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Stefan Kebekus 2024-01-23 09:03:41 +01:00
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\documentclass[12pt,a4paper,twoside]{amsart} % Package for mathematical article
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
% Nice Font and some basic things
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% Nice Referencing
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% Math packages
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% New math commands
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% Bibliography
\usepackage[
backend=biber,
style=alphabetic,
maxnames=10,
maxalphanames=10]{biblatex}
\addbibresource{bibliography.bib}
% TODO functionality:
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% If all todo should be hidden at once just uncomment this line
% \renewcommand\TODO[1]{}
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% Begin of document
\title{THE HODGE DECOMPOSITION FOR COMPACT KÄHLER MANIFOLDS}
\author{Daniel Rath}
\date{}
\setcounter{tocdepth}{2} % Maybe use 1 here
\numberwithin{equation}{section}
\sloppy % Should be better
\makeatletter%---------------------------------------------------
% Customize Table of contents
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\begin{titlepage}
\vspace*{3cm}
\centering{
%\includegraphics[scale=0.35]{../unilogo.png}\\ % Uni Logo is forbidden by guidelines of the institution
\vspace*{1.8cm}
\hrulefill \\
\vspace*{0.8cm}
\textbf{\Large{The Hodge Decomposition for Compact Kähler Manifolds}}\\
\vspace*{0.4cm}
\hrulefill\\
\vspace*{1.8cm}
\large{Bachelor-Arbeit im Fach Mathematik \\
zur Erlangung des akademischen Grades \\
\vspace*{1cm}
,,Bachelor of Science'' \\
\vspace*{1cm}
der Fakultät für Mathematik und Physik \\
der Albert-Ludwigs-Universität Freiburg i.Br.}\\
\vspace*{3cm}
\large{vorgelegt von \\
Daniel Rath} \\
%\vspace*{0.5cm}
\large{Matrikelnummer: 5100169\\}
\vspace*{3cm}
\large{im Sommersemester 2023\\
betreut durch\\
Prof. Dr. Stefan Kebekus\\}
\vspace*{1cm}
}
\end{titlepage}
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\makeheaders
\include{preliminaries}
\vspace*{-0.9cm}
\tableofcontents
\newpage
\pagenumbering{arabic}
\include{local-theory}
\include{kaehler-manifolds}
\include{harmonic-forms}
\include{hodge-decomposition}
\include{german-sum}
\newpage
\thispagestyle{empty}
\printbibliography
\vfill
\newpage
\thispagestyle{empty}
\begin{center}
\textbf{Eigenständigkeitserklärung}
\end{center}
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. \\
\\
\begin{flushleft}
\makebox[.4\textwidth]{\hrulefill}\hfill \makebox[.4\textwidth]{\hrulefill}\\
\makebox[.4\textwidth]{(Ort, Datum)}\hfill
\makebox[.4\textwidth]{Daniel Rath}\\
\end{flushleft}
\vspace{3cm}
\begin{center}
\textbf{Erklärung der Übereinstimmung von digitaler und gedruckter Version}
\end{center}
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. \\
\begin{flushleft}
\makebox[.4\textwidth]{\hrulefill}\hfill \makebox[.4\textwidth]{\hrulefill}\\
\makebox[.4\textwidth]{(Ort, Datum)}\hfill
\makebox[.4\textwidth]{Daniel Rath}\\
\end{flushleft}
\end{document}

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@Article{Mori2007,
author = {Lapo F. Mori},
journal = {PracTex Journal},
title = {Writing a thesis with LaTex},
year = {2007},
abstract = {This article provides useful tools to write a thesis with LATEX. It analyzes the typical problems that arise while writing a thesis with LaTeX and suggests improved solutions by handling easy packages. Many suggestions can be applied to book and article styles, as well.},
file = {:references/mori.pdf:PDF},
groups = {Writing},
keywords = {Tex},
url = {https://tug.org/pracjourn/2008-1/mori/mori.pdf},
}
@Misc{Biquard2008,
author = {Biquard, Oliver and Höring, Andreas},
month = dec,
title = {Kähler Geometry and Hodge Theory},
year = {2008},
comment = {The aim of these lecture notes is to give an introduction to analytic geometry, that is the geometry of complex manifolds, with a focus on the Hodge theory of compact Kähler manifolds. The text comes in two parts that correspond to the distribution of the lectures between the two authors:
- the first part, by Olivier Biquard, is an introduction to Hodge theory, and more generally to the analysis of elliptic operators on compact manifolds.
