Finally, it’s time to get to the definition of a Riemann surface. A Riemann surface is a connected one-dimensional complex manifold. In other words, it’s a Hausdorff space {M} which is locally homeomorphic to {\mathbb{C}} via charts (i.e., homeomorphisms) {\phi_i:U_i \rightarrow V_i} for {U_i \subset M, V_i \subset \mathbb{C}} open and such that {\phi_j \circ \phi_i^{-1}: V_i \cap V_j \rightarrow V_i \cap V_j} is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of {\mathbb{C}} is a Riemann surface in a natural manner.

The Riemann sphere {P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}} or {S^2} is a Riemann sphere with the open sets {U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}} and the charts

\displaystyle \phi_1 =z, \ \phi_2 = \frac{1}{z}.

The transition map is {\frac{1}{z}} and thus holomorphic on {U_1 \cap U_2 = \mathbb{C}^*}.

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair {(f,V)} where {f: V \rightarrow \mathbb{C}} is holomorphic and {V \subset \mathbb{C}} is an open disk. Two function elements {(f,V), (g,W)} are said to be direct analytic continuations of each other if {V \cap W \neq \emptyset} and {f \equiv g } on {V \cap W}. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element {\gamma = (f,V)}, we can consider the totality {X} of all equivalence classes of function elements that can be obtained by continuing {\gamma} along curves in {\mathbb{C}}. Then {X} is actually a Riemann surface.

Indeed, we must first put a topology on {X}. If {(g,W) \in X} with {W=D_r(w_0)} centered at {w_0}, then let a neighborhood of {g} be given by all function elements {(g_w, W')} for {w \in W, W' \subset W}; these form a basis for a suitable topology on {X}. Then the coordinate projections {(g,W) \rightarrow w_0} form appropriate local coordinates. In fact, there is a globally defined map {X \rightarrow \mathbb{C}}, whose image in general will be a proper subset of {\mathbb{C}}.

Basic facts

Since we have local coordinates, we can define a map {f: X \rightarrow Y} of Riemann surfaces to be holomorphic or regular if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map {f: X \rightarrow \mathbb{C}}; for meromorphicity, this becomes {f: X \rightarrow S^2}.

Many of the usual theorems of elementary complex analysis (that is to say, the local ones) transfer immediately to the case of Riemann surfaces. For instance, we can locally get a Laurent expansion, etc.

Theorem 1 Let {f: X \rightarrow Y} be a regular map. If {X} is compact and {f} is nonconstant, then {f} is surjective and {Y} compact.

 

To see this, note that {f(X)} is compact, and by the open mapping theorem, open, so the result follows by connectedness of {Y}.

Complexified differentials

Since a Riemann surface {X} is a smooth 2-dimensional manifold, it is possible to do the usual exterior calculus. We could consider a 1-form to be a section of the (usual) cotangent bundle {T^*(X)}, but it is more natural to take the complexified cotangent bundle {\mathbb{C} \otimes_{\mathbb{R}} T^*(X)}, which I will in the future just abbreviate {T^*(X)}; this will not be confusing since I will only do this when I talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.

I think the natural setting of all this leads to the theory of almost complex manifolds, which sometime soon I will try to learn and/or blog about.

If {z = x+iy} is a local coordinate on {X}, defined say on {U \subset X}, define the (complex) differentials

\displaystyle dz = dx + i dy , \ d\bar{z} = dx - idy.

These form a basis for the complexified cotangent space at each point of {U}. There is also a dual basis

\displaystyle \frac{\partial}{\partial z } := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right), \ \frac{\partial}{\partial \bar{z} } := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)

for the complexified tangent space.

I now claim that we can split the tangent space {T(X) = T^{1,0}(X) + T^{0,1}(X)}, where the former consists of multiples of {\frac{\partial}{\partial z}} and the latter of multiples of {\frac{\partial}{\partial \bar{z}}}; clearly a similar thing is possible for the cotangent space. This is always possible locally, and a holomorphic map preserves the decomposition. One way to see the last claim quickly is that given {g: U \rightarrow \mathbb{C}} for {U \subset \mathbb{C}} open and {0 \in U} (just for convenience), we can write

\displaystyle g(z) = g(0) + Az + A' \bar{z} + o(|z|)

where {A = \frac{\partial g }{\partial z }(0), A' = \frac{\partial g }{\partial \bar{z} }(0)}, which we will often abbreviate as {g_z(0), g_{\bar{z}}(0)}. If {\psi: U' \rightarrow U} is holomorphic and conformal sending {z_0 \in U' \rightarrow 0 \in U}, we have

\displaystyle g(\phi(\zeta)) = g(\phi(0)) + A \phi'(z_0)(\zeta-z_0) + A' \overline{ \phi'(z_0)(\zeta-z_0)} + o(|z|);

in particular, {\phi} preserves the decomposition of {T_0(\mathbb{C})}.

Given {f: X \rightarrow \mathbb{C}} smooth, we can consider the projections of the 1-form {df} onto {T^{1,0}(X)} and {T^{0,1}(X)}, respectively; these will be called {\partial f, \overline{\partial} f}. Similarly, we define the corresponding operators on 1-forms: to define {\partial \omega}, first project onto {T^{0,1}(M)} (the reversal is intentional!) and then apply {d}, and vice versa for {\overline{\partial} \omega}.

In particular, if we write in local coordinates {\omega = u dz + v d\bar{z}}, then

\displaystyle \partial \omega = d( v d \bar{z}) = v_z dz \wedge d\bar{z},

and

\displaystyle \overline{\partial} \omega = d( u dz) = u_{\bar{z}} d\bar{z} \wedge dz.

To see this, we have tacitly observed that {dv = v_z dz + v_{\bar{z}} d\bar{z}}.

Admittedly this isn’t the best place to stop.  Next time, I’ll discuss more about how one can manipulate 1- and 2-forms.  Pretty soon we can then get to some big theorems.

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