Algebraic cycles on an abelian variety

TL;DR

This paper establishes a canonical correspondence between numerical and rational equivalence classes of algebraic cycles on abelian varieties, preserving algebraic operations and morphisms.

Contribution

It introduces a canonical association from numerical to rational equivalence classes of cycles on abelian varieties, respecting algebraic structures.

Findings

Constructs a canonical cycle correspondence respecting algebraic operations.

Ensures compatibility with pullback and pushforward along homomorphisms.

Bridges numerical and rational equivalence in the context of abelian varieties.

Abstract

It is shown that to every Q-linear cycle \bar\alpha modulo numerical equivalence on an abelian variety A there is canonically associated a Q-linear cycle \alpha modulo rational equivalence on A lying above \bar\alpha. The assignment \bar\alpha -> \alpha respects the algebraic operations and pullback and push forward along homomorphisms of abelian varieties.