- the second part, by Andreas Höring, starts with an introduction to complex manifolds and the objects (differential forms, cohomology theories, connections...) naturally attached to them. In Section 4, the analytic results established in the first part are used to prove the existence of the Hodge decomposition on compact Kähler manifolds. Finally in Section 5 we prove the Kodaira vanishing and embedding theorems which establish the link with complex algebraic geometry.},
file = {:references/biquard-höring.pdf:PDF},
groups = {hodge-theory},
ranking = {rank4},
}
@Book{Beauville1996,
author = {Beauville, Arnaud},
publisher = {Cambridge University Press},
title = {Complex algebraic surfaces},
year = {1996},
edition = {2},
isbn = {0521495105},
file = {:references/beauville.pdf:PDF},
groups = {Algebraic Geometry},
ranking = {rank4},
}
@Misc{Sheagren2018,
author = {Calder Sheagren},
note = {Found as a lecturenote online see timestamp for date},
title = {Introduction to Hodge Theory},
year = {2018},
abstract = {We introduce real and complex Hodge theory to study topological invariants using harmonic analysis. To do so, we review Riemannian and complex geometry, intro- duce de Rham cohomology, and give the basic theorems of real and complex Hodge theory. To conclude, we present an application of the complex Hodge decomposition for K ähler manifolds to topology by working out the example of the 2n-torus T2n = Cn/Z2n.},
file = {:references/sheagren.pdf:PDF},
groups = {Hodge Theory},
ranking = {rank4},
relevance = {relevant},
timestamp = {2023-03-23},
url = {http://math.uchicago.edu/~may/REU2018/REUPapers/Sheagren.pdf},
}
@Misc{Park2018,
author = {Peter S. Park},
month = mar,
title = {Hodge Theory},
year = {2018},
abstract = {This exposition of Hodge theory is a slightly retooled version of the author’s Harvard minor thesis, advised by Professor Joe Harris.},
file = {:references/peterspark.pdf:PDF},
groups = {Hodge Theory},
ranking = {rank2},
url = {https://scholar.harvard.edu/files/pspark/files/harvardminorthesis.pdf},
}
@Article{Filippini2014,
author = {Filippini, Sara Angela and Ruddat, Helge and Thompson, Alan},
journal = {Calabi-Yau Varieties: Arithmetic, Geometry and Physics, Fields Inst. Monogr., vol. 34, Springer, 2015, pp. 83-130},
title = {An Introduction to Hodge Structures},
year = {2014},
month = dec,
pages = {83--130},
abstract = {We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid's limiting mixed Hodge structure and Steenbrink's work in the geometric setting. Finally, we give an outlook about Hodge theory in the Gross-Siebert program.},
archiveprefix = {arXiv},
booktitle = {Calabi-Yau Varieties: Arithmetic, Geometry and Physics},
copyright = {arXiv.org perpetual, non-exclusive license},
doi = {10.1007/978-1-4939-2830-9_4},
eprint = {1412.8499},
file = {:http\://arxiv.org/pdf/1412.8499v2:PDF},
groups = {Hodge Theory},
keywords = {Algebraic Geometry (math.AG), FOS: Mathematics, 14C30},
primaryclass = {math.AG},
publisher = {Springer New York},
ranking = {rank3},
}
@Misc{Schwachhoefer20072008,
author = {Viktoria Vilenska},
howpublished = {Homepage of University},
note = {This is a lecture note found online. See timestamp for date.},
title = {Seminar über Kählermannigfaltigkeiten},
year = {2007-2008},
comment = {DO NOT USE:
http://www.mathematik.tu-dortmund.de/~lschwach/WS07/Kaehler-Seminar/},
file = {:references/schwachhöfer.pdf:PDF},
groups = {Hodge Theory},
ranking = {rank1},
timestamp = {2023-03-23},
url = {http://www.mathematik.tu-dortmund.de/~lschwach/WS07/Kaehler-Seminar/Hodge_Theorie.pdf},
}
@Misc{Koch2007,
author = {Artanc Kayacelebi and Jan-Christopher Koch},
howpublished = {Online},
month = dec,
title = {Fast komplexe Mannigfaltigkeiten und Vektorbündel},
year = {2007},
comment = {Do Not Use
There are some usefull definitions of E^(p,q) type differential forms and something about the complexification of the tangent space.},
file = {:references/Koch.pdf:PDF},
ranking = {rank1},
url = {http://www.mathematik.tu-dortmund.de/~lschwach/WS07/Kaehler-Seminar/Fast-Komplex.pdf},
}
@Book{Lee2012,
author = {Lee, John M.},
publisher = {Springer New York},
title = {Introduction to Smooth Manifolds},
year = {2012},
edition = {2},
number = {218},
series = {Graduate Texts in Mathematics},
doi = {10.1007/978-1-4419-9982-5},
file = {:references/lee2012.pdf:PDF},
printed = {printed},
qualityassured = {qualityAssured},
ranking = {rank4},
relevance = {relevant},
}
@Book{Bertin2002,
author = {Jose Bertin and Jean-Pierre Demailly and Luc Illusie and Chris Peters},
publisher = {American Mathematical Society},
title = {Introduction to Hodge theory},
year = {2002},
isbn = {0821820400},
abstract = {Hodge theory is a powerful tool in analytic and algebraic geometry. This book consists of expositions of aspects of modern Hodge theory, with the purpose of providing the nonexpert reader with a clear idea of the current state of the subject. The three main topics are: L2 Hodge theory and vanishing theorems; Hodge theory in characteristic p; and variations of Hodge structures and mirror symmetry. Each section has a detailed introduction and numerous references. Many open problems are also included. The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry. This is the English translation of a volume previously published as volume 3 in the Panoramas et Synthèses series.},
file = {:references/introduction-to-hodge-theory.pdf:PDF},
groups = {Hodge Theroy},
printed = {printed},
qualityassured = {qualityAssured},
ranking = {rank5},
relevance = {relevant},
timestamp = {2023-03-23},
translator = {James Lewis and Chris Peters},
url = {https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hodge-ams.pdf},
