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%% \begin{frame}
%% \title[Calculus III]{Calculus III -- Recitation 1}
%% \author{Prof. Cueto}
%% \institute{The Ohio State University}
%% \date{January 14, 2016}
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%% \title[Calculus III]{Calculus III -- Recitation 1}
%% \author{Prof. Cueto - The Ohio State University}
%% \date{January 14, 2016}
%% \maketitle
\begin{center}
  \Large{\textsf{\textbf{MATH 8140 - Topics in Algebraic Geometry (Riemann surfaces)}}}\\
\vspace{1ex}
  \Large{\textsf{\textbf{Homework 1}}}\\
\vspace{2ex}

\Large{\textbf{\textsf{Basic definitions of Riemann surfaces and holomorphic maps}}}\normalsize

\vspace{1ex}

\end{center}
\normalsize

Please indicate any source in the literature used in finding the solution to a given problem. You are encouraged to work in teams. If you do so, please indicate the name of your collaborator(s) on your submission.
\vspace{1ex}

Solutions to each problem can be uploaded (on Carmen) by at most one student. There is no deadline, so work at your own pace.
\vspace{1ex}

Please use the following name for the file you upload \texttt{HW\#\_Problem\#.pdf}.
\vspace{1ex}


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%% {Topics: {the Tropical semiring, valuations, polyhedra, tropical hypersurfaces.}}\normalsize


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%% 	\mode<presentation>{\frametitle{Vectors and vector-valued functions}}

\vspace{1ex}
\begin{problem} \textbf{(Riemann surfaces from graphs of holomorphic functions)}\\
  Fix a connected open subset $V$ of $\CC$.
  \begin{enumerate}[(i)]
    \item Let $g\in \mathcal{O}(V)$ be a holomorphic function on $V$. Consider the graph $X$ of $g$ inside $\CC^2$, i.e.
  \[X = \{(z,g(z)) | z\in V\}.
  \]
  Show that $(X,\phi:X\to V)$ determines a Riemann structure on $X$.
\item Generalize this construction to the case of multiple holomorphic functions $g_1,\ldots, g_s\in \mathcal{O}(V)$.
  \end{enumerate}
  
  \end{problem}
\medskip


\textbf{Recall:} given two $\RR$-linearly independent vectors $\ww_1$ and $\ww_2$ in $\CC$, consider the rank-2 discrete lattice $\Gamma = \ZZ\ww_1 \oplus \ZZ\ww_2$ and the complex tori $\mathbb{E}=\CC/\Gamma$ constructed in Lecture 1 (called Elliptic curves)
 \vspace{1ex}

 Problems 2, 3 and 4 involve discrete lattices on $\CC$ and their associated complex tori.

  \vspace{2ex}

\begin{problem}
  Show that the map $\psi\colon \CC=\RR\ww_1 \oplus \RR\ww_2 \to \mathbb{S}^1\times \mathbb{S}^1$ defined by \[
  \psi(\lambda_1\ww_1 + \lambda_2\ww_2) = (e^{2\pi \iota \lambda_1}, e^{2\pi \iota \lambda_2})\]
  factors through a homeomorphism $\EE\to \mathbb{S}^1\times \mathbb{S}^1$.

\end{problem}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem} [Forster $\S 1.4$] Let $\Gamma = \ZZ\ww_1\oplus \ZZ\ww_2$ and $\Gamma' = \ZZ\ww_1'\oplus \ZZ\ww_2'$ be two lattices in $\CC$ show that $\Gamma = \Gamma'$ if, and only if, there exists a matrix $A\in \operatorname{SL}(2,\ZZ) = \{A \in \operatorname{GL}(2,\ZZ): \det(A)=1\}$ such that $A\begin{pmatrix} \ww_1\\ \ww_2
  \end{pmatrix} = \begin{pmatrix}\ww_1'\\ \ww_2'
  \end{pmatrix}$.

\end{problem}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}[Forster $\S 1.5$] This problem discusses how to pick lattices giving isomorphic complex structures on tori.  
  \begin{enumerate}[(i)]
  \item Let $\Gamma, \Gamma'$ be two discrete rank-2 lattices in $\CC$ as constructed in Problem 3. Assume there exists $\alpha \in \CC^*$ with $\alpha\Gamma \subseteq \Gamma'$. Show that the map $\CC\to \CC$ with $z\mapsto \alpha z$ induces a holomorphic map $\CC/\Gamma \to \CC/\Gamma'$, which is biholomorphic if, and only if, $\alpha \Gamma = \Gamma'$.
  \item Show that every torus $X=\CC/\Gamma$ is isomorphic to a torus of the form \begin{equation}\label{eq:x_tau} X_{\tau}:=\CC/(\ZZ 1\oplus  \ZZ \tau),\end{equation}
    where $\tau\in \CC$ satisfies $\operatorname{Im}(\tau)>0$.
    \item Suppose $\begin{pmatrix} a & b \\ c& d
    \end{pmatrix}\in \operatorname{SL}(2,\RR)$ and fix $\tau$ in $\CC$ with $\operatorname{Im}(\tau)>0$. Let
      \[
      \tau':=\frac{a\,\tau + b}{c\, \tau + d}.
      \]
      Show that the tori $X(\tau)$ and $X(\tau')$ (see~\eqref{eq:x_tau}) are isomorphic Riemann surfaces.
  \end{enumerate}
\end{problem}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}[Forster $\S 1.2$] \textbf{(Moebius transformations on $\CC$)}\\
  Fix $\begin{pmatrix} a & b \\ c& d
  \end{pmatrix}\in \operatorname{SL}(2,\CC)$ and consider the rational map
  \[f\colon \CC\dashrightarrow \CC \qquad f(z) = \frac{a\,z + b}{c\,z + d}.
  \]
  \begin{enumerate}[(i)]
  \item Show that $f$ can be extended to a meromorphic function $f\colon \pr^1\dashrightarrow \CC$ away from $\{z\in \CC: c\,z +d\neq 0\}$.
  \item Show that $f$ can be extended to a holomorphic function $f\colon \pr^1\to \pr^1$.
    \item Show that $f\colon \pr^1\to \pr^1$ is biholomorphic.
  \end{enumerate}

