In Huybrechts' Complex Geometry 3.3 Lefschetz Theorems p134,we have boundary map $c_1:Pic(X)\rightarrow H^2(X,\mathbb Z)$.Then we can define Jacobian $Pic^0(X)$ is the kernel of the map $c_1$.
We have a corollary(3.3.6):If $X$ is a compact Kahler manifold,then $Pic^0(X)$ is in a natural way a complex torus of dimension $b_1(X)$.
Clearly,the pre-image $Pic^\alpha (X)\subset Pic(X)$ of any element $\alpha\in H^2(X,\mathbb Z)$ in the image of $c_1$ can be identified with $Pic^0(X)$,although not canonically.
Then,we get the claim:Thus,for a compact Kahler manifold,such that $H^2(X,\mathbb Z)$ is torsion free,the Picard group $Pic(X)$ is fibred by tori of complex dimension $b_1(X)$ over the discrete set $H^{1,1}(X, \mathbb Z)$.
Here,$$H^{1,1}(X,\mathbb Z)=Im(H^2(X,\mathbb Z)\rightarrow H^2(X,\mathbb C))\cap H^{1,1}(X),$$also note that for a compact Kahler manifold.The $Pic(X)\rightarrow H^{1,1}(X,\mathbb Z)$ is surjective.
Hence,we have the diagram:
My question is:Why we need the condition: $H^2(X,\mathbb Z)$ is torsion free to get the claim?Can anyone give me some help?Thanks a lot.