In Otto Forster's Lectures on Riemann Surfaces on page 170 Jacobi Variety is defined in 21.6:
Suppose $X$ is a compact Riemann surface of genus $g$ and $ \omega_1,..., \omega_g $ is a basis of $\Omega (X) = H^0(X, \Omega_{X/\mathbb{C}})$ . Then
$$Jac(X):= \mathbb{C}^g/ \operatorname{Per}(\omega_1,..., \omega_g)$$
where $\operatorname{Per}(\omega_1,..., \omega_g)$ consists of all vectors
$$(\int_{\alpha} \omega_1, \int_{\alpha} \omega_2, ... \int_{\alpha} \omega_g)$$
where $\alpha$ runs through the fundamental group $\pi(X)$ (p 168). Moreover Theorem 21.4 proves that $ \operatorname{Per}(\omega_1,..., \omega_g) \subset \mathbb{C}^g$ is a lattice.
The last sentence from the following I not understand:
Here we are considering $Jac(X)$ only as an abelian group. It also has the structure of a compact complex manifold (a complex $g$-dimensional torus). Note that the definition depends on the choice of basis $ \omega_1,..., \omega_g $ but the choice of a different basis leads to an isomorphic $Jac(X)$.
Therefore my question is why two different choices $ \omega_1,..., \omega_g $ and $ \omega' _1,..., \omega' _g $ of the basis of $\Omega (X)$ gives isomorphic Jacobi varieties? Regarded as abelian groups? Or what type of isomorphy Forster here impose? Depending on about which kind of isomorphy Forster talks about there should be different characterizations when $\mathbb{C}^g / \operatorname{Per}(\omega_1,..., \omega_g)$ and $\mathbb{C}^g / \operatorname{Per}(\omega' _1,..., \omega' _g)$ are isomorphic.
and depending on which type of isomorphy we require (I think Forster here means by "$\cong$" iso only as abelian groups)
$\mathbb{C}^g / \operatorname{Per}(\omega_1,..., \omega_g)$ and $\mathbb{C}^g / \operatorname{Per}(\omega' _1,..., \omega' _g)$ are isomorphic in category $\mathcal{C}$ iff there exist a $M \in GL_{2g}(\mathbb{C})$ with $M \operatorname{Per}(\omega_1,..., \omega_g)= \operatorname{Per}(\omega' _1,..., \omega' _g)$ and $M \gamma_i = \gamma' _i$. And the important issue is that depending of which kind of isomorphy we require the matrix $M$ moreover lives restrictionally in a subgroup $H(\mathcal{C}) \subset GL_{2g}(\mathbb{C})$.
I assume that Forster is talking about isomorphy as abelian groups. Then, in which subgroup $H(\mathcal{C}) \subset GL_{2g}(\mathbb{C})$ the trafo matrix $M$ should like? And why it exist?
Clearly, there exist a $G \in GL_{2g}(\mathbb{C})$ with $M \omega_j = \omega' _j$. But of course if $\operatorname{Per}(\omega' _1,..., \omega' _g) = \mathbb{Z} \gamma' _1 + ... + \mathbb{Z} \gamma' _{2g}$ with $\gamma' _j= G \gamma_j$ then $\mathbb{C}^g / \operatorname{Per}(\omega_1,..., \omega_g)$ and $\mathbb{C}^g / \operatorname{Per}(\omega' _1,..., \omega' _g)$ are not isomorphic as abelian groups.