}
@Misc{Bouchard,
author = {Vincent Bouchard},
howpublished = {As a lecture note on the autors personal webpage},
note = {DO NOT USE!! Good explanation of the Hodge Star},
title = {4.8 Hodge Star},
ranking = {rank1},
url = {https://sites.ualberta.ca/~vbouchar/MATH215/section_hodge.html},
}
@Book{Jost2011,
author = {Jost, Jürgen},
publisher = {Springer-Verlag GmbH},
title = {Riemannian Geometry and Geometric Analysis},
year = {2011},
isbn = {9783642212987},
month = jul,
series = {Universitext},
comment = {Capter 3 is about harmonic forms, L2 and the Laplacian operator. Also there could be something about Kähler Manifolds in Chapter 6.},
ean = {9783642212987},
file = {:references/jost.pdf:PDF},
pagetotal = {611},
url = {https://www.ebook.de/de/product/16844179/juergen_jost_riemannian_geometry_and_geometric_analysis.html},
}
@Book{Voisin2002,
author = {Voisin, Claire},
publisher = {Cambridge University Press},
title = {Hodge Theory and Complex Algebraic Geometry I},
year = {2002},
isbn = {9780521718011},
number = {76},
series = {Cambridge Studies in Advanced Mathematics},
doi = {10.1017/cbo9780511615344},
priority = {prio1},
ranking = {rank5},
}
@Misc{4704312,
author = {Michael Albanese (https://math.stackexchange.com/users/39599/michael-albanese)},
howpublished = {Mathematics Stack Exchange},
note = {URL:https://math.stackexchange.com/q/4704312 (version: 2023-05-22)},
title = {Dimensions of underlying real spaces and dimensions of complexifications},
year = {2023},
eprint = {https://math.stackexchange.com/q/4704312},
keywords = {Stack Exchange},
url = {https://math.stackexchange.com/q/4704312},
}
@Book{Wells1986,
author = {Wells, Raymond O'Neil},
publisher = {Springer},
title = {Differential Analysis on Complex Manifolds},
year = {1986},
isbn = {9780387904191},
series = {Graduate Texts in Mathematics},
volume = {65},
comment = {Found in mathematical Library of the mathematical institut of the university of freiburg},
pages = {260},
}
@Book{Huybrechts2004,
author = {Huybrechts, Daniel},
publisher = {Springer},
title = {Complex Geometry},
year = {2004},
isbn = {9783540212904},
series = {Universitext},
doi = {10.1007/b137952},
file = {:references/huybrechts.pdf:PDF},
qualityassured = {qualityAssured},
ranking = {rank4},
subtitle = {An Introduction},
}
@Misc{1417853,
author = {user24142 (https://math.stackexchange.com/users/208255/user24142)},
howpublished = {Mathematics Stack Exchange},
note = {URL:https://math.stackexchange.com/q/1417853 (version: 2015-09-02)},
title = {How do you show that conjugate mapping, $ f(z)=\bar {z}$ isn't linear?},
eprint = {https://math.stackexchange.com/q/1417853},
url = {https://math.stackexchange.com/q/1417853},
}
@Book{Spivak1965,
author = {Spivak, Michael},
publisher = {Addison-Wesley},
title = {Calculus on manifolds},
year = {1965},
edition = {1},
isbn = {0805390219},
file = {:references/spivak-calculus-on-manifolds.pdf:PDF},
subtitle = {a modern approach to classical theorems of advanced calculus.},
}
@Book{Demailly1997,
author = {Demailly, Jean-Pierre},
publisher = {Citeseer},
title = {Complex analytic and differential geometry},
year = {1997},
file = {:references/demailly.pdf:PDF},
url = {https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=88c41281e6910c945cfa79ef6a70d799d6f3a8b4},
}
@Book{Warner1983,
author = {Frank W. Warner},
publisher = {Springer New York},
title = {Foundations of Differentiable Manifolds and Lie Groups},
year = {1983},
edition = {1},
number = {94},
series = {Graduate Texts in Mathematics},
doi = {10.1007/978-1-4757-1799-0},
file = {:references/Warner1983.pdf:PDF},
}
@Misc{Schnell2012,
author = {Christian Schnell},
howpublished = {Lecture notes published online},
title = {Complex manifolds},
year = {2012},
file = {:references/schnell2012.pdf:PDF},
url = {https://www.math.stonybrook.edu/~cschnell/pdf/notes/complex-manifolds.pdf},
}
@Book{Barth1984,
author = {W. Barth and C. Peters and A. Ven},
publisher = {Springer},
title = {Compact Complex Surfaces},
year = {1984},
edition = {1},
number = {3},
series = {Ergebnisse der Mathematik und ihrer Grenzgebiete / A series of modern surveys in mathematics},
volume = {4},
doi = {10.1007/978-3-642-96754-2},
file = {:references/barth1984.pdf:PDF},
}
@Misc{SEMark,
author = {Mark},
howpublished = {Mathematics Stack Exchange},
note = {URL:https://math.stackexchange.com/q/4718945 (version: 2023-06-14)},
title = {Dual of the complexification is complexification of the dual},
url = {https://math.stackexchange.com/q/4718945},
}
@Comment{jabref-meta: databaseType:bibtex;}
@Comment{jabref-meta: fileDirectoryLatex-danielrath-MBP-von-Daniel.fritz.box:/Users/danielrath/Documents/Studium/Bachelorarbeit/bachelor-thesis;}

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\section{\for{toc}{German Summary}\except{toc}{Zusammenfassung}}
\begin{otherlanguage}{ngerman}
Diese Arbeit beschäftigt sich mit der Hodge-Zerlegung für kompakte Kähler-Mannigfaltigkeiten,
welche eine der zentralen Aussagen der Hodge-Theorie ist. Sie liefert eine Zerlegung der
de-Rahm-Kohomologie-Gruppen in passende Dolbeault-Kohomologie-Gruppen und stellt somit eine
Verbindung zwischen den topologischen Eigenschaften und der komplexen Struktur einer kompakten
Kähler-Mannigfaltigkeit her.
Das Ziel dieser Arbeit ist die Ausarbeitung des Beweises dieser Zerlegung. Dafür muss die
erforderlich Theorie eingeführt und erklärt werden. Dabei ist es zunächst sinnvoll, die lokale
Theorie auszuarbeiten. Diese befasst sich hauptsächlich mit den Eigenschaften von euklidischen und
unitären Vektorräumen im Zusammenhang mit der Existenz einer kompatiblen fastkomplexen Struktur.
