When I was learning deformation theory of Kodaira from his book 《complex manifolds》, I feel the definition of deformation of complex structures is rather strange and abstract, is seems like a lot of fibers live in the same base space which satisfies some restrictive conditions, each one called a deformation of others, but from the literal meaning, deformation should only talk about the changing of the space itself, not about the extra space, so I want to visualze the process of deformation, but I failed.
I know if there exist a biholomorphic map between two complex structures, we say they share the same complex structure, I know for 1-dimensional complex projective space $\mathbb CP^1$, there is only one complex structure, but for 1-dimensional complex torus, there are infinite many complex sturctures not biholomorphic to each other, I learned deformation theory a lot, remembering a lot of facts and theorems about deformation theory, but I never feel I really know what we mean when we say a complex structure. So I try to ask myself the simplest question, what's the simplest and most visuable deformation? first I feel that maybe $\mathbb C^1$ is the most visuable one, and I find it surprising that I have never thought about deforming $\mathbb C^1$ before! the simplest case! So I try to deform the complex structure of $\mathbb C^1$, for one person in some place draw a complex coordinate $\mathbb C^1$, and another person in some other place draw another complex coordinate, when can we say these two coordinate share the same complex structure?
For $z\in\mathbb C^1$, add a arbitrary constant or multiply a constant won't change its complex structure, so if the other coordinat is $w=az+b$, we can always say that they share the same complex structure, but if the other person makes his "$1=1,i=2i$" of the original one, they may have different complex structures? So the complex structure may have something to do with ratio of $x$-axis and $y$-axis? And if it is right, from this point of view, I think there are infinite many different complex structures of $\mathbb C^1$, but what's the dimension of the moduli space of the complex structures of $\mathbb C^1$? Has anyone else think the same question before?