\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*{holomorphic vector bundle}

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_complex_algebraic_vector_bundles}{As complex algebraic vector bundles}\dotfill \pageref*{as_complex_algebraic_vector_bundles} \linebreak
\noindent\hyperlink{AsComplexVectorBundlesWithHolomorphicFlatConnection}{As complex vector bundles with holomorphically flat connection}\dotfill \pageref*{AsComplexVectorBundlesWithHolomorphicFlatConnection} \linebreak
\noindent\hyperlink{OverRiemannSurfaces}{Over Riemann surfaces}\dotfill \pageref*{OverRiemannSurfaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_complex_vector_bundles_with_flat_holomorphic_connection}{Relation to complex vector bundles with flat holomorphic connection}\dotfill \pageref*{relation_to_complex_vector_bundles_with_flat_holomorphic_connection} \linebreak


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

A [[complex numbers|complex]] [[vector bundle]] over a [[complex manifold]] such that it admits [[transition functions]] that are [[holomorphic functions]].

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

\hypertarget{as_complex_algebraic_vector_bundles}{}\subsubsection*{{As complex algebraic vector bundles}}\label{as_complex_algebraic_vector_bundles}

By the [[GAGA]]-principle holomorphic vector bundles and more generally analytic [[coherent sheaves]] over a [[projective variety|projective]] [[smooth variety|smooth]] [[complex variety]] coincide with complex [[algebraic vector bundles]]/[[coherent sheaves]].

\hypertarget{AsComplexVectorBundlesWithHolomorphicFlatConnection}{}\subsubsection*{{As complex vector bundles with holomorphically flat connection}}\label{AsComplexVectorBundlesWithHolomorphicFlatConnection}

\begin{theorem}
\label{KoszulMalgrangeTheorem}\hypertarget{KoszulMalgrangeTheorem}{}
\textbf{([[Koszul-Malgrange theorem]])}

Holomorphic vector bundles over a [[complex manifold]] are equivalently [[complex vector bundles]] which are equipped with a holomorphically [[flat connection]] (hence a connection whose holomorphic component vanishes). Under this identification the [[Dolbeault operator]] $\bar \partial$ acting on the [[sections]] of the holomorphic vector bundle is identified with the holomorphic component of the [[covariant derivative]] of the given connection.

The analogous statement is true for generalization of vector bundles to [[chain complexes]] of [[module sheaves]] with [[coherent cohomology]].

\end{theorem}
For [[complex vector bundles]] over [[complex varieties]] this statement is due to [[Alexander Grothendieck]] and (\hyperlink{KoszulMalgrange58}{Koszul-Malgrange 58}), recalled for instance as (\hyperlink{Pali06}{Pali 06, theorem 1}). It may be understood as a special case of the [[Newlander-Nirenberg theorem]], see (\hyperlink{DelzantPy10}{Delzant-Py 10, section 6}), which also generalises the proof to [[infinite-dimensional manifold|infinite-dimensional]] vector bundles. Over [[Riemann surfaces]], see \hyperlink{OverRiemannSurfaces}{below}, the statement was highlighted in (\hyperlink{AtiyahBott83}{Atiyah-Bott 83}) in the context of the [[Narasimhan-Seshadri theorem]].

The generalization from [[vector bundles]] to [[coherent sheaves]] is due to (\hyperlink{Pali06}{Pali 06}). In the genrality of [[(∞,1)-categories of chain complexes]] ([[dg-categories]]) of holomorphic vector bundles the statement is discussed in (\hyperlink{Block05}{Block 05}).

\begin{remark}
\label{}\hypertarget{}{}
The equivalence in theorem \ref{KoszulMalgrangeTheorem} serves to relate a fair bit of [[differential geometry]]/[[differential cohomology]] with constructions in [[algebraic geometry]]. For instance [[intermediate Jacobians]] arise in [[differential geometry]] and [[quantum field theory]] as [[moduli spaces of flat connections]] equipped with [[symplectic structure]] and [[Kähler polarization]], all of which in terms of [[algebraic geometry]] directly comes down [[moduli spaces]] of [[abelian sheaf cohomology]] with [[coefficients]] in the [[structure sheaf]] (and/or some variants of that, under the exponential exact sequence).