Dabei werden vor allem die Werkzeuge aus der Linearen Algebra gebraucht.
Mit den gesammelten Eigenschaften werden dann jeweils der Lefschetz-Operator, der duale
Lefschetz-Operator und der Hodge-Stern-Operator lokal definiert.
Danach wird der Fokus zunehmend auf Mannigfaltigkeiten gelegt. Nachdem hermitesche
Mannigfaltigkeiten definiert wurden, werden einige der zuvor erarbeiteten lokalen Aussagen in
globale Aussagen übersetzt. Außerdem werden die entsprechenden globalen Operatoren definiert. Dabei
wird jedoch angenommen, dass der Leser bereits mit den grundlegenden Begriffen und Eigenschaften von
komplexen und fastkomplexen Mannigfaltigkeiten vertraut ist.
Nachdem formal adjungierte Operatoren mithilfe einer vorher definierten $L^2$-Metrik eingeführt
wurden, werden dann die Kähler-Identitäten behandelt. Diese stellen die zuvor eingeführten globalen
Operatoren in Relation zueinander und sind äußerst essenzielle Eigenschaften von
Kähler-Mannigfaltigkeiten. Diese Kähler-Identitäten werden in dieser Arbeit jedoch nicht bewiesen.
Das nächste Ziel ist der Beweis der Hodge-Isomorphie-Sätze. Dafür wird die Theorie der harmonischen
Differentialformen eingeführt und einige wichtige Eigenschaften werden bewiesen. Dafür werden die
zuvor behandelten Kähler-Identitäten benötigt.
Danach wird mithilfe dieser Isomorphie-Sätze die Hodge-Zerlegung bewiesen. Außerdem wird gezeigt,
dass diese Zerlegung unabhängig von der Wahl der Kähler-Metrik ist. Am Ende wird dann eine nützliche
topologische Anwendung der Hodge-Zerlegung präsentiert.
\end{otherlanguage}

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\section{Harmonic differential forms and cohomology}
In this chapter, we are going to introduce the theory of harmonic differential forms and review the
concepts of the de Rahm and Dolbeault cohomologies. Our primary objective is the proof of two Hodge
Isomorphism theorems, which establish a fundamental relation between the spaces of harmonic forms
and the cohomologies. These theorems will be crucial for our proof of the Hodge Decomposition
theorem later.
This chapter is based on sections 5.1 to 5.3 and 6.1 in the book \emph{Hodge Theory and Complex
Algebraic Geometry} \cite{Voisin2002} written by Claire Voisin. Some of the statements also
originate from chapter \MakeUppercase{\romannumeral 6} in the book \emph{Complex Analytic and
Differential Geometry} \cite{Demailly1997} written by Jean-Pierre Demailly.
For the remainder of this chapter, we will assume the following setting.
\begin{set}
Let $X$ be a compact $m$-dimensional Kähler manifold with Riemannian metric $g$, closed
fundamental form $\omega$ %\in\mathcal{A}^2_\mathbb{R} \cap \mathcal{A}^{1,1}(X)$
and Kähler metric $h:= g -i\omega$.
\end{set}
\subsection{Harmonic forms and the Laplacians}\;
Our first goal in this chapter is the definition of a generalization of the Laplace operator
$\Delta$. This operator is already known for $\mathbb{R}$-valued functions on the euclidean space
$\mathbb{R}^n$ and we will generalize it to obtain the Laplacians $\Delta_d, \Delta_\partial$ and
$\Delta_\opartial$ on the smooth differential forms $\mathcal{A}^\bullet_\mathbb{C}(X)$.
Based on the definition of harmonic functions in the euclidean case, we will use these generalized
Laplacians to define different types of harmonic differential forms. After that, we will further
develop this theory and focus on an important theorem that establishes a relation between the three
Laplacians. In order to prove this theorem, we will need the Kähler identities established in the
last chapter.
\begin{defn}[Laplacians]
For every $k$, we define the \emph{Laplacian of the exterior derivative} $d$ as a linear operator
$\Delta_d: \mathcal{A}^k_\mathbb{C}(X)\rightarrow\mathcal{A}^k_\mathbb{C}(X)$ that
is given as
\begin{equation*}
\Delta_d := d\,d^* + d^*d.
\end{equation*}
Similarly, the \emph{Laplacians of the Dolbeault operators} $\partial$ and $\opartial$ are defined as
\begin{align*}
\Delta_\partial := \partial\,\partial^* + \partial^*\partial \;\enspace \text{ and } \; \enspace
\Delta_{\opartial}:= \opartial\,\opartial^* + \opartial^*\opartial
\end{align*}
and it is by definition that these two operators are linear mappings of type
$\mathcal A^{p,q}(X) \rightarrow\nolinebreak \mathcal A^{p,q}(X)$.
\end{defn}
\begin{rem}[{\cite[p.\,126]{Voisin2002}}]
We already know that $d,\partial$ and $\opartial$ are linear differential operators of order 1, and
the same is true for the formal adjoints $d^*,\partial^*$ and $\opartial^*$. Since the Laplacians
are all defined as a concatenation and sum of these linear differential operators, they are also
linear differential operators but of order 2.
\end{rem}
\begin{defn}[Harmonic differential forms]
Let $\alpha \in \mathcal{A}^k_\mathbb{C}(X)$ be a complex differential form. We are going to call
$\alpha$ \emph{harmonic} if it is $\alpha \in \ker(\Delta_d)$. Furthermore, if $\alpha$ is of type
$(p,q)$, we will call it $\mathit{\Delta_\opartial}$\emph{-harmonic} if
$\alpha \in \ker(\Delta_\opartial)$.
