Let $L$ be a lattice in $\mathbb{C}$ and let $\pi :\mathbb{C}\to X=\mathbb{C}/L$ be a quotient map. Show that the local formula $dz$ in every chart of $\mathbb{C}/L$ is a well-defined holomorphic 1-form on $\mathbb{C}/L$. Compute $\int\int_X dz\wedge d\bar{z}.$
Here is my idea:
Let $z=T(\omega)$ define a holomorphic mapping from the open set $V_1$ to $V_2$. Then $dz=T'(\omega)d\omega$ and $d\bar{z}=\overline{T'(\omega)}d\bar{\omega}$. I'm not sure how to show that the local formula $dz$ in every chart of $\mathbb{C}/L$ is a well-defined holomorphic 1-form on $\mathbb{C}/L$.
Also, $\int\int_X dz\wedge d\bar{z}= \int\int_X T'(\omega)d\omega\wedge \overline{T'(\omega)}d\bar{\omega}= \int\int_X ||T'(\omega)||^2d\omega \wedge d\bar{\omega}$
Now I think I need to use Stoke's theorem to finish the above computation but not sure how to proceed.
Any suggestions/hints will be helpful. Thanks.