Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties

TL;DR

This paper proves the integral Hodge and Tate conjectures for one-cycles on certain abelian varieties, using a lift of the Fourier transform to integral Chow groups, advancing understanding of algebraic cycles.

Contribution

It introduces a lift of the Fourier transform to integral Chow groups and proves the integral Hodge and Tate conjectures for specific abelian varieties.

Findings

Proves the integral Hodge conjecture for one-cycles on principally polarized abelian varieties.

Establishes the integral Tate conjecture for one-cycles on Jacobians of curves of compact type.

Shows that abelian varieties satisfying these conjectures are dense in their moduli space.

Abstract

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a proper curve of compact type over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.