Example. Consider the smooth affine plane curve $X=\{(x,y)\in \mathbb{C}^2\ \vert \ y^2=x^3-1\}$. Is the $1$-form defined as $\omega=dx/y$ where $y\neq 0$ a holomorphic $1$-form on a Riemann surface?
I tried writing out the transition maps but had to choose branches of the square/cube root, and am not sure that I am on the right track.
More generally. Let $f(x,y)$ be a non-singular bivariate polynomial over $\mathbb{C}$ and look at $X=\{f(x,y)=0\}$. How do I decide whether a differential form on $X$ given locally by a single formula $gdx$ (with $g$ a holomorphic function) is holomorphic?
My idea. We know that $\frac{\partial}{\partial x}fdx+\frac{\partial}{\partial y}fdy=0$. So in the above example $3x^2dx=2ydy$. Therefore where $x\neq 0$ and $y\neq 0$ we know that $\frac{dx}{y}=\frac{2}{3x^2}dy$. We conclude that $dx/y$ gives a holomorphic $1$-form on $X\setminus\{(\zeta_1,0),(\zeta_2,0),(\zeta_3,0),(0,i),(0,-i)\}$, where $\zeta_1, \zeta_2, \zeta_3$ are the three third complex roots of unity. Am I right?