\end{defn}
At this point, we will prove an essential equality that will be required for subsequent calculations.
\begin{lm}[{\cite[Lemma 5.12]{Voisin2002}}]
For the Laplacian $\Delta_d$, we have the equality
\begin{align}
\label{harmonic-forms:eq:property-laplacian}
\left(\alpha,\Delta_d\alpha\right)_{L^2} = \left(d\alpha,d\alpha\right)_{L^2} +
\left(d^*\alpha,d^*\alpha\right)_{L^2}.
\end{align}
Also, for the other two Laplacians $\Delta_\partial$ and $\Delta_\opartial$, the analogous
equalities are valid as well.
\end{lm}
\begin{proof}This proof was taken from the given lemma in \cite{Voisin2002}.
With the definition of the Laplacian $\Delta_d$ and the formal adjunction properties \Cref{kaehler-manifolds:eq:formal-adjunction-property,kaehler-manifolds:eq:formal-adjunction-property-2},
we get
\begin{align*}
\left(\alpha,\Delta_d \alpha\right)_{L^2} = \left(\alpha, d\,d^* \alpha + d^*d
\alpha\right)_{L^2}&= \left(\alpha,d\,d^*\alpha\right)_{L^2} + \left(\alpha, d^*d\alpha\right)_{L^2}\\
%\\&= %\left(d^* \alpha, d^*\alpha\right)_{L^2} + \overline{\left(d^*d\alpha, \alpha\right)}_{L^2}
%\\&= (d^*\alpha,d^*\alpha)_{L^2} + \overline{(d\alpha,d\alpha)}_{L^2}
%\overline{\left(d\,d^*\alpha,\alpha\right)}_{L^2} + \left(\alpha,d^*d\alpha\right)_{L^2} \\&=
%\overline{\left(d^*\alpha,d^*\alpha\right)} + \left(d\alpha,d\alpha\right)_{L^2}
&= \left(d^*\alpha,d^*\alpha\right)_{L^2} + \left(d\alpha,d\alpha\right)_{L^2}
\end{align*}
and the analogous calculation can be used for the Laplacians $\Delta_\partial$ and
$\Delta_\opartial$ too.
\end{proof}
Suppose there exists an $\alpha \in \mathcal A^k_\mathbb{C}(X)$ such that $\Delta_d \alpha = 0$.
We can use the equality established in \Cref{harmonic-forms:eq:property-laplacian} to obtain
\begin{align*}
%\label{kaehler-maifolds:eq:laplacian-zero}
0 = (\alpha,\Delta_d\alpha)_{L^2} = \left(d^*\alpha,d^*\alpha\right)_{L^2} +
\left(d\alpha,d\alpha\right)_{L^2}.
\end{align*}
As mentioned in \Cref{kaehler-manifolds:rem:l2-metric-properties}, the hermitian $L^2$-metric is
positive definite, so both of these summands have to be equal to zero, which is only the case if
$d^*\alpha = 0$ and also $d\alpha =0$. If, on the other hand, $d\alpha = 0$ and $d^*\alpha = 0$,
then it is by definition of the Laplacian $\Delta_d \alpha = 0$. This already proves the following corollary.
\begin{cor}[{\cite[Corollary 5.13]{Voisin2002}}]
\label{harmonic-forms:lm:kernel-laplacian}
For the kernel of the Laplacian $\Delta_d$, we have the relation
\begin{align*}
\ker(\Delta_d) = \ker(d) \cap \ker(d^*).
\end{align*}
In particular, $\alpha \in \mathcal{A}^k_\mathbb{C}(X)$ is harmonic if and only if \,
$d\alpha = d^*\alpha = 0$.
\end{cor}
It is also obvious that the same calculation can be repeated for the other two Laplacians
$\Delta_\partial$ and $\Delta_\opartial$. Hence, we also get the same relations for the kernels
of these Laplacians.
\begin{cor}
\label{harmonic-forms:rem:kernel-laplacian-partial-opartial}
For the kernels of the Laplacians $\Delta_\partial$ and $\Delta_\opartial$, we have the relations
\begin{align*}
\ker(\Delta_\partial) = \ker(\partial) \cap \ker(\partial^*) \; \enspace \text{ and } \;
\enspace \ker(\Delta_\opartial) = \ker(\opartial) \cap \ker(\opartial^*).
\end{align*}
In particular, $\alpha \in \mathcal{A}^{p,q}(X)$ is $\Delta_\opartial$-harmonic if and
only if\, $\opartial\alpha = 0$ and $\opartial^*\alpha = 0$.
\end{cor}
\begin{prop}[{\cite[p.\,368]{Demailly1997}}]
\label{harmonic-forms:lm:laplacians-are-self-adjoint}
The Laplacians $\Delta_d, \Delta_\partial$ and $\Delta_\opartial$ are formally self-adjoint.
\end{prop}
\begin{proof}
We calculate for all $\alpha,\beta \in \mathcal{A}^k_\mathbb{C}(X)$
\begin{align*}
(\Delta_d\alpha,\beta)_{L^2} &= (dd^*\alpha + d^*d\alpha,\beta)_{L^2}\\
&= (dd^*\alpha,\beta)_{L^2} + (d^*d\alpha,\beta)_{L^2} \\
&= (d^*\alpha,d^*\beta)_{L^2} + (d\alpha,d\beta)_{L^2}.
