Let $X$ be a compact complex surface and $\omega$ be a holomorphic 1form. $f,g$ are meromorphic function on $X$ such that $\operatorname{trdeg}_{\mathbb{C}}\mathbb{C}(f,g)=1$ and $\omega =g\,df$. Let $C$ be the nonsingular curve corresponding to $\mathbb{C}(f,g)$ and $\phi:X\rightarrow C$ be the rational map induced by $\mathbb{C}(f,g) \rightarrow \mathbb{C}(X)$. Can we view $g\,df$ as a form over $C$. If so, what's the relationship between $g\,df$ and $\omega$ ?
