Arithmetic normal functions and filtrations on Chow groups
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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.References
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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
- MR Author ID: 204180
- Email: lewisjd@ualberta.ca
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2663-2670
- MSC (2010): Primary 14C25; Secondary 14C30, 14C35
- DOI: https://doi.org/10.1090/S0002-9939-2011-11130-4
- MathSciNet review: 2910754