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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 [email protected]
James D. Lewis
Affiliation:
632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, [email protected]
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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 

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Footnotes

with an Appendix by Masanori Asakura

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