Finding a subvariety that is finite over the target.

Question

Given a surjective homomorphism of abelian varieties like $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, is it possible to find a subvariety $Z$ of $A$ such that restriction of $f$ to $Z$ induces a finite surjective morphism $f|_Z:Z\rightarrow B$? (this is equivalent to asking whether there is a cycles $Z$ such that $f_*[Z]$ is an integer in the grothendieck group of $B$)

Answer 1

Claim: "(this is equivalent to asking whether there is a cycles Z such that $f_∗[Z]$ is an integer in the grothendieck group of $B$)"

Comment: If $f: A \rightarrow B$ is a proper morphism of abelian varieties, there is an induced push forward map at the level of Chow groups

$$f_*: A_k(A) \rightarrow A_k(B)$$

and for any cycle $[Z]\in A_k(A)$ it follows $f_*[Z]\in A_k(B)$ is an algebraic cycle and not an integer. The group $A_k(B)$ is an abelian group that is not finitely generated in general. Similarly for the Grothendieck group: The push forward will be a class $\sum_i n_i[E_i]\in K_0(B)$. You must specify what you are talking about.

It seems you want $B$ to be irreducible with $d:=dim(B)$ and hence $A_d(B)$ is generated by $[B]$.

There is if $B:=B_1 \cup \cdots \cup B_l$ are the irreducible components of $B$ with $dim(B_i)=d$ for all $i$ an isomorphism of abelian groups

$$A_d(B)\cong\mathbb{Z}[B_1]\oplus \cdots \oplus \mathbb{Z}[B_l]\cong \mathbb{Z}^l,$$

hence you may not in general identify $A_d(B)$ and $\mathbb{Z}$.

Note: There is the "etale overing lemma" (Mumford, Abelian Varieties page 167) saying that if $f:Y \rightarrow X$ is an etale cover with $X$ an abelian variety, then $Y$ has the structure of abelian variety and $f$ is a separable isogeny. In your case this puts conditions on $Z$ if the induced map $f_Z$ is etale.