We know what complex manifolds are. They’re entities which “locally” look like . We also know about transition functions. However, today we’re going to ask an important question: a question that impedes all progress in modern math- **why** manifolds.
We can sort of understand why manifolds have the condition that each point lies inside a neighbourhood which is homeomorphic to an open set in . This allows us to do a lot of things on the manifold, because we know how to do those things on . Calculus is just the tip of the iceberg. It allows us to establish a metric (at least locally), and we also gain a lot of intuition as to how the manifold “looks” if we zoom in a lot. Which sometimes is enough.
However, why transition functions? Why can we not just “continuously” map neighbourhoods to neighbourhoods to , mapping the intersection of two neighbourhoods to the same points in ?? After all, isn’t a manifold just a slight perturbed, slightly wavy copy of ? Why are we mapping two overlapping sets to humungously different open sets, mapping points in the intersection to different points in , and then just ensuring that is holomorphic? (I realize that I have not specified what and are, but the reader who’s read up on complex manifolds will easily be able to infer this). This is because we want to be able to study objects that locally look like , but are not all slightly perturbed, wavy versions of .
Consider . It is easy to see that it locally looks like . However, there’s a major different between and : is compact while is not. Hence, there is a major global property that differentiates them, and prevents even a homeomorphism between them. We cannot “continuously” map the open neighbourhoods in into . To push an image, imagine “continuously” mapping open neighbourhoods on to . On the ball, you eventually loop around, and move towards where you started. However, on , you just keep going further. These two are incompatible.
Hence, we have to weaken what we can ask for. We need to make smaller demands of our mathematical gods. We cannot have a continuous mapping of neighbourhoods. However, we can at least ensure that the transition functions are holomorphic.