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?