  
  
\end{problem}


\medskip

Recall from the local behavior of holomorphic functions $f\colon X_1\to X_2$ between Riemann surfaces, that given any point $a_1\in X_1$ and a neighborhood $U_0\subset X_1$ of $a_1$, we can find a neighborhood $W\subset X_2$ of $a_2:=f(a_1)$ and an open $U$ with $a_1\in U\subset U_0$ such that $f^{-1}{y}\cap U$ has the same size for each $y\in W\smallsetminus \{a_2\}$. This number is called the \textbf{ramification index} of $f$ at $a_1$ (or the multiplicity of $f$ at $a_1$).

Problems 6, 7 an 8 below involve ramification indices.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem} Fix a degree $N$ polynomial $F$ in $\CC[z]$.
  \begin{enumerate}[(i)]
  \item Show that $F$ defines a holomorphic function $f\colon \pr^1\to \pr^1$, with $f(z)=F(z)$ for each $z\neq \infty$ and $f(\infty)=\infty$.
    \item Show that the ramification index of $f$ is locally constant.
    \item Compute the ramification index of $f$ at each point of $\pr^1$. (\emph{Hint:} Compute this index at $\infty$.)
  \end{enumerate}
\end{problem}

\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem} Show that the ring of meromorphic functions $\mathcal{M}(\pr^1)$ is the field of rational functions $\CC(z)$ in one variable, i.e. every meromorphic function on $\pr^1$ is a quotient of polynomials in $z$.\\
(\emph{Hint:} Show that for each $f\in \mathcal{M}(\pr^1)$ the fiber $f^{-1}(\infty)$ is discrete and finite.)
\end{problem}
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{problem} If $f\colon X_1\to X_2$ is an injective holomorphic function between Riemann surfaces, show that $f$ is biholomorphic onto its image. (\emph{Hint:} Determine the ramification index of $f$.)
\end{problem}

\medskip


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}\textbf{(Stereographic projection)}\\
  Consider the north pole $N=(0,0,1)$  in the unit sphere $\Sb^2$ in $\RR^3$ and let $P=(0,0,-1)$ be the south pole. Fix a complex atlas on $\Sb^2$ via the stereographic projections from the south and north poles, respectively. More precisely, consider the 
  covering $\{U_0,U_1\}$ of $\Sb^2$ and the maps $\phi_0$ and $\phi_1$ with:
  \[
  \phi_0\colon U_0 = \Sb^2\smallsetminus \{P\} \to \CC \quad \text{ and } \quad
  \phi_1\colon U_1 = \Sb^2\smallsetminus \{N\} \to \CC,
  \]
  defined by $\phi_0(x,y,z) =\frac{x-\iota\,y}{1+z}$ and $\phi_1(x,y,z)=\frac{x+\iota\,y}{1-z}$. 
  \begin{enumerate}[(i)]
  \item Show that this defines a complex structure on the unit sphere. We call this the Riemann sphere.
  \item Show that this Riemann sphere is isomorphic to $\pr^1$ by showing $\phi_1$ extends to a unique biholomophism $\rho\colon \Sb^2\to \pr^1$ with $\rho(N) = \infty$.
  \end{enumerate}
\end{problem}
\end{document}




\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}[Forster $\S 1.3$] Identify $\pr^1$ with the unit sphere $\Sb^2\subseteq \RR^3$ under the stereographic projection $\rho\colon \Sb^2\to \CC\cup \{\infty\}\simeq \pr^1$ from Problem 9.

  Let $\operatorname{SO}(3)$ be the group of orthogonal $3 \times 3$ matrices with determinant 1, i.e.
  \[
\operatorname{SO}(3) = \{A \in \operatorname{GL}(3,\RR): A^T\,A = I, \det(A)=1\}.
  \]
  Show that for every $A\in \operatorname{SO}(3)$, the map
  \[\rho\circ A \circ \rho^{-1}\colon \pr^1\to \pr^1\]
  is biholomorphic.\\
  (\emph{Hint:} Factor any matrix in $\operatorname{SO}(3)$ as a product of the matrices of the form  $\begin{pmatrix} 0 & 1 & 0 \\
    0 & 0 & 1\\
    1 & 0 & 0
  \end{pmatrix}$ and  $\begin{pmatrix}B & 0 \\ 0 & 1
  \end{pmatrix}$, where $B\in \operatorname{SO}(2)$.)
\end{problem}
\end{document}
