I'm quiet new to complex geometry (or even complex analysis actually) so I apologize in advanced if this question sounds silly or trivial.
I have been having the following confusion: So we know that, on a compact, connected complex manifold, the only holomorphic functions are the constant ones. But in practice, I don't see why such functions cannot be directly defined? What would stop one from doing so?
For example, if I take the simplest compact, connected complex manifold I know of: $\mathbb{CP^1}$. This is defined by the space $$\{ (z,z') \in \mathbb{C}^2 \ / \ (z,z') \neq 0 \}$$ such that $ (z,z') = \lambda (z,z') $ is identified for every $\lambda \neq 0$. Then we can get an open set $$ U = \{ (z,1) \in \mathbb{CP}^1 \ / \ z \in \mathbb{C} \} $$ and map it holomorphically to $\mathbb{C}$ via $ \varphi(z,1) = z $. Similarly we can do $$ U' = \{ (1,z') \in \mathbb{CP}^1 \ / \ z' \in \mathbb{C} \} $$ so that $ \mathbb{CP^1} = U \cup U' $ and so this is an atlas.
My question is that, why can't I just define locally a function $ f: U \rightarrow \mathbb{C} $ which is holomorphic? For example, I think $ f(z)=f(x+iy) = e^{ix-y} $ could work. Once this is done, then on the intersection of $U \cap U'$ I can just plug in the transition functions to get another representation $ f':U' \rightarrow \mathbb{C} $, where $ f' = f \circ \varphi \circ \varphi' $ and get a globally defined holomorphic function on $M$.
Certainly everything above is nonsense, that's why I'm asking. Please clarify my thoughts!