A bit of a general question, but here goes. Morally, what is the space of germs of a holomorphic function?
I know that a germ is simply an equivalence class of function elements, where we regard two function elements as equivalent at a point if they agree on some open neighbourhood of that point. Moreover I know the definition that the space of germs is simply the union of these equivalence classes for all functions $f$ and points $x$.
This is all a bit abstract at the moment though. I can't see how germs are useful, or how I might calculate them for a concrete function. Has anyone got any nice examples of calculations of the space of germs? And could someone explain the overarching idea behind them in Riemann Surfaces?
Many thanks.
Best Answer
Consider the set of all germs of holomorphic functions at a specific point $x$. If you know of stalks, then this is just $\mathcal{O}_x$. You can think of this set as a collection of all "possible functions" which are holomorphic at that point.
To be precise, you can think of it as the set of all power series which converge in a little neighborhood of $x$. So $\mathcal{O}_x$ is isomorphic to the ring $\mathbb{C}\{z-x\}$ of all convergent power series in $z-x$.
Of course, two functions define the same germ if their power series about $x$ are the same. This gives a method to calculate the germs of a function $f$ at points $x$.
Now the union of all germs of all functions is useful for instance because it allows for the construction of a maximal analytic continuation of a given holomorphic function:
For a Riemann Surface $X$, a point $x\in X$ and a function germ $f$ at $x$, you get a Riemann Surface $Y$ together with an unbranched holomorphic map $Y\rightarrow X$ and a holomorphic function $F$ on all $Y$, such that the germ of $F$ is in some natural way the same as that of $f$.
Now the space of germs allows for the construction of a "maximal $Y$". You can think of it as the largest possible domain of definition for $f$. Because there may be different continuations of a representative of $f$ in different neighborhoods, you have to consider them all and combine them to get your larger space $Y$.
The actual construction and proofs can be read, for instance, in O. Forster's "Lectures on Riemann Surfaces".