I first state the problem:
Q1:Let X be a complex torus $\mathbb C^n/ \Lambda$.Then how to show that $Pic^0(Pic^0(X))$ is isomorphic to $X$?(This is the exercise 3.3.10 of Huybrechts' Complex Geometry).
Here,$Pic^0(X)$denotes the Jacobian of $X$,i.e. the kernel of the map $Pic(X)\rightarrow H^2(X,\mathbb Z)$.
When $n=1$,$X=\mathbb C/\Lambda$:
Notice that for compact Kähler curve,we always have $Alb(X)\cong Pic^0(X)$.And,we also have $Alb(\mathbb C^n/ \Lambda)=\mathbb C^n/\Lambda$.Here,the Albanese torus of a compact Kahler manifold is the complex torus $$Alb(X)=H^0(X,\Omega_X)^*/H_1(X,\mathbb Z).$$
S0,for the dimension one complex torus $X=\mathbb C/\Lambda$,we have $$Pic^0(Pic^0(X))=Pic^0(Alb(X))=Pic^0(X)=Alb(X)=X.$$
But how can I show the claim for higher dimension?
Notice the corolary 3.3.6:If $X$ is a compact Kähler manifold,then $Pic^0(X)$ is in a natural way a complex torus of dimension $\frac {b_1(X)}{2}.$
Q2:If $X$ the special complex torus $\mathbb C^n/\mathbb Z^{2n}$,do we have $Pic^0(X)=X$?
Any references and suggestion are appreciated.Thank you very much!