\end{align*}
Furthermore, we are able to calculate
\begin{align*}
(\alpha,\Delta_d\beta)_{L^2} &= (\alpha,dd^* \beta + d^*d\beta)_{L^2}\\
&= (\alpha,dd^*\beta)_{L^2} + (\alpha, d^*d\beta)_{L^2}\\
&= (d^*\alpha,d^*\beta)_{L^2} + (d\alpha,d\beta)_{L^2}
\end{align*}
This already proves that $\Delta_d$ is formally self-adjoint, and the argument for the other
two Laplacians is exactly the same.
\end{proof}
At this point, we have established the necessary theory to prove the existence of a previously
indicated relation between the different Laplacians. This relation will be fundamental to the proof
of the Hodge Decomposition theorem.
\begin{thm}[{\cite[Theorem 6.7]{Voisin2002}}]
For the Laplacians $\Delta_d,\Delta_\partial$ and $\Delta_\opartial$ on our compact Kähler manifold
$X$, we have the following relation
\begin{align}
\label{harmonic-forms:eq:thm-laplacian-realtion}
\frac{1}{2}\Delta_d = \Delta_\partial = \Delta_\opartial.
\end{align}
In particular, any differential form $\alpha \in \mathcal{A}^k_\mathbb{C}(X) \cap
\mathcal{A}^{p,q}(X)$ is harmonic if and only if it is $\Delta_\opartial$-harmonic.
\end{thm}
\begin{proof}
We provide a similar argument as in the proof of \cite[Theorem 6.7]{Voisin2002} but our approach
will be slightly different. With the splitting of both, the exterior derivative $d = \partial + \opartial$
and the formal adjoint of the exterior derivative $d^* = \partial^* +\,\opartial^*$
(cf. \Cref{kaehler-manifolds:rem:splitting-of-the-formal-adjoint-of-the-exterior-derivative}), the
definition of the Laplacian $\Delta_d$ can be rewritten as follows.
\begin{align*}
\Delta_d &= d\,d^* + d^*d \\
&= \big(\partial + \opartial\big)\big(\partial^* + \opartial^*\big) +
\big(\partial^* + \opartial^*\big) \big(\partial + \opartial\big)\\
&= \big(\partial\partial^* + \partial \opartial^* + \opartial\partial^* + \opartial\,\opartial^*\big) +
\big(\partial^*\partial +\partial^*\opartial +\opartial^* \partial + \opartial^*\opartial\big)\\
&=\partial\partial^* + \partial^*\partial + \partial\opartial^* + \opartial^*\partial +
\opartial\partial^* +\partial^*\opartial + \opartial\,\opartial^* + \opartial^*\opartial%\\&=
%\Delta_\partial + \partial\opartial^* + \opartial^*\partial + \opartial\partial^*
%+\partial^*\opartial + \opartial\,\opartial^* + \opartial^*\opartial
\end{align*}
Now, we are going to use the Kähler identities established in
\Cref{kaehler-manifolds:thm:kaehler-identities}, in particular the equality
$\partial^* = i[\Lambda,\opartial]$, to get
\begin{align*}
\opartial\partial^* + \partial^*\opartial &= \opartial i [\Lambda,\opartial] +
i[\Lambda,\opartial]\opartial \\
&=i\big(\opartial(\Lambda\opartial-\opartial\Lambda)+(\Lambda\opartial-\opartial\Lambda)\opartial\big)\\
&=i\big(\opartial\Lambda\opartial - \opartial^2\Lambda + \Lambda\opartial^2 - \opartial\Lambda\opartial\big)\\
&=i\big(\opartial\Lambda\opartial - \opartial\Lambda\opartial\big) \\
&=0.
\end{align*}
Applying the other Kähler identity $\opartial^* = -i[\Lambda,\partial]$ from
\Cref{kaehler-manifolds:thm:kaehler-identities}, we can do a similar calculation to obtain
\begin{align*}
\partial\opartial^* + \opartial^*\partial &= -i\partial[\Lambda,\partial] -i[\Lambda,\partial]\partial \\
& = -i\big(\partial(\Lambda\partial - \partial\Lambda) + (\Lambda\partial -\partial\Lambda)\partial\big) \\
&= -i\big(\partial\Lambda\partial -\partial^2\Lambda + \Lambda\partial^2 - \partial\Lambda\partial\big) \\
&= -i\big(\partial\Lambda\partial - \partial\Lambda\partial\big) \\
&= 0.
\end{align*}
Thus, the above equation simplifies to
\begin{align*}
\Delta_d =\partial\partial^* + \partial^*\partial + \opartial\,\opartial^* +
\opartial^*\opartial= \Delta_\partial + \Delta_\opartial.
\end{align*}
Therefore, it is enough to prove $\Delta_\partial = \Delta_\opartial$. However, we can calculate
\begin{align*}
\Delta_\partial - \Delta_\opartial &= \partial\partial^* + \partial^*\partial -
\big(\opartial\,\opartial^* + \opartial^*\opartial\big)\\
&=i\big(\partial[\Lambda,\opartial] + [\Lambda,\opartial]\partial + \opartial[\Lambda,\partial]
+ [\Lambda,\partial]\opartial\big)\\
&=i\big(\partial\Lambda\opartial -\partial\opartial\Lambda + \Lambda\opartial\partial -
\opartial\Lambda\partial + \opartial\Lambda\partial -\opartial\partial\Lambda +
\Lambda\partial\opartial - \partial\Lambda\opartial\big)\\
&=i\big(\Lambda\opartial\partial-\partial\opartial\Lambda - \opartial \partial\Lambda +
\Lambda\partial\opartial\big) \\&= i\big(\Lambda(\opartial\partial + \partial\opartial) -
(\partial\opartial + \opartial\partial) \Lambda\big)\\
&= 0.
