\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*{Hodge structure}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - 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{pure_hodge_structure}{Pure Hodge structure}\dotfill \pageref*{pure_hodge_structure} \linebreak
\noindent\hyperlink{mixed_hodge_structure}{Mixed Hodge structure}\dotfill \pageref*{mixed_hodge_structure} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{ForAKaehlerManifold}{On the cohomology of a K\"a{}hler manifold}\dotfill \pageref*{ForAKaehlerManifold} \linebreak
\noindent\hyperlink{ForAComplexAnalyticSpace}{On the cohomology of a complex analytic space}\dotfill \pageref*{ForAComplexAnalyticSpace} \linebreak
\noindent\hyperlink{OnAnAbelianGroup}{Generally on an abelian group}\dotfill \pageref*{OnAnAbelianGroup} \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}

\hypertarget{pure_hodge_structure}{}\subsubsection*{{Pure Hodge structure}}\label{pure_hodge_structure}

A \emph{Hodge structure} (or \emph{pure Hodge structure}, for emphasis) is a (bi-)[[graded object|grading]] structure on [[cohomology groups]] -- called a \emph{Hodge decomposition} -- of the kind that is exhibited by the [[de Rham cohomology]]/complex-[[ordinary cohomology]] of [[compact topological space|compact]] [[Kähler manifolds]], according to the [[Hodge theorem]]. A Hodge structure is said to be \emph{of weight $d$} if it behaves like the cohomology of a K\"a{}hler manifold of [[dimension]] $d$.

If instead of considering a single [[cohomology group]] one considers the cohomology groups of a parameterized collection of spaces -- hence the cohomology \emph{[[sheaves]]/[[stacks]]} -- then one speaks of \emph{variation of Hodge structure} (of a given weight).

By a central theorem of Hodge theory (recalled as theorem \ref{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure} below) the traditional (and original) [[filtered complex|filtration]] on the complex cohomology of a [[Kähler manifold]] induced by the [[harmonic differential forms]] generalizes to a [[filtered complex|filtration]] of the complex-valued [[ordinary cohomology]] of any [[complex analytic space]] which is simply given by the canonical degree-filtration of the [[holomorphic de Rham complex]].

This means that [[ordinary differential cohomology]] in the guise of [[Deligne cohomology]] is nothing but the [[homotopy pullback]] of a stage of the Hodge filtration along the `` [[Chern character]] '' map from integral to complex cohomology. (A point of view highlighted for instance in \hyperlink{PetersSteenbrink08}{Peters-Steenbrink 08, section 7.2}). Viewed this way Hodge structures are filtrations of stages of differential form cycle refinements of [[Chern characters]] that appear in the general definition/characterization of [[differential cohomology]], as discussed at \emph{[[differential cohomology hexagon]]} starting around the section \emph{\href{differential+cohomology+diagram#DeRhamCoefficients}{de Rham coefficients}}

This modern point of view is also crucial for instance in the characterization of an [[intermediate Jacobian]] (see there) as the subgroup of [[Deligne cohomology]] that is in the [[kernel]] of the map to Hodge-filtering stage of ordinary cohomology. See at \emph{\href{intermediate+Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology}{intermediate Jacobian -- characterization as Hodge-trivial Deligne cohomology}}.

\hypertarget{mixed_hodge_structure}{}\subsubsection*{{Mixed Hodge structure}}\label{mixed_hodge_structure}

A \emph{mixed Hodge structure} is a [[filtration]] on [[cohomology groups]] -- called a \emph{[[Hodge filtration]]} -- such that the [[associated graded object]] has pure Hodge structure of weight $k$ in each degree $k$. The archetypical example exhibiting this is the cohomology of [[complex varieties]] that have singularities (\hyperlink{Deligne71}{Deligne 71} \hyperlink{Deligne74}{Deligne 74}).

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

Historically, Hodge structures originate in the special structure induced on the [[de Rham cohomology|de Rham]] [[cohomology groups]] of a compact [[Kähler manifold]] by the existence of [[harmonic differential forms]]. Below we first discuss this canonical Hodge structure

\begin{itemize}%
\item \hyperlink{ForAKaehlerManifold}{on the cohomology of a K\"a{}hler manifold}

\end{itemize}
But it turns out that this Hodge structure only depends on the natural degree-[[filtration]] on the [[holomorphic de Rham complex]] and hence more generally there is canonical Hodge structure

\begin{itemize}%
\item \hyperlink{ForAComplexAnalyticSpace}{on the cohomology of a complex analytic space}.

\end{itemize}
Abstracting from here one defines Hodges structures

\begin{itemize}%
\item \hyperlink{OnAnAbelianGroup}{generally on abelian groups}.

