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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 5}}}\\
\vspace{2ex}

\Large{\textbf{\textsf{\v{C}ech cohomology}}}\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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\begin{problem} [Forster $\S 12.1$] Fix $n$ distinct points $p_1,\ldots, p_n$ on $\CC$ and let $X:=\CC\smallsetminus \{p_1, \ldots, p_n\}$. Show that $H^1(X,\underline{\ZZ})\simeq \ZZ^n$.\\
  (\emph{Hint:} Find a covering $\mathcal{U}:=(U_1,U_2)$ of $X$ by simply connected opens, where $U_1 \cap U_2$ has $n+1$ connected components.)
\end{problem}
\vspace{2ex}


Recall the notation $V\Subset U$ for a \emph{relatively compact} open subset $V$ of $U$, meaning that $\overline{V}$ is compact and $\overline{V}\subset U$.
\vspace{1ex}

\begin{problem} [Forster $\S 12.2$ (a)]
Let $X$ be a differentiable manifold, and let $U$, $V$ be opens with $V\subset U$ and $V \Subset U$. Show that $V$ meets only a finite number of connected components of $U$. 
\end{problem}
\vspace{2ex}

\begin{problem} [Forster $\S 12.2$ (b)]
  Let $X$ be a compact differentiable manifold and pick two open \emph{finite} coverings $\mathcal{U}:=(U_i)_{i\in I}$ and $\mathcal{V}:=(V_j)_{j\in J}$ such that $V_i\Subset U_i$ for all $i\in I$. Show that the image of the restriction map
  \[
  Z^1(\mathcal{U}, \underline{\CC}) \to Z^1(\mathcal{V}, \underline{\CC})
  \]
  is finite-dimensional over $\CC$.
\end{problem}
\vspace{2ex}


\begin{problem} [Forster $\S 12.2$ (c)]
 Let $X$ be a compact Riemann surface. Show that $H^1(X,\CC)$ is a finite dimensional vector space over $\CC$. (\emph{Hint:} Use finite coverings $\mathcal{U}$, $\mathcal{V}$ as is Problem 3, with $U_i$ and $V_i$ homeomorphic to open disks in $\CC$.)
\end{problem}

\vspace{3ex}

\begin{problem} [Forster $\S 12.3$ (a)]
  Let $X$ be a compact Riemann surfaces, and consider the sheaves of locally constant functions $\underline{\ZZ} \subset \underline{\CC}$.
  \begin{enumerate}
    \item Show that this inclusion of sheaves gives an associated map on cohomology:
  \[
\rho\colon H^1(X,\underline{\ZZ})\to H^1(Z,\underline{\CC})
\]
\item Show that $\rho$ is injective.
  \end{enumerate}
\end{problem}

\vspace{3ex}

\begin{problem} [Forster $\S 12.3$ (b)]
  Let $X$ be a compact Riemann surface.
  \begin{enumerate}
  \item Show that $H^1(X,\underline{\ZZ})$ is a finitely generated abelian group using the technique from Problem 4.
  \item Use Problem 5 to conclude that  $H^1(X,\underline{\ZZ})$ is a free  $\ZZ$-module.
  \end{enumerate}
\end{problem}

    \vspace{3ex}

    \begin{problem}    Given a Riemann surface $X$, we let $\Omega$ be the presheaf of holomorphic 1-forms on $\pr^1$. Its sections on an open chart $(U,z)$ are $\Omega(U) := \{f dz\colon f\in \mathcal{O}(U)\}$. Show that, in fact, $\Omega$ is a sheaf on $X$.
    \end{problem}
      \vspace{2ex}
      
    \begin{problem}  [Forster $\S 13.2$]  Consider the standard open covering $\mathcal{U}=(U_0:= \pr^1\smallsetminus \{\infty\}, U_{\infty}:=\pr^1\smallsetminus \{0\})$ of $\pr^1$.
      \begin{enumerate}
      \item Show that $\mathcal{U}$ is a Leray covering for the sheaf $\Omega$ from Problem 7.
        \item Prove that $\Omega(U_0 \cap U_{\infty}) \simeq Z^1(\mathcal{U},\Omega)$ as vector spaces.
        \item Show that $H^1(\pr^1, \Omega)=H^1(\mathcal{U},\Omega)$ is a 1-dimensional vector space, with basis given by the cohomology class of $\frac{d z}{z}$.
      \end{enumerate}

