\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Kähler manifold}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{LinearKählerStructure}{Linear Kähler structure}\dotfill \pageref*{LinearKählerStructure} \linebreak
\noindent\hyperlink{khler_manifolds}{Kähler manifolds}\dotfill \pageref*{khler_manifolds} \linebreak
\noindent\hyperlink{InTermsOfGStructure}{In terms of $G$-Structure}\dotfill \pageref*{InTermsOfGStructure} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_almost_complex_manifold}{Relation to (almost) complex manifold}\dotfill \pageref*{relation_to_almost_complex_manifold} \linebreak
\noindent\hyperlink{relation_to_symplectic_manifolds}{Relation to symplectic manifolds}\dotfill \pageref*{relation_to_symplectic_manifolds} \linebreak
\noindent\hyperlink{relation_to_spinstructures}{Relation to Spin-structures}\dotfill \pageref*{relation_to_spinstructures} \linebreak
\noindent\hyperlink{HodgeStarOperator}{Hodge star operator}\dotfill \pageref*{HodgeStarOperator} \linebreak
\noindent\hyperlink{HodgeStructure}{Hodge structure}\dotfill \pageref*{HodgeStructure} \linebreak
\noindent\hyperlink{as_riemannian_manifolds}{As $\mathbb{C}$-Riemannian manifolds}\dotfill \pageref*{as_riemannian_manifolds} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{K\"a{}hler manifold} is a [[smooth manifold]] compatibly equipped with

\begin{enumerate}%
\item [[complex structure]];


\item [[Riemannian structure]];


\item [[symplectic structure]].



\end{enumerate}
If the symplectic structure is not compatibly present, it is just a [[Hermitian manifold]].

\begin{tabular}{l|l|l}
[[complex structure]]&+ [[Riemannian structure]]&+ [[symplectic structure]]\\
\hline 
[[complex structure]]&[[Hermitian structure]]&[[Kähler structure]]\\
\end{tabular}

Where a Riemannian manifold is a real [[smooth manifold]] equipped with a nondegenerate smooth symmetric 2-form $g$ (the [[Riemannian metric]]), an \textbf{almost K\"a{}hler manifold} is a [[complex manifold|complex holomorphic manifold]] equipped with a nondegenerate hermitian 2-form $h$ (the \textbf{K\"a{}hler $2$-form}). The real [[cotangent bundle]] is replaced with the complex cotangent bundle, and symmetry is replaced with hermitian symmetry. An almost K\"a{}hler manifold is a \textbf{K\"a{}hler manifold} if it satisfies an additional integrability condition.

The K\"a{}hler 2-form can be decomposed as $h = g+i\omega$; here $g$ is a [[Riemannian metric]] and $\omega$ a [[symplectic form]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{LinearKählerStructure}{}\subsubsection*{{Linear Kähler structure}}\label{LinearKählerStructure}

\begin{defn}
\label{KaehlerVectorSpace}\hypertarget{KaehlerVectorSpace}{}
\textbf{([[Kähler vector space]])}

Let $V$ be a [[finite dimensional vector space|finite-dimensional]] [[real vector space]]. Then a \emph{linear Kähler structure} on $V$ is

\begin{enumerate}%
\item a \emph{[[linear complex structure]]} on $V$, namely a [[linear map|linear]] [[endomorphism]]

\begin{displaymath}
J \;\colon\; V \to V
\end{displaymath}
whose [[composition]] with itself is minus the [[identity morphism]]:

\begin{displaymath}
J \circ J = - id_V
\end{displaymath}

\item a skew-symmetric [[bilinear form]]

\begin{displaymath}
\omega \in \wedge^2 V^\ast
\end{displaymath}


\end{enumerate}
such that

\begin{enumerate}%
\item $\omega(J(-),J(-)) = \omega(-,-)$;


\item $g(-,-) \coloneqq \omega(-,J(-))$ is a [[Riemannian metric]], namely

a non-degenerate positive-definite [[bilinear form]] on $V$

(necessarily symmetric, due to the other properties: $g(w,v) = \omega(w,J(v)) = -\omega(J(v),w) = + \omega(J(J(v)), J(w)) = \omega(w,J(w)) = g(v,w)$).



