# Abstract Nonsense

## Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. I)

Point of Post: In this post we discuss a new way of getting Riemann surfaces by taking loci of holomorphic functions on $\mathbb{C}^2$. We necessarily then discuss the (holomorphic) inverse function theorem.

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Motivation

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Just like when studying smooth manifolds, perhaps one of the richest examples of Riemann surfaces come from looking at loci of holomorphic functions. In particular, for us, we shall care most about the zero sets of of holomorphic functions $D\to\mathbb{C}$ with $D$ a domain in $\mathbb{C}^2$. Of course, there are two issues with this. Firstly, what does it mean for a function $D\to\mathbb{C}$ to be holomorphic? Secondly, why is it clear that such a locus is actually a Riemann surface?

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The first is one for which I can only give an unsatisfactory answer for now. For us, a holomorphic function of two variables will be one which is holomorphic in each variable separately. The fact that this definition of holomorphic yields what we expect holomorphic functions to be (functions which are locally power series!) is a relatively deep fact known as (or at least coming from) Hartog’s theorem. This is a theorem whose place is rightfully in the study of Several Complex Variables (SCV), and is something that I can’t do justice to right now. Moreover, it is perhaps the “other branch” (complex analysis joke fully intended) one could take when they want to go to a second graduate course in complex analysis. So, for anyone interested in WHY this is the correct notion of holmorphicity in higher dimensions I recommend taking a look at SCV in general. To make up for my lack of actually explaining these concepts, let me be at least helpful in the narrowing down the length list of SCV books to, what is in my humble opinion, the best four books. The classic books on SCV are [6] and [7]. They are both fantastic books, not only exposition, but in thoroughness–pretty much anything you would want to know is contained in these two books. That said, if one is looking for a compactified, straight to the point, version of these books , I would suggest [8]. Ohsawa definitely is terse, but will give you the necessary facts and theorems that one may want to know–I will warn you that it is definitely an analyst’s book. Lastly, if one is looking for a broader, more geometric approach to SCV I would suggest [9].

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Anyways, coming back from that rant. The second issue about why such a locus is actually a Riemann surface is answered, much like in the case of smooth manifolds, by the implicit function theorem IFT. The intuition for the inverse function theorem is much like the case of the usual IFT Of course, we shall need an adaptation of the usual IFT so as that we get not just a smooth inverse but a holomorphic one. This is where the necessary groan should be emitted by anyone familiar with the proof of the smooth IFT (or at least the equivalent, inverse function theorem). Now, before you decide to skip the proof let me assure you–it’s not bad. In fact, following the general differences between real and complex analysis, not only is the holomorphic IFT much simpler than the real analog, but is much prettier. The basic idea is that by using a slight generalization of the Argument Principle we are able to explicitly right down a function which gives the roots of an equation within a given region. Moreover, we are able to write this function as an integral in holomorphic functions, and thus it will be a holomorphic function itself. See, not so bad?

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Loci of Holomorphic Functions and the IFT

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As we alluded to in the motivation we begin by defining a function $f:U\to\mathbb{C}$, with $U\subseteq\mathbb{C}^2$ open, to be holomorphic if it is holomorphic in each variable. For example, given open sets $W,V\subseteq\mathbb{C}$ we have the open set $W\times V=U\subseteq\mathbb{C}$ and for any pair of holmorphic functions $f:W\to\mathbb{C}$ and $g:V\to\mathbb{C}$ the function $F:U\to\mathbb{C}$ given by $F(w,v)=f(w)g(v)$ is holomorphic. Moreover, since any sum and product of holomorphic functions is obviously holomorphic we get a pretty good representative of holomorphic functions by considering any element of the ring $\mathbb{C}[w,v]$ of two variable polynomials with complex coefficients.

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As was stated earlier, there is an extremely deep theorem, due to Hartog, which is stated as follows:

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Theorem(Hartog’s): Any holomorphic function $F:U\to\mathbb{C}$ with $U\subseteq\mathbb{C}^2$ is continuous.

Proof: See [6] page 28. $\blacksquare$

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Now, you might be yawning or disappointed (or both), by this. I mean, that is the deep theorem that I seem to be paying such deference to? If the machinery needed to prove it doesn’t convince you its deep (admittedly, this is not always the best indication) perhaps this following corollary (sometimes called Hartog’s theorem itself) will convince you:

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Corollary: Any holomorphic function $F:U\to\mathbb{C}$ is locally expressible as a power series.

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Thus, we see that Hartog’s theorem really says that for a function to be holomorphic (analytic) in the traditional sense, it needs only be holomorphic in each variable. There is absolutely no analog of this in real variables. The classic example is given by

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$\displaystyle f(x,y)=\begin{cases}\displaystyle \frac{xy}{x^2+y^2} & \mbox{if}\quad (x,y)\ne(0,0)\\ 0 & \mbox{if}\quad (x,y)=(0,0)\end{cases}$

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which is $C^\infty$ in each variable separately, yet isn’t even continuous (at zero)!

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Anyways, let us denote for an open subset $U\subseteq\mathbb{C}$ the holomorphic functions $U\to\mathbb{C}$ by $\mathcal{O}(U)$. Then, for any $f\in\mathcal{O}(U)$ let us denote $f^{-1}(0)$ by $Z(f)$. Now, clearly $Z(f)$ inherits the subspace topology from $U$, but it is not obvious that $Z(f)$ is a topological manifold, let alone carries a natural complex structure. The crux of this lies in the following theorem:

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Theorem(Holomorphic IFT): Let $U\subseteq\mathbb{C}^2$ be a neighborhood of the origin and let $F\in\mathcal{O}$ be such that $F(0,0)=0$ and $F_v(0,0)\ne 0$. Then, there exists a neighborhood $O$ of $(0,0)\in\mathbb{C}^2$ such that for each $x_0\in\pi_1(O)$ the equation $F(x_0,y)=0$ has a unique solution $\varphi(x_0)$ with $(x_0,\varphi(x_0))\in O$. Moreover, the assignment $x\mapsto \varphi(x)$ is holomorphic $\pi_1(O)\to\varphi(\pi_1(O))$ [note that by the open mapping theorem, $\pi_1(O)$ is necessarily open].

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This tells us that locally loci of holomorphic functions $U\to\mathbb{C}$ look like the graphs of holomorphic functions in one variable. Now, clearly such graphs have canonical charts.

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References:

[1] Varolin, Dror. Riemann Surfaces by Way of Complex Analytic Geometry. Providence, RI: American Mathematical Society, 2011. Print.

[2] Miranda, Rick. Algebraic Curves and Riemann Surfaces. Providence, RI: American Mathematical Society, 1995. Print.

[3] Forster, Otto. Lectures on Riemann Surfaces. New York: Springer-Verlag, 1981. Print.

[4] Conway, John B. Functions of One Complex Variable. New York: Springer-Verlag, 1978. Print.

[5] Gong, Sheng. Concise Complex Analysis. Singapore: World Scientific, 2001. Print

[6] Hörmander, Lars. An Introduction to Complex Analysis in Several Variables. Princeton, NJ: Van Nostrand, 1966. Print.

[7] Krantz, Steven G. Function Theory of Several Complex Variables. New York: Wiley, 1982. Print.

[8] Ohsawa, T. Analysis of Several Complex Variables. Providence, RI: American Mathematical Society, 2002. Print.

[9] Ebeling, Wolfgang. Functions of Several Complex Variables and Their Singularities. Providence, RI: American Mathematical Society, 2007. Print.