Question about the definition of holomorphic maps between riemann surfaces

Question

The definition of holomorphic maps between Riemann surfaces confuses me a little. When we say a (continuous) map between two Riemann surfaces $f: X \to Y$ is holomorphic, we mean that for any pair of charts $\varphi: U_1\subseteq X \to V_1$ and $\psi: U_2 \subseteq Y \to V_2$, the composition $\psi \circ f\circ \varphi^{-1}: \varphi(U_1 \cap f^{-1}(U_2))\to \psi(f(U_1)\cap U_2)$ is differentiable at every point in its domain.

Now the word 'charts' here confuses me. Do we require the charts to be in the complex atlases of $X$ and $Y$? Or do we require the charts to be in the maximal atlases on $X$ and $Y$? Or do we just take arbitrary local homomorphisms $\varphi, \psi$ regardless of the atlases on $X$ and $Y$?

In the first case, will the definition of holomorphic maps depends on the specific atlases (not necessarily maximal) we impose on $X$ and $Y$?

It seems to me that the second case is somehow equivalent to the first one as we can simply write, for any charts $\varphi',\psi$ in the maximal atlases, $$\psi'\circ f\circ\varphi'^{-1}=(\psi'\circ\psi^{-1})\circ(\psi\circ f\circ\varphi^{-1})\circ(\varphi\circ\varphi'^{-1})$$ given some charts $\varphi, \psi$ in the given atlases. But I am not quite sure because even the same author may refer to different things when they mention the word 'chart' in different places. Thanks in advance.

Answer 1

For example, $X=\mathbb{C}$ has two incompatible atlases.

The first analytic structure on $X$ is to just take the usual structure. That is the atlas consists of the charts where $U\to U\subset \mathbb{C}, z\mapsto z$.

The second analytic structure on $X$ is to take conjugates of everything. That is the atlas consists of all the charts where $U\to \mathbb{C}, z\mapsto \bar{z}$. You might think the transition functions are analytic but they are; $\phi\psi^{-1}: \bar{z}\mapsto \bar{z}$, if you follow the definitions. That is $\phi\psi^{-1}$ is an identity map when restricted to where it is defined.

Let's denote $X$ with the first analytic structure $X_1$ and the second structure $X_2$.

So observe that $f:X_1\to X_1,p\mapsto p$ is a valid holomorphic map of complex Riemann surfaaces but $f: X_1\to X_2, p\mapsto p$ is not. That is not to say there is no holomorphic map from $X_1$ to $X_2$.

What we see here is that no, you cannot take arbitrary homeomorphisms (as is case 2 of your question) in your definition of $\phi f \psi^{-1}$. Indeed, even the same space with different a different complex Riemann surface structure will give different answers as to whether this map $f$ is a holomorphic map or not.