\end{enumerate}
\end{defn}
(e.g. \hyperlink{Boalch09}{Boalch 09, p. 26-27})

Linear Kähler space structure may conveniently be encoded in terms of [[Hermitian space]] structure:

\begin{defn}
\label{HermitianForm}\hypertarget{HermitianForm}{}
\textbf{([[Hermitian form]] and [[Hermitian space]])}

Let $V$ be a [[real vector space]] equipped with a [[complex structure]] $J\colon V \to V$. Then a \emph{[[Hermitian form]]} on $V$ is

\begin{itemize}%
\item a complex-valued real-[[bilinear form]]

\begin{displaymath}
h \;\colon\; V \otimes V \longrightarrow \mathbb{C}
\end{displaymath}


\end{itemize}
such that this is \emph{symmetric sesquilinear}, in that:

\begin{enumerate}%
\item $h$ is complex-linear in the first argument;


\item $h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$



\end{enumerate}
where $(-)^\ast$ denotes [[complex conjugation]].

A Hermitian form is \emph{positive definite} (often assumed by default) if for all $v \in V$

\begin{enumerate}%
\item $h(v,v) \geq 0$


\item $h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.



\end{enumerate}
A [[complex vector space]] $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) \emph{[[Hermitian space]]}.

\end{defn}
\begin{prop}
\label{BasicPropertiesOfHermitianForms}\hypertarget{BasicPropertiesOfHermitianForms}{}
\textbf{(basic properties of [[Hermitian forms]])}

Let $((V,J),h)$ be a positive definite [[Hermitian space]] (def. \ref{HermitianForm}). Then

\begin{enumerate}%
\item the [[real part]] of the [[Hermitian form]]

\begin{displaymath}
g(-,-) \;\coloneqq\; Re(h(-,-))
\end{displaymath}
is a [[Riemannian metric]], hence a symmetric positive-definite real-[[bilinear form]]

\begin{displaymath}
g \;\colon\; V \otimes V \to \mathbb{R}
\end{displaymath}

\item the [[imaginary part]] of the [[Hermitian form]]

\begin{displaymath}
\omega(-,-) \;\coloneqq\; -Im(h(-,-))
\end{displaymath}
is a [[symplectic form]], hence a non-degenerate skew-symmetric real-[[bilinear form]]

\begin{displaymath}
\omega \;\colon\; V \wedge V \to \mathbb{R}
  \,.
\end{displaymath}


\end{enumerate}
hence

\begin{displaymath}
h = g - i \omega
  \,.
\end{displaymath}
The two components are related by

\begin{equation}
\omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
\label{RelationBetweennOmegaAndgOnHermitianSpace}\end{equation}
Finally

\begin{displaymath}
h(J(-),J(-)) = h(-,-)
\end{displaymath}
and so the Riemannian metrics $g$ on $V$ appearing from (and fully determining) Hermitian forms $h$ via $h = g - i \omega$ are precisely those for which

\begin{equation}
g(J(-),J(-)) = g(-,-)
  \,.
\label{HermitianMetric}\end{equation}
These are called the \emph{[[Hermitian metrics]]}.

\end{prop}
\begin{proof}
The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:

\begin{displaymath}
\begin{aligned}
    g(w,v)
    & \coloneqq
    Re(h(w,v))
    \\
    & =
    Re\left( h(v,w)^\ast\right)
    \\
    & =
    Re(h(v,w))
    \\
    & =
    g(v,w)
    \,.
  \end{aligned}
\end{displaymath}
That $h$ is invariant under $J$ follows from its sesquilinarity

\begin{displaymath}
\begin{aligned}
    h(J(v),J(w))
    &=
    i h(v,J(w))
    \\
    & =
    i (h(J(w),v))^\ast
    \\
    & =
    i (-i) (h(w,v))^\ast
    \\
    & =
    h(v,w)
  \end{aligned}
\end{displaymath}
and this immediately implies the corresponding invariance of $g$ and $\omega$.

