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. I think this might be equivalent to the claim that the lattices $\operatorname{Per}(\omega_1,..., \omega_g)$ and $\operatorname{Per}(\omega' _1,..., \omega' _g)$ are equivalent, but maybe that's a too strong requirement. From the proof of 21.4 we obtain explicite construction of lattice $\mathbb{Z}$-basis of $\operatorname{Per}(\omega_1,..., \omega_g)= \mathbb{Z} \gamma_1 + ... + \mathbb{Z} \gamma_{2g}$, respectively $\operatorname{Per}(\omega' _1,..., \omega' _g)= \mathbb{Z} \gamma' _1 + ... + \mathbb{Z} \gamma' _{2g}$. Two lattices are equivalent if there exist a $M \in SO_{2g}(\mathbb{Z})$ with  $M \gamma_i = \gamma' _i$. But I not see why such $M$ exist in this case.

Alternatively, which sufficient condition might guaratee that the complex tori $\mathbb{C}^g / \operatorname{Per}(\omega_1,..., \omega_g)$ and $\mathbb{C}^g/ \operatorname{Per}(\omega' _1,..., \omega' _g)$ are isomorphic here?

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.