\end{align*}
For the last step, we have used
$0 = d^2 = \partial^2 + \partial\opartial + \opartial\partial + \opartial^2 = \partial\opartial + \opartial\partial$.
Finally, we combine both of these results to obtain
\begin{align*}
\Delta_d = 2 \Delta_\partial = 2 \Delta_\opartial\,,
\end{align*}
which completes the proof of this theorem.
\end{proof}
%\begin{rem}
% %\TODO{Counter example for this failing on arbitrary hermitian manifolds.}
%\end{rem}
The relation established in \Cref{harmonic-forms:eq:thm-laplacian-realtion}
has some significant implications for the operator $\Delta_d$ and the property
of a differential form being harmonic.
\begin{cor}
\label{harmonic-forms:cor:harmonic-if-and-only-if-delta-opartial-harmonic}
A complex differential form $\alpha \in \mathcal A^k_\mathbb{C}(X) \cap \mathcal A^{p,q}(X)$
is harmonic if and only if it is $\Delta_\opartial$-harmonic.
\end{cor}
\begin{proof}
It is $\Delta_d\alpha = 2\Delta_\opartial\,\alpha$, so either both operators map $\alpha$ to zero
or neither of them.
\end{proof}
When dealing with Kähler manifolds, the last corollary allows us not to distinguish between harmonic
and $\Delta_\opartial$-harmonic forms anymore. Therefore, we will just refer to harmonic forms
from now on.
\begin{cor}[{\cite[Corollary. 6.9]{Voisin2002}}]
\label{harmonic-forms:cor:harmonic-if-and-only-if-components-are-harmonic}
A complex differential form $\alpha \in \mathcal{A}^k_\mathbb{C}(X)$ is harmonic if and only if its
components $\alpha^{p,q}$ of type $(p,q)$ are harmonic.
\end{cor}
\begin{proof}
With the previously used decomposition
\begin{align}
\label{harmonic-forms:eq:decomposition-of-kforms}
\mathcal{A}^k_\mathbb{C}(X) = \bigoplus_{p+q = k} \mathcal{A}^{p,q}(X),
\end{align} (cf. \cite[Corollary 2.6.8]{Huybrechts2004})
we can uniquely write $\alpha = \sum_{p+q=k} \alpha^{p,q}$. As $\Delta_d$ is a linear operator,
the sum of harmonic forms is harmonic again. Therefore, it suffices to prove that $\alpha$
being harmonic already implies that the components $\alpha^{p,q}$ are harmonic.
However, if $\alpha$ is harmonic we can use \Cref{harmonic-forms:eq:thm-laplacian-realtion} to obtain
\begin{align*}
0 = \Delta_d \alpha = \sum_{p+q = k} \Delta_d \alpha^{p,q} =
\sum_{p+q = k} 2\,\Delta_\opartial\,\alpha^{p,q} = 2\sum_{p+q = k} \Delta_\opartial\, \alpha^{p,q}.
\end{align*}
Since $\Delta_\opartial $ maps forms of type $(p,q)$ to forms of type $(p,q)$ and we also have
decomposition \Cref{harmonic-forms:eq:decomposition-of-kforms}, we conclude
that $\Delta_\opartial\,\alpha^{p,q} = 0$ for each $\alpha^{p,q}$. %$p,q \in \mathbb{N}$ with $p+q = k$.
Hence with \Cref{harmonic-forms:cor:harmonic-if-and-only-if-delta-opartial-harmonic}, the
components $\alpha^{p,q}$ are each harmonic.
\end{proof}
\begin{nota}
Let $\mathcal H^k(X) \subset \mathcal{A}_\mathbb{C}^k(X)$ denote the space of complex harmonic
differential $k$-forms. Let also $\mathcal H^{p,q}(X) \subset \mathcal{A}^{p,q}(X)$ denote the space
of complex harmonic differential forms of type $(p,q)$, which is also the space of
$\Delta_\opartial$-harmonic differential forms of type $(p,q)$
(cf. \Cref{harmonic-forms:cor:harmonic-if-and-only-if-delta-opartial-harmonic}).
\end{nota}
With this newly introduced notation, we get the following corollary, which is just a rewritten version
of \Cref{harmonic-forms:cor:harmonic-if-and-only-if-components-are-harmonic} combined with the direct
sum decomposition in \Cref{harmonic-forms:eq:decomposition-of-kforms}.
\begin{cor}[{\cite[Corollary 6.10]{Voisin2002}}]
\label{harmonic-forms:cor:final-harmonic-decomposition}
The harmonic differential $k$-forms decompose as
\begin{align*}
\mathcal H^k(X) = \bigoplus_{p+q=k}\mathcal H^{p,q}(X).
\end{align*}
\end{cor}
\subsection{Elliptic differential operators}\;
Before we can proceed with the harmonic differential forms theory, we need to extend our
understanding of linear differential operators. Therefore, we will introduce elliptic operators and
address their implications on the previously established theory of harmonic differential forms.
Note, however, that we will only provide a very brief overview and refer to other sources because a
more detailed discussion would be beyond the scope of this thesis. For an in-depth discussion of
elliptic differential operators and their properties, see §1. and §2. of chapter VI in
\cite{Demailly1997}.