\end{itemize}
\hypertarget{ForAKaehlerManifold}{}\subsubsection*{{On the cohomology of a K\"a{}hler manifold}}\label{ForAKaehlerManifold}

Let $X$ be a [[compact topological space|compact]] [[Kähler manifold]] and write $H^{p,q}(X)$ for its space of [[harmonic differential forms]], equivalently, via the [[Hodge isomorphism]], its [[Dolbeault cohomology]] in bidegree $(p,q)$.

Notice that by the [[de Rham theorem]] there are canonical maps

\begin{displaymath}
H^{p,q}(X)\to H^{p+q}(X,\mathbb{C})
\end{displaymath}
to [[ordinary cohomology]] of $X$ with complex [[coefficients]].

\begin{defn}
\label{TraditionalHodgeFiltration}\hypertarget{TraditionalHodgeFiltration}{}
The \emph{Hodge filtration} on the cohomology of $X$ is the [[filtered complex]] structure given by the [[direct sum]]

\begin{displaymath}
F^p H^k(X, \mathbb{C})
   \coloneqq
  \underset{k-q \geq p}{\oplus}
  H^{k-q,q}(X)
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The full Hodge filtration of degree-2 cohomology is

\begin{displaymath}
\begin{aligned}
    F^0 H^2(X,\mathbb{C})
    & =
    H^{0,2}(X) \oplus H^{1,1}(X) \oplus H^{2,0}(X)
    \\
    F^1 H^2(X,\mathbb{C})
    & =
    \;\;\;\;\;\;\;
    H^{1,1}(X)
    \oplus
    H^{2,0}(X)
    \\
    F^2 H^2(X,\mathbb{C})
    & = 
    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
    H^{2,0}(X)
 \end{aligned}
\end{displaymath}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
For all $p$ the mid-dimensional Hodge filtration stage in even total degree is

\begin{displaymath}
F^p H^{2p}
  = 
  H^{p,p}(X)
  \oplus
  H^{p+1,p-1}(X)
  \oplus 
  \cdots
  \oplus
  H^{2p,0}(X)
  \,.
\end{displaymath}
\end{example}
\hypertarget{ForAComplexAnalyticSpace}{}\subsubsection*{{On the cohomology of a complex analytic space}}\label{ForAComplexAnalyticSpace}

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[complex analytic space]], write

\begin{displaymath}
\Omega^\bullet_X
  \coloneqq
  (\mathcal{O}_X \stackrel{\partial}{\longrightarrow} \Omega^1_X \stackrel{\partial}{\longrightarrow} \Omega^2(X) \stackrel{\partial}{\longrightarrow} \cdots)
\end{displaymath}
for its [[holomorphic de Rham complex]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Notice the \href{holomorphic+de+Rham+complex#RelationToComplexCohomology}{relation to complex cohomology} given by

\begin{displaymath}
H^k(X,\mathbb{C}) \simeq H^k(X,\Omega^\bullet_X)
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The [[holomorphic de Rham complex]] is naturally [[filtered object|filtered]] by degree with the $p$th filtering stage being

\begin{displaymath}
F^p \Omega^\bullet_X
  \coloneqq
  (0 \to \cdots \to \Omega^p_X \stackrel{\partial}{\longrightarrow}
   \Omega^{p+1} \stackrel{\partial}{\longrightarrow} \cdots)
  \,.
\end{displaymath}
Notice that here $\Omega^p$ is still regarded as sitting in degree $-p$, one just replaces by 0 in the holomorphic de Rham complex the groups of differential forms of degree less than $p$.

\end{remark}
\begin{defn}
\label{HodgeFiltrationForComplexSpace}\hypertarget{HodgeFiltrationForComplexSpace}{}
The \emph{Hodge filtration} on $H^\bullet(X,\mathbb{C})$ is defined to be the [[filtration]] with $p$th stage the [[image]] of the [[hypercohomology|hyper]]-[[abelian sheaf cohomology]] with coefficients in the $p$th filtering stage of the holomorphic de Rham complex inside that with coefficients the full de Rham complex:

\begin{displaymath}
F^p H^k(X,\mathbb{C})
  \coloneqq
  im
  \left(
    H^k(X, F^p \Omega^\bullet_X)
    \to 
    H^k(X, \Omega^\bullet_X)
  \right)
\end{displaymath}
\end{defn}
(e.g. \hyperlink{Voisin02}{Voisin 02, def. 8.2}, \hyperlink{PetersSteenbrink08}{Peters-Steenbrink 08, def. 2.21}).

