June 17: Understanding Riemann Surface

This post is about something from my algebraic geometry class that has helped me to better understand Riemann surface and the ‘structure’ that we can define on a surface.

Given 4 distinct points on the complex plane, why is the Riemann surface structure that we obtain independent of the cutting (i.e. we assume our cutting to be smooth and non-intersecting)?

Let the 4 points be . The Riemann surface that we obtain is the graph of .

Different cuttings correspond to different choices of branch cuts, but topologically, they are all the graphs of the same function.

Now, we need to show that these different cuttings still give us the same Riemann surface structure. Key: what is the structure that we put on the surface?

The structure is given by a holomorphic function that we define on the surface. But how do we such function?

In neighbourhoods away from the branch points, we can take the function to be ; this will be holomorphic and bijective (i.e. away from the branch points, the surface locally resembles the complex plane). Locally, this function (which is holomorphic) generates the space of holomorphic functions that is well-defined on this region. Key: viewing a surface as the graph of a multivalued function automatically gives us a Riemann surface structure.

The following pictures is the graph of restricted to the real plane. For a region away from the branch point, we can actually find 2 holomorphic functions, as shown below:

My question: would the space of holomorphic functions defined on this patch be the ring of functions generated by and ?

Notice at the branch point, we have only ONE choice for our holomorphic function:

Now, returning back to our original question with the 4 distinct points. What happens near a branch point, say ? Since the 4 points are distinct, then in a neighbourhood around . Hence, in this neighbourhood, the surface is isomorphic to near . The local structure near is isomorphic to that of (this will be the same as the example we did above, but shifted). The same could be said for the other 3 branch points.

The isomorphism between the surfaces arises from the fact that the holomorphic functions that we can define on each of the surface are the same (i.e. away from the branch points and in neighbourhoods of branch points). Note: these are not just holomorphic functions that we have found, they are also bijective. Hence, at each patch, we have found ourselves a local isomorphism between the surfaces. We can extend this isomorphism from patch to patch to obtain a global isomorphism.

Something that I’m still thinking about: the isomorphism between Riemann surfaces defined as graphs of multivalued functions is really only dependent of the monodromy group of our graph (removing bad points). It makes sense because from the above, we see that the only parts where the structure really gets interesting is at the branch points. But why the monodromy group though? Why cannot we simply identify using branch points?