Algebraic cycles on an abelian variety

Abstract

It is shown that to every ℚ-linear cycle ᾱ modulo numerical equivalence on an abelian variety A there is canonically associated a ℚ-linear cycle α modulo rational equivalence on A lying above ᾱ, characterised by a condition on the spaces of cycles generated by α on products of A with itself. The assignment ᾱ ↦ α respects the algebraic operations and pullback and push forward along homomorphisms of abelian varieties.