If $C$ is a smooth projective curve of genus $g \geq 2$ and $J(C)$ is the Jacobian of $C,$ then an Abel curve $C \subset J(C)$ is not algebraically equivalent to its image $-C$ under the negation automorphism, even though $C$ is homologically equivalent to $C.$  This was proved by Ceresa in the paper

https://www.jstor.org/stable/2007078