Maximal analytic continuation

Question

I'm currently reading about the concept of a “maximal analytic continuation” from Forster's book Lectures on Riemann Surfaces (see Section 7). There are a bunch of definitions to unpack before I can ask my question, so I'll start with those.

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Definitions

Let $\mathcal{O}$ be the sheaf of germs of holomorphic functions over a Riemann surface $X$. Given a germ $\varphi \in \mathcal{O}_x$ ($\mathcal{O}_x$ denotes the stalk of $\mathcal{O}$ over the point $x \in X$), we define an analytic continuation of $\varphi$ as a quadruplet $(Y, p, f, y)$ satisfying:

1. $Y$ is a Riemann surface, and $p : Y \to X$ is an unbranched holomorphic map (and thus a local diffeomorphism).
2. $f : Y \to \mathbb{C}$ is a holomorphic map on $Y$.
3. $p(y) = x$.
4. $[f \circ (p|_{U_y})^{-1}]_x = \varphi$, where $U_y$ is a small neighborhood about $y$ such that the restriction $p|_{U_y}$ is a diffeomorphism.

Here's an example of an analytic continuation to provide some intuition. Suppose $X = \mathbb{C} \setminus \{0\}$, and suppose $\varphi$ is a germ associated to the complex logarithm at a point $x \in X$. Let $Y$ be the “graph” $$ Y := \{(z, w) : z \in \mathbb{C} \setminus \{0\}, \exp(w) = z\}. $$ ($Y$ is just the set of ordered pairs $(z, \log z)$ over all branches of the logarithm.) Let $p : (z, w) \mapsto z$ and $f : (z, w) \mapsto w$ be the associated projection maps. Setting $y$ to be the value of the germ $\varphi$ at $x$, one can show that $(Y, p, f, y)$ is an analytic continuation of $\varphi$. Viewing $Y$ as a subset of $\mathbb{R}^4$ and projecting it onto its first, second, and fourth coordinates, we see that $Y$ is isomorphic to a helicoid: the Riemann surface that the complex logarithm has classically been defined over.

Forster further defines the notion of the maximal analytic continuation of a germ, which he makes rigorous by defining via a universal property. We can always construct the maximal analytic continuation of a germ $\varphi$ as follows. We define the étalé space $E(\mathcal{O})$ associated to $\mathcal{O}$ as the disjoint union of all stalks of $\mathcal{O}$ (i.e., the set of all germs). The étalé space is a Hausdorff topological space, and has the structure of a Riemann surface via the projection map $\pi : [g]_x \mapsto x$. Let $Y$ be the connected component of the étalé space $E(\mathcal{O})$ that contains the germ $\varphi$. Setting $f([g]_x) := g(x)$, one can show that $(Y, \pi, f, \varphi)$ is the maximal analytic continuation of $\varphi$.

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Question

My issue is that this definition does not seem to be the “correct” definition of analytic continuation. In particular, the maximal analytic continuation of a germ $\varphi$ depends on the choice of the space $X$. This is easily seen from Forster's construction of a maximal analytic continuation. If $\varphi$ is a germ about a point $x \in X$, we can repeat the construction of the maximal analytic continuation with the space $X$ replaced by another space $X' \subseteq X$ that contains the point $x$. However, the sheaf $\mathcal{O}_{X'}$ associated to $X'$ may not be the same as the sheaf $\mathcal{O}_X$ associated to $X$, so this may yield two different maximal analytic continuations! Meanwhile, the germ $\varphi$ is a local property, so it seems nonsensical for its analytic continuation to depend on the space $X$ it lives in.

Is there some purpose to Forster's definition that I'm not understanding? Perhaps, is there some alternative notion of a “maximal analytic continuation” that is independent of the choice of $X$?

Answer 1

I think you're certainly right that one can define the category of quadruples $(Y, p, f, y)$ where

• $Y$ is a Riemann surface and $y \in Y$ is a point,
• $p$ is an equivalence class of open embeddings $U \hookrightarrow Y$ that are biholomorphisms onto the image, where $U$ is a small open neighborhood of $x \in X$ and $p(x) = y$,
• $f$ is a holomorphic function on $Y$ extending the germ in $\mathcal{O}_y$ corresponding to $\varphi \in \mathcal{O}_x$.

Then we can ask whether there exists a final object in this category, and define that to be the maximal analytic continuation of $\varphi \in \mathcal{O}_x$.

It's just that this is completely uninteresting. Let's take the example of $X = \mathbb{C} \setminus \{0\}$ and the germ of the logarithm function at $x = 1$. The final object simply turns out to be $Y = \mathbb{C}$ with $f(x) = x$, where $p$ is the logarithm function around a neighborhood of $1$. Indeed, if $(Y^\prime, p^\prime, f^\prime, y^\prime)$ is any analytic continuation, then the function $f^\prime$ defines the morphism $Y^\prime \to \mathbb{C} = Y$.

What's happening is that equivalence classes of germs $(X, x, \varphi)$ are classified precisely by the value $\varphi(x) \in \mathbb{C}$ and the degree of the first nonzero non-constant term in the Taylor expansion of $\varphi$. When $\varphi^\prime(x) \neq 0$, the terminal object will always be $Y = \mathbb{C}$. It's probably true that when $\varphi^\prime(x) = 0$ there is no terminal object in the category of analytic continuations.