Analogously it follows that $\omega$ is skew symmetric:

\begin{displaymath}
\begin{aligned}
    \omega(w,v)
    & \coloneqq
    -Im(h(w,v))
    \\
    & =
    -Im\left( h(v,w)^\ast\right)
    \\
    & =
    Im(h(v,w))
    \\
    & = - \omega(v,w)
    \,,
  \end{aligned}
\end{displaymath}
and the relation between the two components:

\begin{displaymath}
\begin{aligned}
    \omega(v,w)
    & =
    - Im(h(v,w))
    \\
    & =
    Re(i h(v,w))
    \\
    & =
    Re(h(J(v),w))
    \\
    & = 
    g(J(v),w)
  \end{aligned}
\end{displaymath}
as well as

\begin{displaymath}
\begin{aligned}
    g(v,w)
    & =
    Re(h(v,w)
    \\
    & =
    Im(i h(v,w))
    \\
    & =
    Im(h(J(v),w))
    \\
    & =
    Im(h(J^2(v),J(w)))
    \\
    & = 
    - Im(h(v,J(w)))
    \\
    & = \omega(v,J(w))
    \,.
  \end{aligned}
\end{displaymath}
\end{proof}
As a corollary:

\begin{prop}
\label{RelationBetweenKählerVectorSpacesAndHermitianSpaces}\hypertarget{RelationBetweenKählerVectorSpacesAndHermitianSpaces}{}
\textbf{(relation between [[Kähler vector spaces]] and [[Hermitian spaces]])}

Given a [[real vector space]] $V$ with a [[linear complex structure]] $J$, then the following are equivalent:

\begin{enumerate}%
\item $\omega \in \wedge^2 V^\ast$ is a [[linear Kähler structure]] (def. \ref{KaehlerVectorSpace});


\item $g \in V \otimes V \to  \mathbb{R}$ is a [[Hermitian metric]] \eqref{HermitianMetric}



\end{enumerate}
where $\omega$ and $g$ are related by \eqref{RelationBetweennOmegaAndgOnHermitianSpace}

\begin{displaymath}
\omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
\end{displaymath}
\end{prop}
\hypertarget{khler_manifolds}{}\subsubsection*{{Kähler manifolds}}\label{khler_manifolds}

(\ldots{})

\hypertarget{InTermsOfGStructure}{}\subsubsection*{{In terms of $G$-Structure}}\label{InTermsOfGStructure}

A K\"a{}hler manifold is a [[integrability of G-structure|first-order integrable]] [[almost Hermitian structure]], hence a first order integrable [[G-structure]] for $G = U(n) \hookrightarrow GL(2n,\mathbb{R})$ the [[unitary group]] (e.g. \hyperlink{Verbitsky09}{Verbitsky 09}).

By the fact (see at \emph{\href{unitary+group#RelationToOrthogonalSymplecticAndGeneralLinearGroup}{unitary group -- relation to orthogonal, symplectic and general linear group}}) that $U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})$ this means that a K\"a{}hler manifold structure is precisely a joint [[orthogonal structure]]/[[Riemannian manifold]] structure, [[symplectic manifold]] structure and [[complex manifold]] structure.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

The archetypical elementary example is the following:

\begin{example}
\label{StandardAlmostKaehlerVectorSpaces}\hypertarget{StandardAlmostKaehlerVectorSpaces}{}
\textbf{(standard [[Kähler vector space]])}

Let $V \coloneqq \mathbb{R}^2$ be the 2-dimensional [[real vector space]] equipped with the [[complex structure]] $J$ which is given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, hence, in terms of the canonical [[linear basis]] $(e_i)$ of $\mathbb{R}^2$, this is

\begin{displaymath}
J = (J^i{}_j) 
  \coloneqq 
  \left( 
    \itexarray{
      0 & -1
      \\
      1 & 0
    }
  \right)
  \,.
\end{displaymath}
Moreover let

\begin{displaymath}
\omega = (\omega_{i j}) \coloneqq \left( \itexarray{0 & 1 \\ -1 & 0} \right)
\end{displaymath}
and

\begin{displaymath}
g = (g_{i j}) \coloneqq \left( \itexarray{ 1 & 0 \\ 0 & 1}  \right)
  \,.
\end{displaymath}
Then $(V, J, \omega, g)$ is a [[Kähler vector space]] (def. \ref{KaehlerVectorSpace})

The corresponding [[Kähler manifold]] is $\mathbb{R}^2$ regarded as a [[smooth manifold]] in the standard way and equipped with the [[bilinear forms]] $J, \omega g$ extended as constant rank-2 [[tensors]] over this manifold.

