250 lines
7.5 KiB
TeX
250 lines
7.5 KiB
TeX
\documentclass[12pt,a4paper,twoside]{amsart} % Package for mathematical article
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\usepackage[T1]{fontenc}
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\usepackage[utf8]{inputenc}
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% Nice Font and some basic things
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\usepackage{libertine}
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\usepackage[libertine]{newtxmath}
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\usepackage{newunicodechar}
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\usepackage[nospace]{varioref}
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\usepackage[ngerman,english]{babel}
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\usepackage{graphicx}
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\usepackage{mathrsfs} % For using mathsrc and not overwriting mathcal
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\usepackage[top=4cm,bottom=3.5cm,left=3.5cm,right=3cm]{geometry}
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\usepackage{parskip}
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\setlength{\parskip}{0.3\baselineskip plus 2pt}
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\raggedbottom
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\usepackage{etoolbox}
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\AtBeginEnvironment{proof}{\vspace*{-2\parskip}}
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%\setlength{\parindent}{0pt}
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%\setlength{\parskip}{0.13 cm}
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% Nice Referencing
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\usepackage{hyperref}
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\urlstyle{same}
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\usepackage[noabbrev]{cleveref}
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% Math packages
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\usepackage{mathtools}
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\usepackage{amsthm}
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\usepackage{derivative}
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% New math commands
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\DeclareMathOperator{\img}{Im}
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\DeclareMathOperator{\sign}{sign}
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\DeclareMathOperator{\id}{Id}
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\DeclareMathOperator{\vol}{vol}
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\DeclareMathOperator{\hodgestar}{\ast}
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\DeclareMathOperator{\Hom}{Hom}
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%\DeclareMathOperator{\opartial}{{\overline{\partial}}}
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\newcommand{\interior}[1]{{\kern0pt#1}^{\mathrm{o}}}
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\newcommand{\into}{\hookrightarrow}
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\newcommand{\onto}{\twoheadrightarrow}
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\newcommand{\opartial}{{\overline{\partial}}}
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\usepackage{enumitem} % Control the format of \enumerate \itemize and \description
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\renewcommand\labelitemi{--} % This doesn't belong to package enumitem but can be used to customize itemize
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% Definitions for theorems of amsthm
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\theoremstyle{plain}
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\newtheorem{thm}{Theorem}[subsection]
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\newtheorem*{thm*}{Theorem}
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\newtheorem{lm}[thm]{Lemma}
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\newtheorem{prop}[thm]{Proposition}
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\newtheorem{cor}[thm]{Corollary}
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\newtheorem{fact}[thm]{Fact}
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\newtheorem{q}[thm]{Question}
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\theoremstyle{definition}
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\newtheorem{defn}[thm]{Definition}
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\newtheorem{nota}[thm]{Notation}
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\newtheorem{exmp}[thm]{Example}
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\newtheorem{xca}[thm]{Exercise}
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\newtheorem*{set}{Setting}
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\theoremstyle{remark}
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\newtheorem{rem}[thm]{Remark}
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\Crefname{thm}{Theorem}{Theorems}
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\Crefname{lm}{Lemma}{Lemmas}
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\Crefname{prop}{Proposition}{Propositions}
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\Crefname{cor}{Corollary}{Corollaries}
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\Crefname{fact}{Fact}{Facts}
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\Crefname{q}{Question}{Questions}
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\Crefname{defn}{Definition}{Definitions}
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\Crefname{nota}{Notation}{Notations}
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\Crefname{exmp}{Example}{Examples}
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\Crefname{xca}{Exercise}{Exercises}
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\Crefname{rem}{Remark}{Remarks}
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\Crefname{enumi}{Property}{Properties}
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\newcommand{\creflmpart}[2]{%
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\nameCref{#1}~\hyperref[#2]{\labelcref*{#1}~(\ref*{#2})}%
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}
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\Crefname{equation}{}{} % Set equation to empy
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\usepackage{tikz-cd}
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\usetikzlibrary{decorations.markings}
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\usepackage{float}
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\renewcommand\thepart{\rm \Roman{part}}
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\setdescription{labelindent=1 em, leftmargin=2 em}
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\setitemize[1]{labelindent=1 em, leftmargin=2 em}
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\setenumerate[1]{labelindent=0cm, leftmargin=*, widest=iiii}
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% Bibliography
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\usepackage[
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backend=biber,
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style=alphabetic,
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maxnames=10,
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maxalphanames=10]{biblatex}
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\addbibresource{bibliography.bib}
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% TODO functionality:
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\usepackage{xcolor}
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%\newcommand\TODO[1]{\textcolor{red}{#1}}
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% If all todo should be hidden at once just uncomment this line
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% \renewcommand\TODO[1]{}
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%\newcommand\CITATION[1]{\textcolor{green}{#1}}
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% Begin of document
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\title{THE HODGE DECOMPOSITION FOR COMPACT KÄHLER MANIFOLDS}
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\author{Daniel Rath}
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\date{}
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\setcounter{tocdepth}{2} % Maybe use 1 here
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\numberwithin{equation}{section}
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\sloppy % Should be better
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\makeatletter%---------------------------------------------------
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% Customize Table of contents
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\def\l@section{\@tocline{1}{7pt}{1pc}{}{}}
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\def\l@subsection{\@tocline{2}{0pt}{1.6pc}{}{}}
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\renewcommand{\tocsection}[3]{%
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\indentlabel{\@ifnotempty{#2}{\makebox[1.4em][l]{%
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\bfseries\ignorespaces#1 #2.\hfill}}}\bfseries#3}
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\renewcommand{\tocsubsection}[3]{%
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\indentlabel{\@ifnotempty{#2}{\makebox[2em][l]{%
