I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page 289.


Theorem 10.14.
The Albanese map
\begin{equation\*}
 alb\_X:CH\_0(X)\_{hom}\rightarrow Alb(X)
\end{equation\*}
Induces an isomorphism on torsion points.


What does it mean that the isomorphism is on torsion points? what is an intuitive way to think about torsion points?



Meaning of torsion points in a Roitman's theorem
It occurs to me that your question shouldn't be taken literally, and is simply asking about the meaning of the theorem. To appreciate it, one can ask what $CH\_0(X)$ looks like. As a first attempt, map it to something more concrete like the Albanese (which is just a complex torus over $\mathbb{C}$). It is easy to see that the map is surjective, and an isomorphism for curves. In higher dimensions, Mumford showed that the kernel can be huge. Given this, Roitman's theorem that the Albanese map restricted to the torsion subgroups yields an isomorphism is pretty surprising. For example if $H\_1(X,\mathbb{Z})=0$, the theorem would imply that the torsion subgroup of $CH\_0(X)$ is trivial.