If we write

\begin{displaymath}
x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
\end{displaymath}
for the standard [[coordinate functions]] on $\mathbb{R}^2$ with

\begin{displaymath}
z \coloneqq x + i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
\end{displaymath}
and

\begin{displaymath}
\overline{z} \coloneqq x - i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
\end{displaymath}
for the corresponding complex coordinates, then this translates to

\begin{displaymath}
\omega
  \in
  \Omega^2(\mathbb{R}^2)
\end{displaymath}
being the [[differential 2-form]] given by

\begin{displaymath}
\begin{aligned}
    \omega
    & =
    d x \wedge d y
    \\
    & =
    \tfrac{i}{2} d z \wedge d \overline{z}
  \end{aligned}
\end{displaymath}
and with [[Riemannian metric]] [[tensor]] given by

\begin{displaymath}
g = d x \otimes d x + d y \otimes d y
  \,.
\end{displaymath}
The [[Hermitian form]] is given by

\begin{displaymath}
\begin{aligned}
    h 
    & = 
    g - i \omega
    \\
    & = 
    d z \otimes d \overline{z}
    \,.
  \end{aligned}
\end{displaymath}
\end{example}
\begin{proof}
This is elementary, but, for the record, here is one way to make it fully explicit (we use [[Einstein summation convention]] and ``$\cdot$'' denotes [[matrix multiplication]]):

\begin{displaymath}
\begin{aligned}
    \omega_{i j'} J^{j'}{}_j
    & =
    \left(
       \itexarray{
         0 & 1
         \\
         -1 & 0
       }
    \right)
      \cdot
    \left(
      \itexarray{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \\
    & 
    =
    \left(
      \itexarray{
        1 & 0
        \\
        0 & 1
      }
    \right)
    \\
    & =
    g_{i j}
  \end{aligned}
\end{displaymath}
and similarly

\begin{displaymath}
\begin{aligned}
    \omega(J(-),J(-))_{i j}
    & =
    \omega_{i' j'} J^{i'}{}_{i} J^{j'}{}_{j}
    \\
    & =
    (J^t \cdot \omega \cdot J)_{i j}
    \\
    & =
    \left(
    \left(
      \itexarray{
        0 & 1
        \\
        -1 & 0
      }
    \right)
    \cdot
    \left(
      \itexarray{
        0 & 1
        \\
        -1 & 0
      }
    \right)
      \cdot
    \left(
      \itexarray{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \right)_{i j}
    \\
    & =
    \left(
    \left(
      \itexarray{
        -1 & 0
        \\
        0 & -1
      }
    \right)
    \cdot
    \left(
      \itexarray{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \right)_{i j}
    \\
    & = 
    \left(
      \itexarray{
        0 & 1
        \\
        -1 & 0
      }
    \right)_{i j}
    \\
    & = 
    \omega_{i j}
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{example}
\label{}\hypertarget{}{}
\textbf{([[Fubini-Study metric]])}

There is a unique (up to a scalar) hermitian metric on [[complex projective space]] (which may be normalized), the \emph{[[Fubini-Study metric]]}.

All analytic subvarieties of a complex projective space are in fact [[algebraic variety|algebraic subvarieties]] and they inherit the K\"a{}hler structure from the projective space.

Examples include [[complex tori]] $\mathbb{C}^n/L$ where $L$ is a lattice in $\mathbb{C}^n$, [[K3-surfaces]], compact [[Calabi-Yau manifolds]], quadrics, products of projective spaces and so on.

