Let $\omega$ be a meromorphic 1-form defined globally on $X\setminus\{N\}$ (that is, on the neighborhood of $U_j$, $\mathcal{N}(U_j),$ for all $j$) where $\omega=gdz\in\mathcal{M}\Omega(X;\mathbb{C})\subset\Omega^1(X;\mathbb{C}),$ for $g$ meromorphic, $U_j\subset \mathbb{C}$ an open subset of the complex plane, and $X={\mathbb{CP}}^1$. Here, $\mathcal{M}\Omega(X)$ and $\Omega^1(X)$ denote the space of meromorphic 1-forms and the space of continuous 1-forms on $X$, respectively. Then define a diffeomorphism $f:{\mathbb{CP}}^1\setminus\{N\}\to\mathbb{C}$ from the complex projective space modulo the antipodal point $\{N\}$ onto the complex plane $\mathbb{C}.$ The chart for the topological space ${\mathbb{CP}}^1\setminus\{N\}$ is defined by the homeomorphism $f:{\mathbb{CP}}^1\setminus\{N\}\to\mathbb{C}$, where we endow $X\setminus\{N\}$ with a weak topology, such that $\bigcup_j X_j =X.$ Then consider $\bar\sigma_j:=\int_{\partial U_j}\omega=\int_{\partial X_j}f^{*}\omega,$ defined via a pullback $f^{*}:\Omega^1(\mathbb{C})\to \Omega^1(X\setminus\{N\}).$ By extension, we define a Fourier-like expression, $$\sigma_j(\chi):=\int_{\partial X_j}e^{-i\pi \langle\xi,\chi \rangle}f^{*}\omega$$ for $\chi,\xi \in X\setminus\{N\}.$
If it was shown that $\sigma_j(\chi)\not=0$ ($\sigma_j$ does not have compact support on $X_j$), then would it follow that $f^{*}\omega$ has compact support on each $X_j$ $\left(i.e. \int_{\partial X_j}f^{*}\omega=0\right)$ and, consequently, $\int_{\partial U_j}\omega=0$?
Thanks in advance! Any feedback is greatly appreciated.