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I've taken a small course in Riemann surfaces, and there is one part that I still don't understand (and I've been unable to find a reference that explains this rigorously and in detail).

It is about the construction of a Riemann surface associated to an analytic germ.

By analytic germ we mean a couple $(z_0, (a_n)_{n \in \mathbb{N}})$ where $z_0$ and the $a_i$ are in the complex plane, and the series $\sum a_n (z-z_0)^n$ has strictly positive (but typically finite) radius of convergence.

Then (without getting into technical details) the Riemann surface associated to such a germ is the connected component of that germ in the space of all analytic germs, equipped with a certain topology. I get that it defines a Riemann surface.

I also understand "intuitively" what it does on the classic examples : for example, the Riemann surface associated to $(1,\log)$ is a sort of "spiral surface", since when we turn around zero, we add (if turning counter-clockwise) a $2 i \pi$ to the principal determination (so there are countably many sheaves).

In the case of the germ of the square root at $z_0=1$, it is a two-sheaves surface.

However I have no idea how one would go to determine this properly. In this example, I know that this surface is biholomorphic to $\mathbb{C}$ in the case of the log but I have no idea how to prove that.

Can anyone either give me a detailed reference or a sketch of the proof (but with all the key arguments) ?

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2 Answers 2

What you are describing is more or less Weierstrass's construction of the Riemann surface of a multi-valued function. It is a very intuitive construction, but, as you point out, not convenient to do concrete calculations with. If you want to explicitly describe the Riemann surface obtained by analytic continuation of a given germ, there is not much choice other than to go back to the way in which the particular germ was obtained - the behavior of a power series under analytic continuation can be very complicated and can't readily be predicted simply by looking at the coefficients $\{a_n\}$.

For example, if you want to show that $\log z$ increases by $2\pi i$ with every counter-clockwise turn of $z$ around the origin, it is much better to go back to the definition of $\log z$ than to start from the power series expansion of its principal branch around $z=1$, although a brute-force calculation is certainly possible. What is $\log z$? By definition, $\log z = \int_1^z dt/t$. Of course, this is only well-defined up to the choice of path from $1$ to $z$, which results in an ambiguity of $2\pi i n$ in the value of the function. What we see, in fact, is that $\log z$ is well-defined as a function on the universal cover of the punctured plane, as soon as we specify its value at any given point. This universal cover $E$ can be intuitively thought of as a helicoid, as you point out. And, as you say, it is biholomorphic to $\mathbb C$. The function $\log$, well-defined on $E$, is the isomorphism you are looking for. Think about it!

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well ok, but then how do yo see that the covering surface is simply connected ? also how would you do that if instead of log, which as you point out may be defined using a multivalues integral, the germ is something like $\sqrt{(z-a)(z-b)}$ ? also is it clear is this last example that all points different from a and b may be base points ? (i.e. the covered surface is the twice punctured plane) –  Glougloubarbaki Nov 29 '11 at 20:25
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The problem is that this is AFAIK treated in older texts, those which are still influenced by the first book by Weyl ("The Concept of a Riemann Surface").

You coud try e.g.

  • George Springer: "Introduction to Riemann Surfaces", chapter 3: "Riemann Surface of an Analytic Function"
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thanks ! I've begun reading it and it seems it covers this in detail. –  Glougloubarbaki Nov 29 '11 at 11:18
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