\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*{Dolbeault complex}

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

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

[[!include complex geometry - contents]]

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

[[!include synthetic differential 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{dolbeault_complex}{Dolbeault complex}\dotfill \pageref*{dolbeault_complex} \linebreak
\noindent\hyperlink{holomorphic_differential_forms}{Holomorphic differential forms}\dotfill \pageref*{holomorphic_differential_forms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{DolbeaultTheorem}{Dolbeault theorem}\dotfill \pageref*{DolbeaultTheorem} \linebreak
\noindent\hyperlink{on_stein_manifolds}{On Stein manifolds}\dotfill \pageref*{on_stein_manifolds} \linebreak
\noindent\hyperlink{todd_genus}{Todd genus}\dotfill \pageref*{todd_genus} \linebreak
\noindent\hyperlink{relation_to_structures}{Relation to $Spin^c$-structures}\dotfill \pageref*{relation_to_structures} \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}

The \emph{Dolbeault complex} is the analog of the [[de Rham complex]] in [[complex geometry]].

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

\hypertarget{dolbeault_complex}{}\subsubsection*{{Dolbeault complex}}\label{dolbeault_complex}

On a [[complex manifold]] $X$ the [[de Rham complex]] $\Omega^\bullet(X)$ refines to a [[bigraded object|bigraded]] complex $\Omega^{\bullet, \bullet}(X)$, where a [[differential form]] of bidegree $(p,q)$ has holomorphic degree $p$ and antiholomorphic degree $q$, hence is given on a local [[coordinate chart]] by an expression of the form

\begin{displaymath}
\omega = \sum f_{I J} d z_{i_1} \wedge \cdots \wedge d z_{i_p} \wedge d \bar z_{j_1} \wedge \cdots \wedge d \bar z_{j_q}
 \,.
\end{displaymath}
Moreover, the [[de Rham differential]] $\mathbf{d}$ decomposes as

\begin{displaymath}
\mathbf{d} = \partial + \bar \partial
  \,,
\end{displaymath}
where $\partial \colon \Omega^{\bullet, \bullet}\to \Omega^{\bullet + 1, \bullet}$ and $\bar \partial \colon \Omega^{\bullet, \bullet}\to \Omega^{\bullet, \bullet + 1}$.

The \emph{Dolbeault complex} of $X$ is the [[chain complex]] $(\Omega^{\bullet, \bullet}(X), \bar \partial)$. The \emph{[[Dolbeault cohomology]]} of $X$ is the [[cochain cohomology]] of this complex.

\hypertarget{holomorphic_differential_forms}{}\subsubsection*{{Holomorphic differential forms}}\label{holomorphic_differential_forms}

Here $\Omega^{p,0}(X)$ defines a [[holomorphic vector bundle]] and a [[holomorphic section]] is a differential form with local expression as above, such that the coefficient functions $f_{I J}$ are [[holomorphic functions]]. This is called a \emph{holomorphic differential form}.

For $p \lt dim_{\mathbb{C}}(X)$ equivalently this is a differential form in the [[kernel]] of the antiholomorphic Dolbeault operator $\bar \partial$.

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

\hypertarget{DolbeaultTheorem}{}\subsubsection*{{Dolbeault theorem}}\label{DolbeaultTheorem}

The complex analog of the [[de Rham theorem]] is the [[Dolbeault theorem]]:

for $X$ a [[complex manifold]] then ints [[Dolbeault cohomology]] in bi-degree $(p,q)$ is [[natural isomorphism|naturally isomorphic]] to the [[abelian sheaf cohomology]] in degree $q$ of the [[abelian sheaf]] $\Omega^p \coloneqq \Omega^{p,0}$ of [[holomorphic p-forms]]

\begin{displaymath}
H^{p,q}(X)\simeq H^q(X,\Omega^p)
  \,.
\end{displaymath}
(\ldots{})

Let $Disk_{compl}$ be the [[category]] of complex [[polydiscs]] in $\mathbb{C}^n$ and [[holomorphic functions]] between them.

For $p \in \mathbb{N}$ write $\Omega^p \colon Disk_{complex}^{op} \to Set$ for the [[sheaf]] of holomorphic [[differential n-form|differential p-forms]].

\begin{prop}
\label{}\hypertarget{}{}
For $X$ a [[complex manifold]], let $\{U_i \to X\}$ be a holomorphic good open cover. Then the [[Cech cohomology]] of this cover with [[coefficients]] in $\Omega^p$ in degree $q$ is the [[Dolbeault cohomology]] in bidegree $(p,q)$

\begin{displaymath}
H^{p,q}(X) \simeq \pi_0 sPSh(Disk_{comp})(C(\{U_i\}, \Omega^p[q]))
  \,.
\end{displaymath}
\end{prop}
For instance (\hyperlink{Maddock}{Maddock, theorem 1.0.1}).

