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On the image of the $l$-adic Abel-Jacobi map for a variety over the algebraic closure of a finite field


Author: Chad Schoen
Journal: J. Amer. Math. Soc. 12 (1999), 795-838
MSC (1991): Primary 14C25, 14G15
DOI: https://doi.org/10.1090/S0894-0347-99-00303-3
Published electronically: April 23, 1999
MathSciNet review: 1672878
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Abstract: Let $Y$ be a smooth projective variety of dimension at most 4 defined over the algebraic closure of a finite field of characteristic $>2$. It is shown that the Tate conjecture implies the surjectivity of the $l$-adic Abel-Jacobi map, $\mathbf {a}^{r}_{Y,l}:CH^{r}_{hom}(Y)\to H^{2r-1}(Y,\mathbb Z_l (r))\otimes \mathbb Q_l /\mathbb Z_l$, for all $r$ and almost all $l$. For a special class of threefolds the surjectivity of $\mathbf {a}^{2}_{Y,l}$ is proved without assuming any conjectures.


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Additional Information

Chad Schoen
Affiliation: Department of Mathematics, Duke University, Box 90320, Durham, North Carolina 27708-0320
Email: schoen@math.duke.edu

Keywords: Algebraic cycles, $l$-adic Abel-Jacobi map
Received by editor(s): June 24, 1997
Received by editor(s) in revised form: January 5, 1999
Published electronically: April 23, 1999
Additional Notes: This research was partially supported by NSF grants DMS-90-14954, DMS-93-06733.
Article copyright: © Copyright 1999 American Mathematical Society