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Beilinson's Hodge Conjecture for Smooth Varieties
Published online by Cambridge University Press: 06 March 2013
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.
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- Research Article
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- Copyright © ISOPP 2013
Footnotes
with an Appendix by Masanori Asakura
References
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