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Section 2.2 Riemann Surfaces I (Ricky)

Subsection 2.2.1 Definitions

Definition 2.2.1.

A topological surface is a Hausdorff space \(X\) wich has a collection of charts

\begin{equation*} \{\phi_i \colon U_i \xrightarrow\sim \phi_i(U_i) \subseteq \CC,\,\text{open}\}_{i\in I} \end{equation*}

such that

\begin{equation*} X= \bigcup_{i\in I} U_i\text{.} \end{equation*}

We call \(X\) a Riemann surface if the transition functions \(\phi_i\circ \phi_j^{-1}\) are holomorphic.

Subsection 2.2.2 Examples

Example 2.2.2.

Open subsets of \(\CC\text{,}\) e.g.

\begin{equation*} \CC \end{equation*}
\begin{equation*} \mathbf D = \{z\in \CC : |z| \lt 1 \} \end{equation*}
\begin{equation*} \HH = \{z\in \CC : \im z \gt 0 \}\text{.} \end{equation*}
Example 2.2.3.

\(\hat \CC = \) Riemann sphere \(= \CC\cup \{\infty\}\text{.}\) A basis of neighborhoods of \(\infty\) is given by

\begin{equation*} \{z\in \CC : |z| \gt R \} \cup \{\infty\}\text{.} \end{equation*}
Example 2.2.4.
\begin{equation*} \PP^1(\CC )= \{ [z_0 :z_1 ] : (z_0,z_1) \ne (0,0)\} \end{equation*}
\begin{equation*} U_0 =\{[z_0,z_1] : z_0 \ne 0\}\to \CC \end{equation*}
\begin{equation*} [z_0:z_1] \mapsto \frac{z_1}{z_0} \end{equation*}
\begin{equation*} U_1 =\{[z_0,z_1] : z_1 \ne 0\}\to \CC \end{equation*}
\begin{equation*} [z_0:z_1] \mapsto \frac{z_0}{z_1}\text{.} \end{equation*}
Example 2.2.5.

Let \(\Lambda = \ZZ \oplus \ZZ i \subseteq \CC\) then \(X = \CC/\Lambda\) is a Riemann surface.

Subsection 2.2.3 Morphisms

Definition 2.2.6. (Holo/Mero)-morphisms of Riemann surfaces.

A morphism of Riemann surfaces is a continuous map

\begin{equation*} f\colon S\to S' \end{equation*}

such that for all charts \(\phi, \psi\) on \(S, S'\) respectively we have \(\psi \circ f \circ \phi\inv\) is holomorphic.

We call a morphism \(f\colon S\to \CC\) a holomorphic function on \(S\text{.}\)

We say \(f \colon S \to \CC\) is a meromorphic function is \(f\circ \phi\inv\) is meromorphic.

The set of meromorphic functions on a Riemann surface form a field.

We denote the field of meromorphic functions by \(\mathcal M (S)\text{.}\)

Let \(f\colon \hat\CC \to \CC\) be meromorphic. Then the number of poles of \(f\) is finite say at \(a_1, \ldots, a_n\text{.}\) So, locally at \(a_i\) we can write

\begin{equation*} f(z) = \sum_{j=1}^{j_i} \frac{\lambda_{j,i}}{(z-a_i)^j} + h_i(z) \end{equation*}

with \(h_i\) holomorphic. Then

\begin{equation*} f(z) - \sum_{i=1}^n \sum_{j=1}^{j_i} \frac{\lambda_{j,i}}{(z-a_i)^j} \end{equation*}

is holomorphic everywhere. By Liouville's theorem this is constant.

We say \(S,S'\) are isomorphic if \(\exists f\colon S\to S'\text{,}\) \(g\colon S'\to S\) morphisms such that \(f\circ g = \id_{S'}\text{,}\) \(g\circ f = \id_{S}\text{.}\)

Show that

\begin{equation*} \hat \CC \simeq \PP^1(\CC)\text{.} \end{equation*}
Remark 2.2.10.

\(\CC \not\simeq \mathbf D\) by Liouville.

If \(S, S'\) are connected compact Riemann surfaces, then any nonconstant morphism \(f\colon S \to S'\) is surjective. (Nonconstant holomorphic maps are open)

Subsection 2.2.4 Ramification

Definition 2.2.11. Orders of vanishing.

The order of vanishing at \(P\in S\) of a holomorphic function on \(S\) is defined as follows: For \(\phi\) a chart centered at \(P\) write

\begin{equation*} f\circ \phi\inv (z) = a_n z^n + a_{n+1}z^{n+1} + \cdots,\,a_n\ne 0 \end{equation*}

then \(\ord_P(f) = n\text{.}\)

More generally, for \(f\colon S \to S'\) we can define \(m_P(f)\) (multiplicity of \(f\) at \(P\)) by using a chart \(\psi\) on \(S'\) and setting

\begin{equation*} m_P(f) = \ord_P(\psi\circ f)\text{.} \end{equation*}

If \(m_P(f)\ge 2\) then we call \(P\) a branch point of \(f\) and call \(f\) ramified at \(P\text{.}\)

Example 2.2.12.
\begin{equation*} f\colon \CC \to \CC,\,f(z) = z^2\text{.} \end{equation*}

The chart \(\phi_a(z) = z-a\) is centered at \(a \in \CC\text{.}\) Then to compute \(m_a(f)\) we compute

\begin{equation*} f\circ \phi\inv_a(z) = a^2 + 2az + z^2 \end{equation*}

hence

\begin{equation*} \ord_a(f) =\begin{cases} 0,\amp \text{ if } a\ne 0\\ 2,\amp\text{ if } a= 0\end{cases}\text{.} \end{equation*}

Subsection 2.2.5 Genus

Given such an oriented polygon coming from a Riemann surface, we can associate a word \(w\) to it from travelling around the perimeter.

Example 2.2.15.

For the sphere \(w = a\inv ab\inv bc\inv c\text{.}\)

Example 2.2.17.

\(w_1 = a_1b_1 a_1\inv b_1 \inv\text{.}\)

\(w_2 = a_1b_1 a_1\inv b_1a_2b_2 a_2\inv b_2 \inv\text{.}\)