Complex and Kahler Geometry

\title{Lecture Notes on Complex Geometry and Kahler Geometry}
\begin{abstract}
Our goals of this lecture notes are
\begin{itemize}
\item Calabi-Yau Theorem
\item Existence of Hermitian-Einstein metric (Donaldson-Uhlenbeck-Yau Theorem or Hitchin-Kobayashi Correspondence)
\item The Existence of Kahler-Einstein Metric (related with stability)
\end{itemize}
\end{abstract}

\tableofcontents
\section{Basic Concepts in Complex Geometry}
\subsection{Holomorphic Map}
\begin{defn}
A complex-valued function $f(z)$ defined on a connected open domain $W\subseteq\C^n$ is called \iemph{holomorphic} if for each $a=(a_1,\ldots, a_n)\in W$, $f(z)$ can be represented as a power series
\[
\sum_{k_1\geq0,\cdots,k_n\geq0}^\infty C_{k_1\cdots k_n}(z_1-a_1)^{k_1}\cdots (z_n-a_n)^k_n
\]
in some neighbourhood of $a$.
\end{defn}
We have the following equivalent definition
\begin{defn}
Let $f(z)$ be a (continuously) differentiable function on an open set $W\subseteq\C^n$. Then $f(z)$ is \emph{holomorphic} iff $\frac{\pt{f}}{\pt\bar z_\nu}=0$, $1\leq\nu\leq n$.
\end{defn}
\begin{proof}
The proof is based on the Cauchy integral theorem and called “\iemph{Osgood Theorem}”, cf. \cite{morrow1971complex} for detail.
\end{proof}
Recall that a complex-valued function of $n$ complex variables can be considered as a function of $2n$ real variables, since $\C^n\simeq \R^{2n}$, related by $z_\nu=x_\nu+i y_\nu$, $i=\sqrt{-1}$, $x_\nu,y_\nu\in\R$. We have the following
\begin{alignat*}{2}
\rd z_\nu&=\rd x_\nu+i\rd y_\nu,
&\qquad
\rd \bar z_\nu&=\rd x_\nu-i\rd y_\nu\\
\ppt{z_\nu}&=\frac{1}{2}\left(\ppt{x_\nu}-i\ppt{y_\nu}\right),
&\qquad
\ppt{\bar z_\nu}&=\frac{1}{2}\left(\ppt{x_\nu}+i\ppt{y_\nu}\right)\\
\rd x_\nu&=\frac{1}{2}(\rd z_\nu+\rd\bar z_\nu),
&\qquad
\rd y_\nu&=\frac{1}{2i}(\rd z_\nu-\rd\bar z_\nu)\\
\ppt{x_\nu}&=\ppt{z_\nu}+\ppt{\bar z_\nu},
&\qquad
\ppt{y_\nu}&=i\left(\ppt{z_\nu}-\ppt{\bar z_\nu}\right).
\end{alignat*}
With the help of the above relation, the \iemph{total differential} of a complex-$n$ valued function $f$ is
\begin{align*}
\rd f &=\frac{\pt f}{\pt x_\nu}\rd x_\nu+\frac{\pt f}{\pt y_\nu}\rd y_\nu\\
&=\left(
\frac{\pt f}{\pt z_\nu}+\frac{\pt f}{\pt \bar z_\nu}\right)\frac{1}{2}(\rd z_\nu+\rd\bar z_\nu)+
i\left(
\frac{\pt f}{\pt z_\nu}-\frac{\pt f}{\pt \bar z_\nu}\right)
\frac{1}{2i}(\rd z_\nu-\rd\bar z_\nu)\\
&=\frac{\pt f}{\pt z_\nu}\rd z_\nu+\frac{\pt f}{\pt \bar z_\nu}\rd \bar z_\nu\\
&\eqdef \pt f+\bar\pt f.
\end{align*}
We shall call $\pt f$ and $\bar\pt f$ the \emph{holomorphic part}\index{holomorphic!part} and \emph{Anti-holomorphic part}\index{holomorphic!Anti-!part} of $f$, respectively. \begin{rem}
$\bar\pt f=0$ is the \iemph{Cauchy-Riemann Equation}.
\end{rem}
\subsection{Complex Manifold and Examples}
\subsection{Almost Complex Structure and Almost Complex Manifolds}
\subsection{Newlander-Nirenberg Theorem}
\subsection{Complex and Holomorphic Vector Bundle}
\subsection{Hermitian Metric and Kahler Metric on Complex Manifolds}
\subsection{Curvature Tensor of Kahler Manifolds}
\subsection{Laplace Operator and Harmonic Form}
\subsection{Sheaf Cohomology Theory}
\subsubsection{De Rahm Theorem}
\subsubsection{Dolbeault Theorem}
\subsubsection{Hodge Theorem}
\subsection{Chern Class}
\subsection{Vanishing Theorem}
\section{Hermitian-Einstein Metric}
\subsection{Stability by Mumford}
\subsection{Hermitian-Einstein Connection}
Existence of H-E => semi-stable or Direct sum of stable(Kobayashi), we will use Donaldson’s heat flow method to proof this. Note that Uhlenbeck-Yau has show this with the method of continuous.
\subsection{Donaldson Functional}
\subsection{Donaldson Heat Flow and Yang-Mills Flow}
\subsection{The Proof of Donaldson-Uhlenbeck-Yau Theorem}
\section{Calabi-Yau Theorem}
\subsection{Complex Monge-Ampere Equation}
\subsection{$C^0$-Estimate}
\subsection{$C^2$-Estimate}
\subsection{Kahler-Einstein Metric and Kahler-Ricci Flow}
the case of Chern class $<0$ is solved, $>0$ is still open. Note also that we can use Kahler-Ricci Flow to prove Calabi-Yau Theorem, refer Huai-Dong, Cao‘s work.

The \emph{Handdraft} notes is not available anymore.

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