Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Nondivisible cycles on products of very general Abelian varieties


Author: Humberto A. Diaz
Journal: J. Algebraic Geom. 30 (2021), 407-432
DOI: https://doi.org/10.1090/jag/775
Published electronically: December 14, 2020
MathSciNet review: 4283547
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Abstract | References | Additional Information

Abstract: In this paper, we give a recipe for producing infinitely many nondivisible codimension $2$ cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of “field of definition” for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension $2$ cycles, that we call “primitive cycles”, the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori’s isogeny method (modified by Totaro) to produce infinitely many nondivisible cycles. As an application, we prove the Chow group modulo a prime of a product of three or more very general elliptic curves is infinite, generalizing work of Schoen.


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Humberto A. Diaz
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
MR Author ID: 1174879
Email: humberto@wustl.edu

Received by editor(s): December 6, 2018
Received by editor(s) in revised form: July 24, 2019, September 10, 2019, December 6, 2019, March 9, 2020, July 13, 2020, August 10, 2020, August 17, 2020, September 30, 2020, and October 5, 2020
Published electronically: December 14, 2020
Article copyright: © Copyright 2020 University Press, Inc.