I would like to ask the following question:

Let us consider the following branched cover $X_n$ of $\mathbb{P}^1$ that can be thought as a hyper-surface of $\mathbb{P}^1 \times \mathbb{P}^1$: \begin{equation} y^n={(x-x_1)(x-x_3) \over (x-x_2)(x-x_4)} \end{equation} where $(x,y)$ are coordinates on $\mathbb{P}^1 \times \mathbb{P}^1$.

When $n$ is an integer, this is a Riemann surface of genus $(n-1)$, and its homology cycles and cohomology elements are well described. The homology cycles are straightforward to find and the $2(n-1)$ elements of $H^1(X)$ can be explicitly written out in local coordinates as holomorphic differentials $\omega_k$ and their complex conjugates: \begin{equation} \omega_k ={y^k \over (x-x_1)(x-x_3)} dx, \qquad k=1,\cdots,(n-1) . \end{equation} This makes it easy to compute various properties of $X_n$ for integral $n$ such as period matrices, etc.

Now comes the question. I would like to understand what happens when we analytically continue $n$. That is, whether we can make sense of $X_n$ when $n$ is an irrational number or even a complex number. $X_n$ would become a branched cover of $\mathbb{P}^1$ of infinite degree. I was made aware by a friend that this surface is a non-compact, non-algebraic surface when $n$ is an irrational number.

If this is the case, how can we describe the (compact) homology and cohomology of this surface for irrational, or possibly complex $n$? In particular,

(1) What are the (compact) homology cycles?

(2) What are the differential 1-forms (with compact support)?

(3) Given two differential 1-forms $\omega$ and $\eta$ in the (compact) cohomology ring, is there a simple expression for \begin{equation} \int\int \omega \wedge \eta \end{equation} as in the compact case?

(4) Are such surfaces well studied? If so, which references do you recommend?

Thanks for your responses in advance.