\begin{theorem}
\label{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure}\hypertarget{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure}{}
When the compact [[complex manifold]] $X$ happens to have the structure of a [[Kähler manifold]] then the [[Frölicher spectral sequence]] degenerates at the $E_1$ page which implies that def. \ref{HodgeFiltrationForComplexSpace} coincides with the traditional definition via [[harmonic differential forms]], def. \ref{TraditionalHodgeFiltration}:

\begin{displaymath}
\underset{k-q \geq p}{\oplus}
  H^{k-q,q}(X)
  \simeq
  im
  \left(
    H^k(X, F^p \Omega^\bullet_X)
    \to 
    H^k(X, \Omega^\bullet_X)
  \right)
  \,.
\end{displaymath}
\end{theorem}
(e.g. \hyperlink{Voisin02}{Voisin 02, remark 8.29}, \hyperlink{Voisin}{Voisin, 1.1.2} \hyperlink{PetersSteenbrink08}{Peters-Steenbrink 08, prop 2.22}).

\begin{remark}
\label{FrolicherEquivalenceInComponents}\hypertarget{FrolicherEquivalenceInComponents}{}
The equivalence in theorem \ref{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure} is exhibited by the following morphism.

Write

\begin{displaymath}
tot(\Omega^{\bullet \geq p, \bullet}, \mathbf{d}= \partial + \bar \partial)
\end{displaymath}
for the holomorphically truncated [[de Rham complex]], as indicated, thought of as the [[total complex]] of the [[Dolbeault complex|Dolbeault]] [[double complex]]

\begin{displaymath}
\itexarray{
    \Omega^{p,0} &\stackrel{\bar \partial}{\to}& \Omega^{p-1,1} &\stackrel{\bar \partial}{\to}& \cdots
    \\
    \downarrow^{\mathrlap{\partial }}
    &&
    \downarrow^{\mathrlap{\partial }}
    \\
    \Omega^{p+1,0} &\stackrel{\bar \partial}{\to}& \Omega^{p,1} &\stackrel{\bar \partial}{\to}& \cdots
    \\
    \downarrow^{\mathrlap{\partial }}
    &&
    \downarrow^{\mathrlap{\partial }}
    \\
    \vdots && \vdots
  }
  \,.
\end{displaymath}
Since this is in each row the [[Dolbeault resolution]] of the given sheaf of [[holomorphic differential forms]], this total complex is indeed [[quasi-isomorphism|quasi-isomorphic]] to the (truncated) [[holomorphic de Rham complex]].

The total complex is in degree $-k$ given by $\underset{k-q \geq p}{\oplus} \Omega^{k-q, q}$ and hence globally defined closed $(k-q \geq p,q)$-forms naturally inject into

\begin{displaymath}
H^k(X, tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial) )
  \simeq
  H^0(X,tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial)[-k])
  \,.
\end{displaymath}
Therefore given a representative $\alpha \in \Omega^{p,q}_{cl}(X)$ of $[\alpha] \in H^{p,q}(X)$ it is canonically sent along

\begin{displaymath}
\underset{k-q\geq p}{\oplus} \Omega^{p,q}_{cl}(X)
  \simeq
  \underset{k-q\geq p}{\oplus} 
   H^0(X, \Omega^{p,q}_{cl})
  \to
  H^0(X,tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial)[-k])
  \,.
\end{displaymath}
This map exhibits the equivalence in theorem \ref{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure} (e.g. \hyperlink{Voisin}{Voisin, section 1.1.2}).

Dually,

\begin{displaymath}
\Omega_{hol}^{\leq k}
  \simeq
  tot(  \Omega^{\bullet \leq k, \bullet})
  \,.
\end{displaymath}
This plays a role in the discussion of [[intermediate Jacobians]], where for $dim_{\mathbb{C}}(X)= k+1$ we have

\begin{displaymath}
H^{2k+1}(X,\mathbb{R})
  \simeq
  H^{2k+1}(X,\mathbb{C}) / F^{k+1} H^{2k+1}(X,\mathbb{C})
  \simeq
  H^{2k+1}(X, \Omega_{hol}^{\bullet \leq k})
  \,.
\end{displaymath}
Here a real differential $(2k+1)$-form

\begin{displaymath}
\alpha 
  = 
  \overline{\alpha^{0,2k+1}}
  +
  \overline{\alpha^{1, 2k}}
  + 
  \cdots
  +
  \alpha^{1, 2k}
  + 
  \alpha^{0,2k+1}
\end{displaymath}
injects via its pieces in

\begin{displaymath}
\Omega^{p \leq k, 2k+1-p}(X)
  \simeq
  H^0(X, \Omega^{p \leq k, 2k+1-p})
  \to
  H^0(X, tot(\Omega^{\bullet \leq k, \bullet})[-k])
  \simeq
  H^k(\Omega_{hol}^{\bullet\leq k})
  \,.
\end{displaymath}
\end{remark}
\hypertarget{OnAnAbelianGroup}{}\subsubsection*{{Generally on an abelian group}}\label{OnAnAbelianGroup}

