Looking for examples of not injective maps and not surjective maps of the form $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $

Question

Here: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf, on pages: $ 1 $ and $ 2 $, we find the following paragraph:

For any scheme of finite type over a ground field and any integer $ k>0 $, we will define the so-called Chow groups $ A_k (X) $ whose elements are formal linear combination of $ k $ -dimentional closed subvarieties of $ X $, modulo ''boundaries'' in a suitable sense. The formal properties of this groups $ A_k (X) $ will be similar to those of homology groups. Il the ground field is $ \mathbb{C} $, you might even thought of the $ A_k (X) $ as being ''something like'' $ H_{2k} (X , \mathbb{Z} ) $, although these groups are usually different. But, there is a map $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $, so you can think of elements in the Chow groups as something that determines a homology class, but this map is in general neither injective nor surjective.

Questions,

• After reading this block, and since it is said in this block that, in general, the morphism $ A_k (X) \to H_{2k} (X, \mathbb {Z}) $ is neither injective, nor surjective, can you give me some examples of $ A_k (X) \to H_{2k} (X, \mathbb{Z}) $ maps that are not injective, or that they are not surjective where : $ 4 \leq 2k \leq n-4 $ and $ n - 4 > 0 $, and $ n $ is the dimension of $ X $ ?

Thanks in advance for you help.

Answer 1

If $C$ is a smooth projective curve of genus $g \geq 3$ 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

EDIT: Ceresa also shows in this paper that if $C$ is generic and $1 \leq k \leq g-2$, the cycles $W_k$ and $-W_k$ in $J(C)$ (recall that $W_k$ is the cycle parametrizing effective line bundles of degree $k$ on $C$) are algebraically independent (although they are homologically equivalent). This gives examples of non-injectivity for higher-dimensional cycles.