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

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.

• Well, the image of the cycle class map has to be contained in $H^{(k,k)}(X,\mathbb{Z}). Feb 3 '20 at 15:44 • For obstructions to infectivity, look up the generalized Bloch conjecture. For obstructions to surjectivity @Qfwfq's comment should be the only rational obstruction; there are other failures of surjectivity known (look up papers giving counterexamples to the integral Hodge conjecture). At the end of the day, assuming Hodge and generalized Bloch, there is a complete characterization of when injectivity/surjectivity fails rationally. – dhy Feb 3 '20 at 20:26 • Thank you Qfwfq and dhy. It's clear now. :-) – YoYo Feb 3 '20 at 20:51 ## 1 Answer 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. • It should be$g\ge 3\$. Feb 3 '20 at 16:35
• @AGlearner Fixed, thanks! Feb 3 '20 at 16:46
• Thank you @Yusuf. :-)
– YoYo
Feb 3 '20 at 20:52