Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-23T19:06:40.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Beilinson's Hodge Conjecture for Smooth Varieties

Published online by Cambridge University Press:  06 March 2013

Rob de Jeu  and
James D. Lewis
Rob de Jeu
Affiliation:
Faculteit Exacte Wetenschappen, Afdeling Wiskunde, VU University Amsterdam, The Netherlandsjeu@few.vu.nl
James D. Lewis
Affiliation:
632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADAlewisjd@ualberta.ca
Get access

Abstract

Let U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, and

clr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))

the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

Keywords

Chow groupalgebraic cycleregulatorBeilinson-Hodge conjectureAbel-Jacobi mapDeligne cohomologyPrimary: 14C2519E15Secondary: 14C30
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 2 , April 2013 , pp. 243 - 282
Copyright
Copyright © ISOPP 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

with an Appendix by Masanori Asakura

References

A-K.Arapura, D., Kumar, M., Beilinson-Hodge cycles on semiabelian varieties. Math. Res. Lett. 16 (4) (2009), 557–562.Google Scholar
A-S.Asakura, M., Saito, S., Beilinson's Hodge conjecture with coefficients for open complete intersections, in Algebraic Cycles and Motives 2, Edited by Nagel, J. and Peters, C., LNS 344 (2007), 3–37, London Math. Soc.Google Scholar
Be.Beilinson, A., Notes on absolute Hodge cohomology, Contemporary Mathematics 55, (1) (1986), 35–68.Google Scholar
Bl.Bloch, S., Algebraic cycles and higher K-theory, Advances in Math. 61 (1986), 267–304.Google Scholar
Bl.Bloch, S., The moving lemma for higher Chow groups, Jour. of Alg. Geometry 3 (1994), 537–568.Google Scholar
CS.Coombes, K., Srinivas, V., A remark on K1 of an algebraic surface, Math. Ann. 265 (1983), 335–342.Google Scholar
De.Deligne, P., Théorie de Hodge II, Publ. Math. IHES 40 (1971), 5–58.Google Scholar
dJ.de Jeu, R., Notes on Beilinson's Hodge conjecture for surfaces.Google Scholar
EV.Esnault, H. and Viehweg, E., Deligne-Beilinson cohomology, in Beilinson's Conjectures on Special Values of L-Functions, (Rapoport, , Schappacher, , Schneider, , eds.), Perspect. Math. 4 (1988), 43–91, Academic Press,Google Scholar
Ja1.Jannsen, U., Mixed motives and Algebraic K-Theory, Springer Lect. Notes. 1400, Springer-Verlag.Google Scholar
Ja2.Jannsen, U. Deligne homology, Deligne homology, Hodge-D-conjecture, and motives, in “Beilinson's Conjectures on Special Values of L-functions” (Academic Press, Boston, MA, 1988), 305–372.Google Scholar
Ja3.Jannsen, U.Equivalence relations on algebraic cycles, in: The Arithmetic and Geometry of Algebraic Cycles, Proceedings of the CRM Summer School, June 7–19, 1998, Banff, Alberta, Canada (Editors: Gordon, B., Lewis, J., Müller-Stach, S., Saito, S. and Yui, N.), NATO Science Series 548 (2000), 225–260, Kluwer Academic Publishers.Google Scholar
KLM.Kerr, M., Lewis, J. D., Müller-Stach, S., The Abel-Jacobi map for higher Chow groups, Comp. Math. 142 (2006), 374–396.Google Scholar
K-L.Kerr, M., Lewis, J. D., The Abel-Jacobi map for higher Chow groups, II, Invent. Math. 170 (2) (2007), 355–420.CrossRefGoogle Scholar
SJK-L.Kang, S. J., Lewis, J. D., Beilinson's Hodge conjecture for K1 revisited, in Cycles, Motives and Varieties, Shimura, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai (2010), 197–215.Google Scholar
Lew.Lewis, J. D., A Survey of the Hodge Conjecture, CRM Monograph Series 10, Second edition, American Mathematical Society, 1991.Google Scholar
Mi.Milne, J. S., Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, Princeton, N.J., 1980.Google Scholar
MS.Müller-Stach, S., Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Alg. Geom. 6 (1997), 513–543.Google Scholar
MSa.Saito, M., Hodge-type conjecture for higher Chow groups, Pure and Applied Mathematics Quarterly 5 (3) (2009), 947–976.Google Scholar
Sa.Saito, S., Beilinson's Hodge and Tate conjectures, in Transcendental Aspects of Algebraic Cycles, Edited by Müller-Stach, S. and Peters, C., LNS 313 (2004), 276–290, London Math. Soc.Google Scholar
Su.Suslin, A., Algebraic K-theory and the norm residue homomorphism, J. Soviet Math. 30 (1985), 2556–2611.Google Scholar
We.Weibel, C., The norm residue isomorphism theorem, Topology 2 (2009), 346–372.CrossRefGoogle Scholar