Let us start with a multiple cover $C$ of the $x$-plane branched at $z$ and $-z$, and so described by an equation $y^N = x^2 - z^2$.
For $N=2$, it is known that there are globally-defined holomorphic vector fields on $C$ that are of the form $V_n = u^{n+1} \frac{d}{du} $ where $u= x \pm y$ for any $n\in \mathbb{Z}$.
Can we also construct such well-defined vector fields on $C$ for any $N$?
Naively, they can be generalized to $V_n = u^{n+1} \frac{d}{du} $ where $u= \left(x^{\frac 2N} - w^i y\right)^{\frac N2}$ for $i=0,1,..,N-1$ and $w$ is the N-th root of unity. Am I missing something here?
I am a theoretical physicist, and not good at math. Need your help.
Thank you very much in advance.