent bundle, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1667–1681.
  • Uwe Jannsen, Motivic sheaves and filtrations on Chow groups, In Motives (Seattle, WA, 1991), 245–302, Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
  • Uwe Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107 (1992), no. 3, 447–452.
  • B. Kahn and R. Sujatha, Birational motives, I pure birational motives, Preprint, February 27, 2009.
  • Shun-Ichi Kimura, Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), no. 1, 173–201.
  • Steven L. Kleiman, The standard conjectures, In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos. Pure Math., pages 3–20. Amer. Math. Soc., Providence, RI, 1994.
  • 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.
  • A. J. Scholl, Classical motives, In Motives (Seattle, WA, 1991), 163–187, Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
  • Ronnie Sebastian, Smash nilpotent cycles on varieties dominated by products of curves, Compositio Math. 149 (2013), 1511–1518.
  • Charles Vial, Chow-Kuenneth decomposition for $3$- and $4$-folds fibred by varieties with trivial Chow group of zero-cycles, J. Algebraic Geom. 24 (2015), 51–80.
  • Charles Vial, Pure motives with representable Chow groups, C. R. Math. Acad. Sci. Paris 348 (2010), no. 21–22, 1191–1195.
  • Charles Vial, Niveau and coniveau filtrations on cohomology groups and Chow groups, Proc. Lond. Math. Soc. (3) 106 (2013), no. 2, 410–444.
  • Kejian Xu and Ze Xu, Remarks on Murre's conjecture on Chow groups, J. K-Theory 12 (2013), no. 1, 3–14.