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\begin{document}

\vorlesungA{D-MATH}{FS 2016}{Prof.~Peter S. Jossen}{Complex Abelian Varieties}{Exercise Sheet}{8}
\begin{aufgaben}
%\item \input{q_4.tex}
\item \input q_1.tex
\item \input{q_3.tex}
%%\input a_1.tex
%%\pagebreak
%\item (\emph{Morphisms to $\mathbb{P}^N$})\footnote{For a more general treatment, see R. Hartshorne, \emph{Algebraic Geometry}, II, Theorem 7.1.} Let $X$ be a complex variety \textcolor{red}{or maybe complex manifold?}. Show:
%\begin{unteraufgaben}
%\item If $\varphi:X\lra \mathbb{P}^N$ is a morphism of \textcolor{red}{what?}, then $\varphi^*(\mathcal{O}(1))$ is a line bundle on $X$, generated by the global sections $\varphi^*(x_0),\dotsc,\varphi^*(x_N)$.
%\item Conversely, if $\mathcal{L}$ is a line bundle on $X$ and $s_0,\dotsc,s_N\in \Gamma(X,\mathcal{L})$ are global sections generating $\mathcal{L}$, then there exists a unique morphism of \textcolor{red}{what?} $\varphi:X\lra \mathbb{P}^N$ such that $\mathcal{L}\cong \varphi^*(\mathcal{O}(1))$ and $s_j=\varphi^*(x_j)$ for each $j=0,\dotsc,N$ under this isomorphism.
%\end{unteraufgaben}
\item \input{q_2.tex}
%\input a_2.tex
%\pagebreak
%\item
%Let $n\in\ZZ_{>0}$.
%\begin{enumerate}[(a)]
%\item  Check that $||z||=\sqrt{\sum_{j=0}^n z_j\bar{z}_j}$ defines a real, type (1,1) differential $2$-form on $\PP^n$, the Fubini-Study form $\omega_{\text{FS}}$.
%\item Prove that
%\begin{align*}
%\omega_{\text{FS}}=\frac{i}{2\pi}\partial\bar{\partial}\log||z||^2.
%\end{align*}
%\item Prove  that the differential form
%\begin{align*}
%\gamma=\frac{i}{2\pi}\frac{||z||^2\left(\sum_{j=0}^n dz_j\wedge d\bar{z}_j\right)-\left(\sum_{j=0}^n \bar{z}_j dz_j\right)\wedge \left(\sum_{k=0}^n z_k d\bar{z}_k\right)}{||z||^4}
%\end{align*}
%is closed and invariant under the action of the unitary group $U(n+1)$.
%\end{enumerate}
%\item \input q_3.tex
%\input a_3.tex
\item \input q_5.tex
%\input a_4.tex
%\item \input q_5.tex
%\input a_5.tex3
%\item \input q_6.tex
%\input a_6.tex
\end{aufgaben}

%\begin{aufgaben}
%\addtocounter{aufgabe}{5}
%\item
%\end{aufgaben}


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