\begin{defn}
\label{HodgeStructureOfWeightk}\hypertarget{HodgeStructureOfWeightk}{}
For $H_{\mathbb{Z}}$ an [[abelian group]], a \emph{Hodge structure} of \emph{weight} $k \in \mathbb{Z}$ on $H_{\mathbb{Z}}$ is a [[direct sum]] decomposition of its [[complexification]]

\begin{displaymath}
H_{\mathbb{C}}\coloneqq H_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{C}
\end{displaymath}
into [[complex vector spaces]] $H^{p,q}$ with $p +q = k$ of the form

\begin{displaymath}
H_{\mathbb{C}} \simeq \underset{p+q = k}{\oplus} H^{p,q}
\end{displaymath}
such that $H^{q,p}$ is the [[complex conjugation|complex conjugate]] of $H^{p,q}$:

\begin{displaymath}
H^{p,q} = \overline{H^{q,p}}
  \,.
\end{displaymath}
This is an equality of the underlying sets of the complex vector spaces.

\end{defn}
With this the above def. \ref{TraditionalHodgeFiltration} has the following verbatim generalization

\begin{defn}
\label{GeneralHodgeFiltration}\hypertarget{GeneralHodgeFiltration}{}
Given a Hodge structure $H_{\mathbb{Z}}, \{H^{p,q}\}$ of weight $k$, def. \ref{HodgeStructureOfWeightk}, then the associated \emph{Hodge filtration} on $H_{\mathbb{C}}$ is the [[filtered complex]] structure given by the [[direct sum]]

\begin{displaymath}
F^p H_{\mathbb{C}}
   \coloneqq
  \underset{k-q \geq p}{\oplus}
  H^{k-q,q}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hodge theory]],

[[nonabelian Hodge theory]], [[noncommutative Hodge theory]]


\item [[Hodge symmetry]]


\item [[Hodge cycle]]


\item [[Hodge cohomology]]


\item [[intermediate Jacobian]]


\item [[Lefschetz decomposition]]



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

Textbook accounts include

\begin{itemize}%
\item [[Claire Voisin]], section 7 of \emph{[[Hodge theory and Complex algebraic geometry]] I,II}, Cambridge Stud. in Adv. Math. \textbf{76, 77}, 2002/3


\item [[Claire Voisin]], \emph{Hodge theory and the topology of compact K\"a{}hler and complex projective manifolds} (\href{http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf}{pdf})


\item [[Chris Peters]], [[Jozef Steenbrink]], \emph{[[Mixed Hodge Structures]]}, Ergebisse der Mathematik (2008) (\href{http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck}{pdf})



\end{itemize}
The notion of mixed Hodge structures was introduced in

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Th\'e{}orie de Hodge II}, Publ. Math. I.H.E.S, 40, 5--58 (1971)


\item [[Pierre Deligne]], \emph{Th\'e{}orie de Hodge III}, Publ. Math., I. H. E. S, 44, 5-77 (1974)



\end{itemize}
A review is in section 8.4 of

\begin{itemize}%
\item [[Alain Connes]], [[Matilde Marcolli]], \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]}

\end{itemize}
See also

\begin{itemize}%
\item [[Jozef Steenbrink]], S. Zucker, \emph{Variation of mixed Hodge structure I}, Invent. Math. \textbf{80} (1985), 489-542.


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Hodge_structure}{Hodge structure},}


\item [[Donu Arapura]], \emph{Mixed Hodge Structures Associated to Geometric Variations} (\href{http://www.math.purdue.edu/~dvb/preprints/tifr.pdf}{pdf})


\item [[eom]],

A.I. Ovseevich \emph{\href{http://eom.springer.de/H/h047470.htm}{Hodge structure}}, \emph{\href{http://eom.springer.de/p/p072140.htm}{Period mapping}},

[[Jozef Steenbrink]], \emph{\href{http://eom.springer.de/v/v096170.htm}{Variation of Hodge structure}}


\item [[Michael Hopkins]], [[Gereon Quick]], \emph{Hodge filtered complex bordism} (\href{http://arxiv.org/abs/1212.2173}{arXiv:1212.2173})



\end{itemize}
[[!redirects Hodge structures]]

[[!redirects pure Hodge structure]] [[!redirects pure Hodge structures]]

[[!redirects mixed Hodge structure]] [[!redirects mixed Hodge structures]]

[[!redirects Hodge filtration]] [[!redirects Hodge filtrations]] [[!redirects Hodge decomposition]] [[!redirects Hodge decompositions]]

[[!redirects Hodge filtering]] [[!redirects Hodge filterings]]

[[!redirects variation of Hodge structure]] [[!redirects variation of Hodge structures]] [[!redirects variations of Hodge structure]] [[!redirects variations of Hodge structures]]



\end{document}