2 Definitions of Holomorphic functions on Riemann surfaces

Question

In a lecture that I currently attend we defined Riemann surfaces and holomorphic mappings on it somewhat different than in another lecture that I attended a year ago.

My question is: Are these definitions of Riemann surfaces and holomorphic mappings between them (what I'm especially interested in) equivalent? I fail to show that. It would be nice, if that is true, since I want to use some results form the old lecture.

Let me write down the definitions.

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Definitions in lecture 1

Let $X$ be a one-dimensional complex manifold.

• A chart of $X$ is a homeomorphism $\phi: U \to V$, where $U \subset X$ open and $V \subset \mathbb{R}^n$ or $\mathbb{C}^n$ open.
• $A = \{ \phi_i: U_i \to V_i \ \text{ chart of } X | i \in I \text{ index set} \}$ with $X = \bigcup_{i \in I} U_i$ is called an atlas on $X$.
• Two charts $\phi_i: U_i \to V_i$ of $X$ with $i = 1,2$ is called holomorphically compatible if the transition map $\phi_2 \circ \phi_1^{-1}: \phi_1(U_1 \cap U_2 ) \to \phi_2(U_1 \cap U_2)$ is biholomorphic.
• We call an atlas analytical if the charts are pairwise biholomorphic compatible.
• Two analytical atlases $A_1,A_2$ are called analytically equivalent if every chart of $A_1$ is holomorphically compatible with every chart of $A_2$.
• The analytically equivalence of conformal atlases is an equivalence relation. We call an equivalence class under this relation a complex structure.

A Riemann surface is a pair $(X,S)$ with a connected one-dimensional complex manifold $X$ and a complex structure $S$.

Let $X$ and $Y$ be Riemann surfaces. A continuous mapping $X \to Y$ is called holomorphic, if for every pair of charts $\phi: U \to V$ of $X$ and $\psi: U' \to V'$ of $Y$ with $f(U) \subset U'$ the mapping $\psi \circ f \circ \phi^{-1}: V \to V'$ is holomorphic in the usual sense.

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Definitions in lecture 2 (current lecture)

• A geometric structure $O_X$ on a topological space $X$ is a collection of subrings $O_X(U) \subset C(U) := \{ f:U \to \mathbb{C} \}$ where $U$ runs through all open subsets such that the following conditions are satisfied:

  1. The constant functions are in $O(U)$.
  2. If $V \subset U$ are open sets then: $f\in O(U) \implies f|_V \in O(V)$
  3. Let $(U_i)_{i \in I}$ be a system of open subsets and $f_i \in O(U_i)$ such that $f_i|_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$ for all $(i,j)$ then there exists $f \in O(U)$ where $U = \bigcup _{i \in I} U_i$ with the property $f|_{U_i} = f_i$ for all $i$.

• A geometric space is a pair (X,O) consisting of a topological space and a geometric structure.

• A morphism $f:(X,O_X) \to (Y,O_Y)$ of geometric spaces is a continuous map $f:X \to Y$ with the following additional property. If $V \subset Y$ is open and $g \in O_Y(V)$, then $g \circ f$ is contained in $O_X (f^{-1}(V))$.

• A morphism $f:(X,O_X) \to (Y,O_Y)$ of geometric spaces is called an isomorphism if $f$ is topological and if $f^{-1}: (Y,O_Y) \to (X,O_X)$ is also a morphism. This means that the rings $O_X(U)$ and $O_Y(f(U))$ are naturally isomorphic.

A Riemann surface is a geometric space $(X,O_X)$ such that for every point there exists an open neighborhood $U$ and an open subset $V \subset \mathbb{C}$ such that the geometric spaces $(U,O_X|_U)$ and $(V,O_V)$ are isomorphic geometric spaces. We always assume that $X$ is a Hausdorff space with countable basis of topology.

A map $f:(X,O_X) \to (Y,O_Y)$ between Riemann surfaces is called holomorphic if it is a morphism of geometric spaces.

• A chart on a Riemann surface $X$ is a topological mao from an open subset $U \subset X$ onto an open subset $V \subset \mathbb{C}$. The chart is called analytic if its moreover biholomorphic, i.e. an isomorphism of geometric spaces $(U,O_X|_U) \to (V,O_V)$.

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Answer 1

First notice that in your first definition of Riemann surface you shouldn't start with the hypothesis that $X$ is a 1-dimensional complex manifold, since this means that you are already considering, implicitely, a fixed maximal analytical atlas on it. In other words, Riemann surface and (connected) 1-dimensional complex manifold are synonyms. Just assume that $X$ is a Hausdorff second-countable topological space (or, if you prefer, a topological or smooth 1-manifold).

So let $X$ be a connected Hausdorff second-countable topological space.

If we have an analytical atlas, then, as you wrote, we also have a notion of holomorphic functions $\varphi:U\subset X\rightarrow\mathbb{C}$. Defining \begin{equation} O_X(U):=\{\varphi:U\rightarrow\mathbb{C}\ \ holomorphic\} \end{equation} for each $U\subset X$ open, this gives a geometric space $(X,O_X)$, just because constant functions are holomorphic and being holomorphic is a local property.

Viceversa, assume that you have a geometric space $(X,O_X)$. We will construct from it an analytic atlas on $X$. First you need to notice that you have a model geometric structure $O_\mathbb{C}$ on $\mathbb{C}$ made by holomorphic functions, and that if $V_1,V_2\subset\mathbb{C}$ are open sets, then a map $f:V_1\rightarrow V_2$ is holomorphic if and only if it induces a morphism $f:(V_1,O_\mathbb{C}|_{V_1})\rightarrow(V_2,O_\mathbb{C}|_{V_2})$ of geometric structures.

For each $x\in X$, take an open neighbourhood $U_x\subset X$ of $x$ and an isomorphism $\phi^{(x)}:(U_x,O_X|_{U_x})\rightarrow(\phi^{(x)}(U_x),O_\mathbb{C}|_{\phi^{(x)}(U_x)})$, with $\phi^{(x)}(U_x)\subset\mathbb{C}$ open. In particular, $\phi^{(x)}:U_x\rightarrow \phi^{(x)}(U_x)$ is a chart on $X$. When two open sets $U_x$ and $U_{x'}$ overlap, you get an isomorphism \begin{equation} (\phi^{(x)}(U_x\cap U_{x'}),O_\mathbb{C}|_{\phi^{(x)}(U_x\cap U_{x'})})\rightarrow(U_x\cap U_{x'},O_X|_{U_x\cap U_{x'}})\rightarrow(\phi^{(x')}(U_x\cap U_{x'}),O_\mathbb{C}|_{\phi^{(x')}(U_x\cap U_{x'})})\,, \end{equation} which means that the transition map $\phi^{(x)}(U_x\cap U_{x'})\rightarrow \phi^{(x')}(U_x\cap U_{x'})$ between the two charts is biholomorphic. Thus $\{(U_x,\phi^{(x)})\}$ is an analytic atlas on $X$.

You can easily see that these two constructions are one the inverse of the other. Finally, at this point it is clear that a map $f:X_1\rightarrow X_2$ between two Riemann surfaces is holomorphic according to the first definition (i.e. it is represented by holomorphic maps in analytic charts) if and only if it is holomorphic according to the second definition (i.e. it pulls back holomorphic functions on $X_2$ to holomorphic functions on $X_1$).