Consider a torus $T^2$ made by gluing a rectangular with a real coordinate $0 \leq x^1, x^2 \leq 1$. Define $\tau = \frac{e_2}{e_1}$ where $e_1, e_2$ are ($\mathbf{C}$ numbers corresponding to) basis of the rectangular. Then we define a complex coordinate $z = x^1 + \tau x^2$ ($z \sim z +1, z+\tau$), and the volume form is given by \begin{align} \frac{i}{2} dz \wedge d\bar{z}. \end{align} Integral of this over $T^2$ gives $+1$, which determines the orientation of the torus.
Then we consider an object $A(z) = {\rm Im}(\bar{z} dz)$.
Now if we compare it on identified two points, we get $A(z+1) - A(z) = {\rm Im} dz \equiv d\chi_1, A(z+\tau) - A(z) = {\rm Im} \bar{\tau }dz \equiv d\chi_2$.
Since there is the identification $z \sim z +1, z+\tau$, we may interpret them as contour integrals: \begin{align} \chi_1 = \oint_{\gamma_1} A(z),\quad\chi_2 = \oint_{\gamma_2} A(z), \end{align} where $\gamma_1, \gamma_2$ correspond to $((0, x^2) \rightarrow (1, x^2))$ and $((x^1, 0) \rightarrow (x^1, 1))$.
However, I calculated these integrals but I obtained $-\chi_1, -\chi_2$ respectively.
I may have made a mistake, especially in the orientation of the integrals.
Could you clarify my mistake, please?