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

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.

23 pages, final version, to appear in Compositio Mathematica

References in corpus (2)

• Some elementary theorems about divisibility of 0-cycles on abelian varieties defined over finite fields
• The integral Hodge conjecture for two-dimensional Calabi-Yau categories