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_almost_complex_manifold}{}\subsubsection*{{Relation to (almost) complex manifold}}\label{relation_to_almost_complex_manifold}

\begin{quote}%
The following based on \href{http://mathoverflow.net/a/73330/381}{this MO comment} by [[Spiro Karigiannis]]


\end{quote}
When $(X, J)$ is an [[almost complex manifold]], then there is a notion of smooth complex-valued [[differential forms]] of type $(p,q)$. A complex valued $2$-form $\omega$ is of type $(1,1)$ precisely if it satisfies

\begin{displaymath}
\omega(J v,J w) = \omega(v,w)
\end{displaymath}
for all smooth [[vector fields]] $v,w$ on $X$. Here $\omega$ is a \emph{real} $2$-form of type $(1,1)$, if $\overline \omega = \omega$. Setting

\begin{displaymath}
g(v,w) = \omega(v, J w),
\end{displaymath}
defines a smooth symmetric rank $(2,0)$ [[tensor field]]. This is a [[Riemannian metric]] precisely if it is fiberwise a [[positive definite bilinear form]]. If it $g(-,-) = \omega(-,J -)$ is hence a Riemannian metric, then $\omega(-,-)$ is called positive definite, too.

The triple of data $(J, \omega, g)$, where $J$ is an [[almost complex structure]], $\omega$ is a real positive $(1,1)$-[[differential form]], and $g$ is the associated [[Riemannian metric]] this way define an \emph{[[almost Hermitian manifold]]}.

Now the condition for $X$ to be a K\"a{}hler is that $X$ be a [[complex manifold]] ($J$ is integrable) and that $d\omega = 0$. Equivalently that for the [[Levi-Civita connection]] $\nabla$ of $G$ we have $\nabla \omega = 0$ or $\nabla J = 0$.

Hence given a [[complex manifold]] $X$, together with a \emph{closed} real $2$-form $\omega$, the only additional condition required to ensure that it defines a K\"a{}hler metric is that it be a positive $(1,1)$-form.

\hypertarget{relation_to_symplectic_manifolds}{}\subsubsection*{{Relation to symplectic manifolds}}\label{relation_to_symplectic_manifolds}

Lifting a [[symplectic manifold]] structure to a K\"a{}hler manifold structure is also called choosing a \emph{[[Kähler polarization]]}.

\hypertarget{relation_to_spinstructures}{}\subsubsection*{{Relation to Spin-structures}}\label{relation_to_spinstructures}

\begin{prop}
\label{}\hypertarget{}{}
A spin structure on a [[compact topological space|compact]] [[Hermitian manifold]] ([[Kähler manifold]]) $X$ of complex [[dimension]] $n$ exists precisely if, equivalently

\begin{itemize}%
\item there is a choice of [[square root]] $\sqrt{\Omega^{n,0}}$ of the [[canonical line bundle]] $\Omega^{n,0}$ (a ``[[Theta characteristic]]'');


\item there is a trivialization of the [[first Chern class]] $c_1(T X)$ of the [[tangent bundle]].



\end{itemize}
\end{prop}
In this case one has:

\begin{prop}
\label{}\hypertarget{}{}
There is a [[natural isomorphism]]

\begin{displaymath}
S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}}_X
\end{displaymath}
of the [[sheaf]] of [[sections]] of the [[spinor bundle]] $S_X$ on $X$ with the [[tensor product]] of the [[Dolbeault complex]] with the corresponding [[Theta characteristic]];

Moreover, the corresponding [[Dirac operator]] is the [[Dolbeault-Dirac operator]] $\overline{\partial} + \overline{\partial}^\ast$.

\end{prop}
This is due to (\hyperlink{Hitchin74}{Hitchin 74}). A textbook account is for instance in (\hyperlink{Friedrich74}{Friedrich 74, around p. 79 and p. 82}).

\hypertarget{HodgeStarOperator}{}\subsubsection*{{Hodge star operator}}\label{HodgeStarOperator}

On a K\"a{}hler manifold $\Sigma$ of [[dimension]] $dim_{\mathbb{C}}(\Sigma) = n$ the [[Hodge star operator]] acts on the [[Dolbeault complex]] as

\begin{displaymath}
\star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X)
  \,.
\end{displaymath}
(notice the exchange of the role of $p$ and $q$) See e.g. (\hyperlink{BiquerdHoering08}{BiquerdH\"o{}ring 08, p. 79}).