    \end{problem}

\end{document}


Recall that $\ConvSC$ denotes the field of convergent power series in a neighborhood of the origin.
\vspace{1ex}
\begin{problem} [Forster $\S 8.3$] Let $F(z,w):=w^2-z^3w+z \in \ConvSC[w]$.
  \begin{enumerate}[(i)]
  \item Show that $F$ is irreducible over $\ConvSC[w]$.
    \item Determine the Puiseux expansion of a solution $w(\eta)$ to  $F(\eta^2, w(\eta))=0$. 
  \end{enumerate}
\end{problem}

\vspace{1ex}

\begin{problem} Fix a Riemann surface $X$  and open $U\subset X$ and let $k=0$ or 1. Show that $F^*(w)$  for $w\in \mathcal{E}^{(k)}(U)$ is independent of the expressions defining $w$ (i.e. $w=\sum_{j} f_j \,dg_j$ with $f_j,g_j\in \mathcal{E}(U)$ for $k=1$, and $w=\sum_j f_j \,d g_j\wedge dh_j$ with  $f_j,g_j,h_j\in \mathcal{E}(U)$ for $k=2$, respectively).
\end{problem}

\vspace{1ex}

\begin{problem} Consider the Riemann surface $\CC^*$ and the (closed) holomorphic 1-form $w = \frac{dz}{z}$.
  \begin{enumerate}[(i)]
  \item Determine the period homomorphism $\rho_w\colon \pi_1(\CC^*)\to \CC$.
  \item Compute $p^*w$ where $p=\exp\colon \CC\to \CC^*$ is the universal covering of $\CC^*$.
  \end{enumerate}
\end{problem}
\vspace{2ex}

  \noindent \emph{Definition:} Given a Riemann surface $X$ and its universal covering $\pi\colon \tilde{X}\to X$, we say a function $f\in \mathcal{E}(\tilde{X}))$ is \emph{additively automorphic with constant summands of automorphy} if for each $\sigma\in \operatorname{Deck}(\tilde{X}|X)$ the function
\[
a_\sigma:= f - \sigma \cdot f\in \mathcal{E}(\tilde{X})
\] is constant.

\vspace{1ex}
  \begin{problem} Consider the holomorphic 1-form $w=\frac{dz}{z}$ on $\CC^*$ from Problem 4. Fix any primitive $F$ to $p^*w$ on $\CC$.
\begin{enumerate}[(i)]
\item Show that $F$ is an additively  automorphic function with constant summands of automorphy.
  \item Show that $F$ is not constant along the fibers of the universal cover $p=exp\colon \CC\to \CC^*$.
\end{enumerate}
  \end{problem}

  \vspace{1ex}
  
  \begin{problem}  [Forster $\S 9.2$]
    Consider the  holomorphic 1-form $w = dz / (1+z^2)$ on $\CC\smallsetminus\{\pm i\}$.
    \begin{enumerate}[(i)]
    \item Show that $w$ can be extended to a holomorphic 1-form on $\pr^1\smallsetminus \{\pm i\}$.
      \item Determine $p^*w$, where $p=tan \colon \CC\to \pr^1\smallsetminus \{\pm i\}$  is the universal covering.
    \end{enumerate}
  \end{problem}
    \vspace{1ex}

    The next problem shows how to determine the periods of an elliptic curve and highlights the relevance of the construction of the summands of automorphy.
\vspace{1ex}
    
    \begin{problem} Consider a rank-two lattice $\Gamma \subset \CC$ and the associated Riemann surface $X=\CC/\Gamma$. 
      \begin{enumerate}[(i)]
      \item  Show that $\operatorname{Deck}(\CC|X)\simeq \Gamma$.
      \item Show that the identity function $f=\operatorname{id}\colon \CC\to \CC$ is an  additively  automorphic function with constant summands of automorphy. Furthermore, confirm that $a_{\sigma} = \sigma$ for each $\sigma \in \Gamma$.
      \item Build a closed 1-form $w$ on $X$ with $dz=\pi^*(w)$, where $\pi\colon \CC\to X$ is the natural projection.
        \item Determine $\rho_w(\sigma)$ for each $\sigma \in \Gamma$.
      \end{enumerate}
      \end{problem}


    
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