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.
Published electronically: December 14, 2020
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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

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