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Arithmetic normal functions and filtrations on Chow groups

Author: James D. Lewis
Journal: Proc. Amer. Math. Soc. 140 (2012), 2663-2670
MSC (2010): Primary 14C25; Secondary 14C30, 14C35
Published electronically: December 27, 2011
MathSciNet review: 2910754
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Abstract: Let $ X/\mathbb{C}$ be a smooth projective variety, and let $ \textrm {CH}^r(X,m)$ be the higher Chow group defined by Bloch. Saito and Asakura defined a descending candidate Bloch-Beilinson filtration $ \textrm {CH}^r(X,m;\mathbb{Q}) = F^0\supset \cdots \supset F^r\supset F^{r+1} = F^{r+2}=\cdots $, using the language of mixed Hodge modules. Another more geometrically defined filtration is constructed by Kerr and Lewis in terms of germs of normal functions. We show that under the assumptions (i) $ X/\mathbb{C} = X_0\times \mathbb{C}$ where $ X_0$ is defined over $ \overline {\mathbb{Q}}$, and (ii) the general Hodge conjecture, that $ F^{\bullet }\textrm {CH}^r(X,m;\mathbb{Q})$ coincides with the aforementioned geometric filtration. More specifically, it is characterized in terms of germs of reduced arithmetic normal functions.

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Additional Information

James D. Lewis
Affiliation: Department of Mathematical and Statistical Sciences, 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Keywords: Bloch-Beilinson filtration, arithmetic normal function, Chow group
Received by editor(s): March 28, 2010
Received by editor(s) in revised form: March 16, 2011
Published electronically: December 27, 2011
Additional Notes: The author was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
Communicated by: Lev Borisov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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