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\ignorespaces#1 #2.\hfill}}}#3}
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\renewcommand\section{\@startsection{section}{1}%
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\z@{.7\linespacing\@plus\linespacing}{.8\linespacing}%
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{\normalfont\scshape\centering}}
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\newcommand{\makeheaders}{%
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\begingroup
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\toks@\@xp{\shortauthors}\@temptokena\@xp{\shorttitle}%
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\toks4{\def\\{ \ignorespaces}}% defend against questionable usage
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\edef\@tempa{%
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\@nx\markboth{\the\toks4
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\@nx\MakeUppercase{\the\toks@}}{\the\@temptokena}}%
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\@tempa
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\endgroup
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}
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\hypersetup{
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pdfauthor={\authors},
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pdftitle={\@title},
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colorlinks,
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linkcolor=[rgb]{0.2,0.2,0.6},
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citecolor=[rgb]{0.2,0.6,0.2},
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urlcolor=[rgb]{0.6,0.2,0.2}}
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\makeatother
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%----------------------------------------------------
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% commands for temporarily enabling and disabling toc entires
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\let\oldaddcontentsline\addcontentsline
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\newcommand{\stoptocentries}{\renewcommand{\addcontentsline}[3]{}}
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\newcommand{\starttocentries}{\let\addcontentsline\oldaddcontentsline}
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% Begin of document ----------------------------------------
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\begin{document}
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\newgeometry{
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includeall,
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bindingoffset=0cm,
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margin=2cm,
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marginparsep=0cm,
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marginparwidth=0cm,
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top=0cm
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}
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\begin{titlepage}
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\vspace*{3cm}
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\centering{
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%\includegraphics[scale=0.35]{../unilogo.png}\\ % Uni Logo is forbidden by guidelines of the institution
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\vspace*{1.8cm}
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\hrulefill \\
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\vspace*{0.8cm}
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\textbf{\Large{The Hodge Decomposition for Compact Kähler Manifolds}}\\
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\vspace*{0.4cm}
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\hrulefill\\
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\vspace*{1.8cm}
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\large{Bachelor-Arbeit im Fach Mathematik \\
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zur Erlangung des akademischen Grades \\
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\vspace*{1cm}
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,,Bachelor of Science'' \\
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\vspace*{1cm}
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der Fakultät für Mathematik und Physik \\
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der Albert-Ludwigs-Universität Freiburg i.Br.}\\
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\vspace*{3cm}
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\large{vorgelegt von \\
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Daniel Rath} \\
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%\vspace*{0.5cm}
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\large{Matrikelnummer: 5100169\\}
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\vspace*{3cm}
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\large{im Sommersemester 2023\\
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betreut durch\\
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Prof. Dr. Stefan Kebekus\\}
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\vspace*{1cm}
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}
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\end{titlepage}
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\restoregeometry
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%\newpage
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\thispagestyle{empty}
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\cleardoublepage\null
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\pagenumbering{roman}
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\addtocounter{page}{-1}
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\makeheaders
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\include{preliminaries}
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\vspace*{-0.9cm}
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\tableofcontents
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\newpage
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\pagenumbering{arabic}
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\include{local-theory}
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\include{kaehler-manifolds}
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\include{harmonic-forms}
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\include{hodge-decomposition}
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\include{german-sum}
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\newpage
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\thispagestyle{empty}
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\printbibliography
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\vfill
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\newpage
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\thispagestyle{empty}
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\begin{center}
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\textbf{Eigenständigkeitserklärung}
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\end{center}
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Hiermit erkläre ich, dass ich diese Abschlussarbeit mit dem Titel \glqq{\emph{The Hodge Decomposition for Compact Kähler Manifolds}}\grqq{} selbstständig verfasst habe, keine anderen als die angegebenen Quellen verwendet habe und alle Stellen, die wörtlich oder sinngemäß aus veröffentlichten Schriften entnommen wurden, als solche kenntlich gemacht habe. Darüber hinaus erkläre ich, dass diese Abschlussarbeit nicht, auch nicht auszugsweise, bereits für eine andere Prüfung angefertigt wurde. \\
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\\
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\begin{flushleft}
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\makebox[.4\textwidth]{\hrulefill}\hfill \makebox[.4\textwidth]{\hrulefill}\\
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\makebox[.4\textwidth]{(Ort, Datum)}\hfill
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\makebox[.4\textwidth]{Daniel Rath}\\
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\end{flushleft}
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\vspace{3cm}
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\begin{center}
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\textbf{Erklärung der Übereinstimmung von digitaler und gedruckter Version}
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\end{center}
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Hiermit versichere ich, dass die eingereichte elektronische und gedruckte Version meiner Bachelorarbeit mit dem Titel \glqq{\emph{The Hodge Decomposition for Compact Kähler Manifolds}}\grqq{} über\-ein\-stim\-men. Insbesondere stimmen die beiden gedruckten Exemplare überein. \\
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\begin{flushleft}
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\makebox[.4\textwidth]{\hrulefill}\hfill \makebox[.4\textwidth]{\hrulefill}\\
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\makebox[.4\textwidth]{(Ort, Datum)}\hfill
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\makebox[.4\textwidth]{Daniel Rath}\\
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\end{flushleft}
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\end{document}
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