I'm a graduate student struggling to find a topic for his doctoral dissertation which hasn't already been explored. At present, my hope was to see if classical results about Jacobian Varieties, Abelian Functions, Lattès maps, Theta functions, and the like, which are known for smooth projective algebraic curves could be extended to smooth affine transcendental curves (Ex: $y^{2}=3^{x}-2^{x}+\frac{1}{6}x^{2}$).
After searching for such things (“jacobian varieties of transcendental curves”, etc.), I've come across a variety (pun possibly intended) of related material and subject areas. Otto Forster's Lectures on Riemann Surfaces had a chapter on non-compact Riemann surfaces which more or less boiled down to explaining how classical complex analysis essentially holds over non-compact Riemann surfaces (proved using functional analytic/distributional methods).
I also caught wind of something called “Complex Analytic sets” and “Complex analytic varieties” (viz. this textbook, among other things). These are defined as the study of sets which are the vanishing locus of one or more holomorphic functions. I cannot, however, find a straight-forward answer to exactly what kinds of functions this entails. I have seen that it encompasses polynomials of several complex variables, but I have not yet found any indication that it also deals with varieties defined by the zero loci of one or more transcendental functions. Knowing whether or not varieties defined by transcendental functions are among those studied in Complex analytic varieties is obviously essential if I am to have any chance of getting to proceed with studying them on my own.
Today, I stumbled upon an AMS book review of Riemann Surfaces of Infinite Genus by Feldman, Knörrer, and Trubowitz. Much to my grief-joy, many of the ideas I had under consideration were realized in it. This is especially potent seeing as, only yesterday, I had what (at least from my perspective) constituted a possible breakthrough in my research. Skimming through the book, I didn't notice the appearance of my idea, and so, I wonder if it might be sufficiently novel to be worth exploring, even if a possibly-closely-related approach to the issue already exists.
For the sake of simplicity, let's work with a non-singular curve $C$ defined by the zero locus of the function $f\left(x,y\right)=y^{\nu}-T\left(x\right)$ over $\mathbb{C}$, where $\nu$ is an integer $\geq2$ and where $T\left(x\right)$ is a transcendental entire function of $x$. The classical methods for constructing a basis of holomorphic differential $1$-forms on a compact Riemann surface (as explained in these absolutely spectacular notes of Bertrand Eynard's) generalize without any trouble to surfaces of infinite genus, such as $C$.
Let: $$T_{N}\left(x\right)=\sum_{n=0}^{N\nu}\frac{T^{\left(n\right)}\left(0\right)}{n!}x^{n}$$ be an $N\nu$th degree taylor polynomial of $T$ about $0$, and let $C_{N}$ be the curve defined by the zero locus of the function $f_{N}\left(x,y\right)=y^{\nu}-T_{N}\left(x\right)$. Using the newton polygon method, it is easy to show that a basis for $\mathcal{O}^{1}\left(C_{N}\right)$ (the vector space of holomorphic $1$-forms on $C_{N}$) is given by: $$\left\{ \frac{x^{j-1}}{\left(T_{N}\left(x\right)\right)^{\frac{\nu-k}{\nu}}}dx:\begin{array}{c} 1\leq k\leq\nu-1\\ 1\leq j\leq N\nu\left(\nu+k\right)-1 \end{array}\right\}$$ and that $C_{N}$ will therefore be of genus $g_{N}=\frac{1}{2}\left(3\nu^{2}N-2\right)\left(\nu-1\right)$.
Now, for each $N$, let $\alpha_{N,1},\beta_{N,1},\ldots,\alpha_{N,g_{N}},\beta_{N,g_{N}}$ be a symplectic basis for the homology group $H_{1}\left(C_{N},\mathbb{Z}\right)\cong\mathbb{Z}^{2g_{N}}$, and write: $$\alpha_{N,m}\left(\omega\right)=\int_{\alpha_{N,m}}\omega$$
$$\beta_{N,m}\left(\omega\right)=\int_{\beta_{N,m}}\omega$$ for any $\omega\in\mathcal{O}^{1}\left(C_{N}\right)$.
It was while trying to verify by analytical estimates that the Riemann Bilinear Relations: $$\int_{C_{N}}\omega\wedge\pi=\sum_{m=1}^{g_{N}}\left(\alpha_{N,m}\left(\omega\right)\beta_{N,m}\left(\pi\right)-\alpha_{N,m}\left(\pi\right)\beta_{N,m}\left(\omega\right)\right)$$ (for any $\omega,\pi\in\mathcal{O}^{1}\left(C_{N}\right)$) hold as $N\rightarrow\infty$ for particular choices of $T\left(x\right)$ didn't seem to be going anywhere (I will likely try again some other time, but that's a question for another day), that I had my “big idea”.
I observed that given surfaces $S_{N}$ and $S_{N^{\prime}}$ of geni $g_{N}$ and $g_{N^{\prime}}$ (where $g_{N^{\prime}}>g_{N}$), there is then a surface $S$ of genus $g_{N^{\prime}}-g_{N}$ so that $S_{N^{\prime}}$ is homeomorphic to the connected sum of $S_{N}$ and $S$ (denoted $S_{N}\#S$). This connected sum induces an inclusion of the homology group of $S_{N}$ into that of $S_{N^{\prime}}$.
As shown here, since Riemann surfaces are compact oriented real manifolds of dimension $2$, it follows that: $$H_{1}\left(X\#Y\right)=H_{1}\left(X\right)\oplus H_{1}\left(Y\right)$$ holds for all compact riemann surfaces $X,Y$.
So, if we consider the $C_{N}$s as pointed topological spaces (with the point in question being the designated basepoint of the fundamental group/homology group) and then have an ascending chain of base-point preserving inclusions:
$$C_{1}\hookrightarrow C_{2}\hookrightarrow C_{3}\hookrightarrow\cdots$$
which shows that for every $N$, there is a Riemann surface $S_{N}$ so that $C_{N+1}$ is homeomorphic (conformally equivalent?) to the connected sum of $C_{N}$ and $S_{N}$; that is: $$C_{N+1}\cong C_{N}\#S_{N}$$
Consequently, I can define $H_{1}\left(C,\mathbb{Z}\right)$ as the direct limit of the $H_{1}\left(C_{N},\mathbb{Z}\right)$s. My hope is that I can then use this to show that Riemann matrix of (normalized) periods for $C_{N}$, the Jacobian Variety of $C_{N}$, the Riemann Theta function for $C_{N}$, and the like, then can be extended via direct limit to analogues for the infinite-genus curve $C$.
As I remarked earlier, exactly these conclusions seem to be reached by Feldman, Knörrer, and Trubowitz. Consequently, I was wondering if (a) my direct limit approach could end up being useful, and, (b) whether or not it is novel (and thus, a suitable basis for my dissertation).
Finally, I guess I should point out that I'm an analyst by training and by habit, so too much geometry or algebra at once will likely go over my head.
Anyhow, thanks in advance for any assistance anyone can give.
