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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.

edited Jan 31 2012 at 0:48

asked Jan 31 2012 at 0:37

D. S. Park
113●5

1

Would it be better to choose a new variable $w$ to stand in for $\log y$, and change $y^n$ to $e^{nw}$ on the left? – S. Carnahan♦ Jan 31 2012 at 0:59

Yes. It almost certainly would. Then you could embed it in $\mathbb C \times \mathbb C$ and study it there. Note that if $n$ is not a rational number, there is nothing we can add to $w$ to keep both $y$ and $y^n$ fixed. Therefore, it does not make sense to look at some quotient of this surface as the actual corresponding branch cover, so it makes no sense to think about $y$ at all, and one should just work with $n \log y$, so you have a single equation $e^w=f(x)$ – Will Sawin Jan 31 2012 at 3:19

Thanks for the comments. I am not so sure whether working with $n \log y$ is the thing to do. If one does that, I do not understand how the structure (cohomology) of the surface would depend on $n$. Ideally, I would like some way to treat these surfaces so that they reproduce the data of the compact Riemann surface when $n$ is integral, which means the structure should have $n$ dependence. – D. S. Park Jan 31 2012 at 17:10

Branched Covers of a Two-sphere with Infinite Degree

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