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Let $X$ be a compact complex surface and $\omega$ be a holomorphic 1-form. $f,g$ 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 $\omega$ ?

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$gdf$ is a meromorphic form on $C$ and $\omega=gdf$ is just the pull back of this form via $\phi$. –  Mohan Oct 13 '13 at 21:45

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