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.
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:
- $Y$ is a Riemann surface, and $p : Y \to X$ is an unbranched holomorphic map (and thus a local diffeomorphism).
- $f : Y \to \mathbb{C}$ is a holomorphic map on $Y$.
- $p(y) = x$.
- $[f \circ (p|_{U_y})^{-1}]_x = \varphi$, 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 $\varphi$.
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 $X$?
