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

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