\end{remark}
\hypertarget{OverRiemannSurfaces}{}\subsubsection*{{Over Riemann surfaces}}\label{OverRiemannSurfaces}

Over [[Riemann surfaces]] holomorphic vector bundles are a central part of the theory of the [[moduli space of flat connections]]. See at \emph{[[Narasimhan-Seshadri theorem]]}.

A key observation here is (\hyperlink{AtiyahBott83}{Atiyah-Bott 83, section 7}), that a $U(n)$-[[principal connection]] induces a holomorphic structure on the [[associated bundle|associated]] [[complex vector bundle]] by taking the $(0,1)$-part of the connection 1-form as the [[Dolbeault operator]]. For review of the statement and its proof see (\hyperlink{Evans}{Evans, lecture 10}).

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

\begin{itemize}%
\item [[Kodaira vanishing theorem]]


\item [[Chern connection]]


\item [[Kähler polarization]]


\item [[stable vector bundle]]


\item [[moduli space of bundles]]


\item [[Deligne line bundle]]


\item [[Higgs bundle]]


\item [[algebraic line bundle]]


\item [[holomorphic line 2-bundle]], [[holomorphic line n-bundle]]


\item [[complex analytic stack]]


\item [[pseudoholomorphic vector bundle]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Holomorphic_vector_bundle}{holomorphic vector bundle}}

\end{itemize}
\hypertarget{relation_to_complex_vector_bundles_with_flat_holomorphic_connection}{}\subsubsection*{{Relation to complex vector bundles with flat holomorphic connection}}\label{relation_to_complex_vector_bundles_with_flat_holomorphic_connection}

The classical statement of theorem \ref{KoszulMalgrangeTheorem} is due to [[Alexander Grothendieck]] and

\begin{itemize}%
\item [[Jean-Louis Koszul]], [[Bernard Malgrange]], \emph{Sur certaine structures fibr\'e{}es complexes}, arch. mat, vol IX, 1958

\end{itemize}
Over [[Riemann surfaces]] and in the context of the [[moduli space of flat connections]]:

\begin{itemize}%
\item [[Michael Atiyah]], [[Raoul Bott]], \emph{The Yang-Mills equations over Riemann surfaces}, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (\href{http://www.jstor.org/stable/37156}{jstor}, \href{http://math.stackexchange.com/a/295505/58526}{lighning summary})


\item [[Jonathan Evans]], \emph{\href{http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm}{Aspects of Yang-Mills theory}}, lecture notes, (\href{http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture10.pdf}{lecture 10}, \href{http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture11.pdf}{lecture 11}, \href{http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture12.pdf}{lecture 12})



\end{itemize}
Generalization to [[coherent sheaves]] is due to

\begin{itemize}%
\item N. Pali, \emph{Faisceaux $\bar \partial$-cohrents sur les vari\'e{}t\'e{} complexes} ( \emph{$\bar \partial$-Coherent sheaves on complex manifolds) Math. Ann. 336 (2006), no. 3, 571--615 (\href{http://arxiv.org/abs/math/0305422}{arXiv:math/0305422})}

\end{itemize}
Further Generalization to [[chain complexes]] of holomorphic vector bundles is discussed in

\begin{itemize}%
\item [[Jonathan Block]], \emph{Duality and equivalence of module categories in noncommutative geometry I} (\href{http://arxiv.org/abs/math/0509284}{arXiv:0509284})

\end{itemize}
in terms of [[Lie infinity-algebroid representations]] of the holomorphic [[tangent Lie algebroid]].

Generalization to [[infinite-dimensional manifold|infinite-dimensional]] vector bundles is in

\begin{itemize}%
\item Thomas Delzant, Pierre Py, \emph{K\"a{}hler groups, real hyperbolic spaces and the Cremona group}, Compositio Math. 148, no. 1 (2012), 153--184 (\href{http://arxiv.org/abs/1012.1585}{arXiv:1012.1585})

\end{itemize}
[[!redirects holomorphic vector bundles]]

[[!redirects holomorphic line bundle]] [[!redirects holomorphic line bundles]]

[[!redirects holomorphic section]] [[!redirects holomorphic sections]]



\end{document}