\hypertarget{on_stein_manifolds}{}\subsubsection*{{On Stein manifolds}}\label{on_stein_manifolds}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{([[Cartan theorem B]])}

For $X$ a [[Stein manifold]],

\begin{displaymath}
H^k(\Omega^{p,\bullet}(X), \bar \partial)
  =
  \left\{
    \itexarray{  
      0 & k \neq 0 
      \\
      \Omega^p_{hol}(X) & k = 0
    }
  \right.
  \,.
\end{displaymath}
\end{prop}
For instance (\hyperlink{GunningRossi}{Gunning-Rossi}).

\begin{prop}
\label{}\hypertarget{}{}
For $X$ a [[Stein manifold]] of complex [[dimension]] $n$, the [[compactly supported cohomology|compactly supported]] Dolbeault [[cohomology]] is

\begin{displaymath}
H^k(\Omega_c^{p, \bullet}(X), \bar \partial)
  = 
  \left\{
    \itexarray{
      0 , & k \neq n
      \\
      (\Omega_{hol}^{n-p}(X))^\ast
    }
  \right.
  \,,
\end{displaymath}
where on the right $(-)^\ast$ denotes the continuous linear dual.

\end{prop}
First noticed in (\hyperlink{Serre}{Serre}).

\hypertarget{todd_genus}{}\subsubsection*{{Todd genus}}\label{todd_genus}

By the [[Hirzebruch-Riemann-Roch theorem]] the [[index]] of the Dolbeault operator is the [[Todd genus]].

\hypertarget{relation_to_structures}{}\subsubsection*{{Relation to $Spin^c$-structures}}\label{relation_to_structures}

A [[complex manifold]], being in particular an [[almost complex manifold]], carries a canonical [[spin{\tt \symbol{94}}c structure]]. The corresponding [[Spin{\tt \symbol{94}}c Dirac operator]] identifies with the Dolbeault operator under the identification of the [[spinor bundle]] with that of [[holomorphic differential forms]]

\begin{displaymath}
S(X) \simeq \wedge^{0,\bullet} T^\ast X
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Dolbeault cohomology]]


\item [[Dolbeault-Dirac operator]]


\item [[chiral Dolbeault complex]]


\item [[holomorphic de Rham complex]]



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

\begin{itemize}%
\item [[Claire Voisin]], section 2.3 of \emph{[[Hodge theory and Complex algebraic geometry]] I,II}, Cambridge Stud. in Adv. Math. \textbf{76, 77}, 2002/3


\item Zachary Maddock, \emph{Dolbeault cohomology} (\href{http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf}{pdf})


\item Robert C. Gunning and Hugo Rossi, \emph{Analytic functions of several complex variables}, Prentice-Hall Inc., Englewood Cliffs, N.J., (1965)


\item [[Jean-Pierre Serre]], \emph{Quelques probl\`e{}mes globaux relatifs aux vari\'e{}t\'e{}s de Stein}, Colloque sur les fonctions de plusieurs variables, tenu \`a{} Bruxelles, 1953, Georges Thone, Li\`e{}ge, 1953, pp. 57--68. MR 0064155 (16,235b)



\end{itemize}
A [[formal geometry]] version:

\begin{itemize}%
\item Shilin Yu, \emph{Dolbeault dga of a formal neighborhood}, \href{http://arxiv.org/abs/1206.5155}{arxiv/1206.5155}; \emph{The Dolbeault dga of the formal neighborhood of a diagonal}, \href{http://arxiv.org/abs/1211.1567}{arxiv/1211.1567}

\end{itemize}
[[!redirects Dolbeault cohomology]]

[[!redirects holomorphic smooth function]] [[!redirects holomorphic function]] [[!redirects holomorphic smooth functions]] [[!redirects holomorphic functions]]

[[!redirects holomorphic differential 1-form]] [[!redirects holomorphic 1-form]] [[!redirects holomorphic differential form]]

[[!redirects holomorphic differential 1-forms]] [[!redirects holomorphic 1-forms]] [[!redirects holomorphic differential forms]]

[[!redirects holomorphic form]] [[!redirects holomorphic forms]]

[[!redirects holomorphic differential p-form]] [[!redirects holomorphic differential p-forms]] [[!redirects holomorphic p-form]] [[!redirects holomorphic p-forms]]

[[!redirects Dolbeault differential]] [[!redirects Dolbeault differentials]]

[[!redirects Dolbeault operator]] [[!redirects Dolbeault operators]]



\end{document}