Consider $X:=\overline{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$. We know that $X$ is a compact Riemann surface. Let $\omega$ be a holomorphic form on the entire surface $X$. I want to prove that $\omega$ is constant. To do that assume that $\omega$ is not constant. We know that $$ \mathcal{A} := \{(U=\mathbb{C},Id),(V=X \setminus \{0\},w = 1/z)\} $$ is a holomorphic structure on $X$. By definition $$ \omega = fdz, \quad \omega = g dw $$ where $f$ is entire and $g:V \to \mathbb{C}$ is holomorphic on $V$. Changing coordinates
$$ f(z) = -\dfrac{1}{z^2}g(1/z), \quad \forall z \in \mathbb{C} \setminus \{0\}. $$ Then $$ f(0) = \lim_{z \to 0}-\dfrac{1}{z^2}g(1/z) $$ As $g(\infty) \in \mathbb{C}$, we deduce that $\lim_{z \to 0}-\dfrac{1}{z^2}g(1/z) = \infty$. Then, $f$ has a pole in the point $z = 0$, a contradiction.
Is this argument correct?