\hypertarget{HodgeStructure}{}\subsubsection*{{Hodge structure}}\label{HodgeStructure}

The [[Hodge theorem]] asserts that for a compact K\"a{}hler manifold, the canonical $(p,q)$-grading of its [[differential forms]] descends to its [[de Rham cohomology]]/[[ordinary cohomology]]. The resulting structure is called a \emph{[[Hodge structure]]}, and is indeed the archetypical example of such.

\hypertarget{as_riemannian_manifolds}{}\subsubsection*{{As $\mathbb{C}$-Riemannian manifolds}}\label{as_riemannian_manifolds}

[[!include normed division algebra Riemannian geometry -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[bilinear form]], [[quadratic form]], [[sesquilinear form]]


\item [[symplectic form]], [[Hermitian form]]


\item \textbf{K\"a{}hler manifold}, [[hyper-Kähler manifold]], [[quaternionic Kähler manifold]]

\begin{itemize}%
\item [[Kähler potential]]

\end{itemize}

\item [[Kähler-Einstein manifold]]


\item [[symplectic manifold]]


\item [[Lefschetz decomposition]]


\item [[Kähler polarization]], [[almost Kähler geometric quantization]],


\item [[Fedosov deformation quantization]]


\item [[Hodge filtration]], [[Frölicher spectral sequence]]


\item [[Hodge manifold]]


\item [[Sasakian manifold]]



\end{itemize}
[[!include special holonomy table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

K\"a{}hler manifolds were first introduced and studied by P. A. Shirokov (cf. \href{http://dx.doi.org/10.1070/RM1995v050n02ABEH002098}{a historical article}) and later independently by K\"a{}hler.

Textbook accounts include

\begin{itemize}%
\item [[Claire Voisin]], section 3 of \emph{[[Hodge theory and Complex algebraic geometry]] I,II}, Cambridge Stud. in Adv. Math. \textbf{76, 77}, 2002/3

\end{itemize}
Lecture notes include

\begin{itemize}%
\item Andrei Moroianu, \emph{Lectures on K\"a{}hler Geometry} ()


\item Philip Boalch, \emph{Noncompact complex symplectic and hyperkähler manifolds}, 2009 (\href{https://www.math.u-psud.fr/~boalch/cours09/hk.pdf}{pdf})



\end{itemize}
Discussion in terms of [[integrability of G-structure|first-order integrable]] [[G-structure]] include

\begin{itemize}%
\item Misha Verbitsky, \emph{K\"a{}hler manifolds}, lecture notes 2009 (\href{http://verbit.ru/MATH/TALKS/Unicamp-kahler-1.pdf}{pdf})

\end{itemize}
Discussion of [[spin structures]] in K\"a{}hler manifolds is for instance in

\begin{itemize}%
\item [[Thomas Friedrich]], \emph{Dirac operators in Riemannian geometry}, Graduate studies in mathematics 25, AMS (1997)

\end{itemize}
Discussion of [[Hodge theory]] on K\"a{}hler manifolds is in

\begin{itemize}%
\item O. Biquard, A. H\"o{}ring, \emph{K\"a{}hler geometry and Hodge theory}, 2008 (\href{http://math.unice.fr/~hoering/hodge/hodge.pdf}{pdf})

\end{itemize}
[[!redirects Kähler manifolds]]

[[!redirects Kahler manifold]] [[!redirects Kahler 2-form]] [[!redirects almost Kahler manifold]] [[!redirects Kähler 2-form]] [[!redirects almost Kähler manifold]]

[[!redirects Kähler form]] [[!redirects Kähler forms]]

[[!redirects Kähler geometry]]

[[!redirects Kähler manifold]]

[[!redirects Kähler structure]] [[!redirects Kähler structures]] [[!redirects Kaehler structure]] [[!redirects Kaehler structures]]

[[!redirects almost Kähler manifold]] [[!redirects almost Kähler manifolds]]

[[!redirects almost Kaehler manifold]] [[!redirects almost Kaehler manifolds]]

[[!redirects almost Kähler structure]] [[!redirects almost Kähler structures]] [[!redirects almost Kaehler structure]] [[!redirects almost Kaehler structures]]

[[!redirects Kähler manifold structure]] [[!redirects Kähler manifold structures]]

[[!redirects Kaehler manifold structure]] [[!redirects Kaehler manifold structures]]



\end{document}