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Higher bivariant Chow groups and motivic filtrations


Author: Abhishek Banerjee
Journal: Trans. Amer. Math. Soc. 363 (2011), 5943-5969
MSC (2010): Primary 14C15, 14C25
DOI: https://doi.org/10.1090/S0002-9947-2011-05300-6
Published electronically: May 25, 2011
MathSciNet review: 2817416
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Abstract: The purpose of this paper is twofold: first, we extend Saito's filtration on Chow groups, which is a candidate for the conjectural Bloch Beilinson filtration on the Chow groups of a smooth projective variety, from Chow groups to the bivariant Chow groups. In order to do this, we construct cycle class maps from the bivariant Chow groups to bivariant cohomology groups. Secondly, we use our methods to define a bivariant version of Bloch's higher Chow groups.


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  • 1. Beilinson, A. A. Height pairing between algebraic cycles. $ K$-theory, arithmetic and geometry (Moscow, 1984-1986), 1-25, Lecture Notes in Math., 1289, Springer, Berlin, 1987. MR 923131 (89h:11027)
  • 2. Bloch, S. Some notes on elementary properties of higher Chow groups, including functoriality properties and cubical Chow groups. Preprint.
  • 3. Bloch, S. Algebraic cycles and higher $ K$-theory. Adv. in Math. 61 (1986), no. 3, 267-304. MR 852815 (88f:18010)
  • 4. Brasselet, J.P., J. Schürmann and S. Yokura. On Grothendieck transformations in Fulton-MacPherson's bivariant theory. J. Pure Appl. Algebra 211 (2007), no. 3, 665-684. MR 2344222 (2008j:14038)
  • 5. Brasselet, J.P., J. Schürmann and S. Yokura. On the uniqueness of bivariant Chern class and bivariant Riemann-Roch transformations. Adv. in Math. 210 (2007), no. 2, 797-812. MR 2303240 (2008f:14009)
  • 6. Fulton, W. Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Grenz- gebiete. 3rd Series 2. Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
  • 7. Fulton, W. and R. MacPherson. Categorical framework for the study of singular spaces. Mem. Amer. Math. Soc. 31 (1981), no. 243. MR 609831 (83a:55015)
  • 8. Ginzburg, V. Geometric methods in the representation theory of Hecke algebras and quantum groups, A. Broer and A. Daigneault (eds.), Theories and Algebraic Geometry (Montreal, PQ, 1997), Kluwer Acad. Publ., Dordrecht, 1998, pp. 127-183. MR 1649626 (99j:17020)
  • 9. Ginzburg, V. $ {\mathfrak{G}}$-modules, Springer's representations and bivariant Chern classes. Adv. in Math. 61 (1986), no. 1, 1-48. MR 847727 (87k:17014)
  • 10. Saito, S. Motives and filtrations on Chow groups. II. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 321-346, NATO Sci. Ser. C Math. Phys. Sci., 548, Kluwer Acad. Publ., Dordrecht, 2000. MR 1744952 (2001d:14025)
  • 11. Saito, S. Motives and filtrations on Chow groups. Invent. Math. 125 (1996), no. 1, 149-196. MR 1389964 (97i:14002)
  • 12. Yokura, S. On Ginzburg's bivariant Chern classes. II. Geom. Dedicata 101 (2003), 185-201. MR 2017902 (2004m:14003)
  • 13. Yokura, S. On Ginzburg's bivariant Chern classes. Trans. Amer. Math. Soc. 355 (2003), no. 6, 2501-2521. MR 1974000 (2004f:14012)

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Additional Information

Abhishek Banerjee
Affiliation: Institut des Hautes Études Scientifiques, Le Bois-Marie 35, Route de Chartres 91440, Bures sur Yvette, France
Address at time of publication: Department of Mathematics, Ohio State University, 231 W. 18th Avenue, 100 Math Tower, Columbus, Ohio 43210
Email: abhishekbanerjee1313@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2011-05300-6
Received by editor(s): September 19, 2009
Received by editor(s) in revised form: January 16, 2010
Published electronically: May 25, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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