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Chow groups are finite dimensional, in some sense

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Abstract.

When S is a surface with p g (S)>0, Mumford proved that its Chow group A*S is not “finite dimensional” in some sense. In this paper, we propose another definition of “finite dimensionality” for the Chow groups. Using this new definition, at least the Chow group of some surface S with p g (S)>0 (for example, the product of two curves) becomes finite dimensional. The finite dimensionality of the Chow groups follows from the finite dimensionality of the Chow motives. It turns out that the finite dimensionality of the Chow motives is a very strong property. For example, we can prove Bloch’s conjecture (representability of the Chow groups of surfaces with p g (S)=0) under the assumption that the Chow motive of S is finite dimensional.

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References

  1. André, Y.: Motifs de dimension finie. Séminaire Bourbaki, 2004

  2. Beauville, A.: Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273, 647–651 (1986)

    MathSciNet  MATH  Google Scholar 

  3. Bloch, S.: Some elementary theorems about algebraic cycles on abelian varieties. Invent. Math. 37, 215–228 (1976)

    MATH  Google Scholar 

  4. Bloch, S., Kas, A., Lieberman, D.: Zero cycles on surfaces with p g =0. Compositio Math. 33, 135–145 (1976)

    MATH  Google Scholar 

  5. Deninger, C., Murre, J.: Motivic decomposition of Abelian schemes and the Fourier transform. J. Reine. Angew. Math. 422, 201–219 (1991)

    MathSciNet  MATH  Google Scholar 

  6. Fulton, W.: Intersection Theory. Springer, Berlin, 1984, pp. xi+470

  7. Fulton, W., Harris, J.: Representation Theory. Springer GTM 129, New York, 1991, pp. xvi+551

  8. Guletskii, V.: A remark on nilpotent correspondences, K-theory preprint archive 651 (2003), http://www.math.uiuc.edu/K-theory

  9. Kimura, S.: Fractional Intersection and Bivariant Theory. Commun. Algebra 20, 285–302 (1992)

    MathSciNet  MATH  Google Scholar 

  10. Kimura, S.: On Varieties whose Chow Groups have Intersection Products with -coefficients. Thesis, University of Chicago, 1990

  11. Kimura, S.: A cohomological characterization of Alexander schemes. Invent. math. 137, 575–611 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kimura, S., Vistoli, A.: Chow rings of infinite symmetric products. Duke Math. J. 85, 411–430 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Kleiman, S.: The standard conjectures. In: Motives (Seattle, WA, 1991). Proc. Sympos. Pure Math. Vol 55, Amer. Math. Soc. pp. 3–20, 1994

  14. Knutson, D.: λ-Rings and the Representation Theory of the Symmetric Group. Lecture Notes in Math. 308, (Springer, Berlin 1973) iv+203 pp

  15. Mazza, C.: Schur functors and motives. K-theory preprint archive 641 (2003), http://www.math.uiuc.edu/K-theory/

  16. Mumford, D.: Rational equivalences of 0-cycles on surfaces. J. Math. Kyoto Univ. 9, 195–204 (1969)

    MATH  Google Scholar 

  17. Roitman, A.A.: The torsion of the group of 0-cycles modulo rational equivalence. Ann. Math. 111, 553–569 (1980)

    MathSciNet  Google Scholar 

  18. Scwarzenberger, R.L.E.: Jacobians and Symmetric Products. Illinois J. Math. 7, 257–268 (1963)

    Google Scholar 

  19. Shermenev, A.M.: The motive of an abelian variety. Funct. Anal. 8, 47–53 (1974)

    Google Scholar 

  20. Voevodsky, V.: Nilpotence theorem for cycles algebraically equivalent to zero. Internat. Math. Res. Notices pp. 187–198 (1995)

  21. Vistoli, A.: Intersection theory on algebraic stacks and on their moduli spaces. Invent. math. 97, 613–670 (1989)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, 739-8526, Japan

    Shun-Ichi Kimura

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  1. Shun-Ichi Kimura
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Correspondence to Shun-Ichi Kimura.

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Mathematics Subject Classification (2000): 14C

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Kimura, SI. Chow groups are finite dimensional, in some sense. Math. Ann. 331, 173–201 (2005). https://doi.org/10.1007/s00208-004-0577-3

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