At first, we have the following definition, motivated by the discussion in \cite[Section 15]{Schnell2012}
and the explanations at the beginning of Section 5.2.1 in \cite{Voisin2002}.
\begin{defn}
In the same setting as in \Cref{kaehler-manifolds:defn:differential-opperators}, for every
$D_{U_j}$, we define a function
$P_j: U_j \times \mathbb{C}^m \rightarrow \mathbb{C}^{r_1\times r_2}$ as
\begin{align*}
P_j(p,\xi) := \sum_{S} \big(P_{r,S,t}(p)\big)_{rt} \xi^S
\end{align*}
with $\big(P_{r,S,t}(p)\big)_{rt} \in \mathbb{C}^{r_1\times r_2}$ being the matrix defined by the
coefficients $P_{r,S,t}(p) \in \mathbb{C}$, $S := (s_1,\dots,s_m)$ and
$\xi^S:= \xi_1^{s_1} \cdots \xi_m^{s_m}$. The functions $P_j$ define the \emph{symbol} of the
operator $D$.
\end{defn}
\begin{defn}[Elliptic differential operator]
In the setting of \Cref{kaehler-manifolds:defn:differential-opperators}, the linear differential
operator $D$ is said to be \emph{elliptic} if $r_1 = r_2$ and for any local symbol $P_j$, the
image $P_j(p,\xi)$ is an invertible matrix for all $p \in U_j$ and $\xi \in \mathbb{C}^n\setminus\{0\}$.
\end{defn}
\begin{rem}
The usual definition of an elliptic operator only requires the matrices to define injective
homomorphisms (cf. \cite[Ch.\,VI §1. Definition 1.8]{Demailly1997}). However, the ranks $r_1$ and
$r_2$ will always be equal in our setting. Therefore, if we additionally require these ranks to be
equal, these definitions are equivalent for our purposes.
\end{rem}
Through an extensive calculation (cf. \cite[Lemmas 5.18 and 5.19]{Voisin2002}), it is possible to
calculate the symbols of the three Laplacians and to conclude the following proposition.
\begin{prop}[{\cite[Corollary 5.20]{Voisin2002}}]
\label{harmonic-forms:lm:laplacians-are-elliptic}The Laplacians $\Delta_d,\Delta_\partial$ and
$\Delta_\opartial$ are each elliptic differential operators.
\end{prop}
After establishing this property, we are able to apply a fundamental result from the theory of
Sobolev spaces. There is a more general version with proof in {\cite[Ch.\,VI §2. Corollary 2.4]{Demailly1997}}
but in our setting, this theorem states the following.
\begin{thm}
\label{harmonic-forms:thm:fundamental-theorem}
Let $D: \mathcal{A}^k_\mathbb{C}(X) \rightarrow \mathcal{A}^k_\mathbb{C}(X)$ be an elliptic linear
differential operator. Also let
$D^*: \mathcal{A}^k_\mathbb{C}(X)\rightarrow \mathcal{A}^k_\mathbb{C}(X)$
be its formal adjoint with respect to the hermitian $L^2$-metric. Then, $D$ has the following
properties:
\begin{enumerate}
\item $\mathit{ker(D)}$ is of finite dimension and
$\mathit{Im(D)}\subset\mathcal{A}^k_\mathbb{C}(X)$ is closed and of finite codimension.
\label{harmonic-forms:prop:fundamental-theorem-1}
\item There is a decomposition
\begin{align*}
\mathcal{A}^k_\mathbb{C}(X) = Im(D) \oplus \ker(D^*),
\end{align*}
which is an orthogonal direct sum decomposition with respect to the $L^2$-metric.
\end{enumerate}
\end{thm}
\begin{rem}
\label{harmonic-forms:rem:elliptic-theorem}
For $D: \mathcal{A}^{p,q}(X) \rightarrow \mathcal{A}^{p,q}(X)$ an elliptic operator with formal
adjoint $D^*: \mathcal{A}^{p,q}(X) \rightarrow \mathcal{A}^{p,q}(X)$, we get a similar theorem,
but it is instead $\img(D) \subset \nolinebreak \mathcal{A}^{p,q}(X)$ closed and the
decomposition is given as
\begin{align*}
\mathcal{A}^{p,q}(X) = \img(D) \oplus \ker(D^*).
\end{align*}
\end{rem}
At this point, we have two corollaries resulting from this theorem together with the previously
established fact that the Laplacians are elliptic operators.
\begin{cor}[{\cite[p.\,58]{Schnell2012}}]
\label{harmonic-forms:lm:finite-dimensional}
If regarded as complex vector spaces, it is $\dim_\mathbb{C} \mathcal{H}^k(X) < \infty$ and
$\dim_\mathbb{C} \mathcal{H}^{p,q}(X) < \infty$.
\end{cor}
\begin{proof}
This is a direct consequence of \Cref{harmonic-forms:lm:laplacians-are-elliptic}
and \creflmpart{harmonic-forms:thm:fundamental-theorem}{harmonic-forms:prop:fundamental-theorem-1}.
% Note custom command as defined in main_thesis
\end{proof}
\begin{cor}[{\cite[p.\,129]{Voisin2002}}]
For the differential $k$-forms, we have the orthogonal decomposition
\begin{align}
\label{harmonic-forms:eq:decomposition-of-the-k-forms}
\mathcal{A}^k_\mathbb{C}(X) = \Delta_d(\mathcal{A}^k_\mathbb{C}(X)) \oplus \mathcal H^k(X).
\end{align}
At the same time, the differential forms of type $(p,q)$ decompose orthogonally as
\begin{align}
\label{harmonic-forms:eq:decomposition-of-the-pq-forms}
\mathcal{A}^{p,q}(X) = \Delta_\opartial(\mathcal{A}^{p,q}(X)) \oplus \mathcal H^{p,q}(X).
\end{align}
\end{cor}
\begin{proof}
If we combine \Cref{harmonic-forms:lm:laplacians-are-elliptic},
\Cref{harmonic-forms:thm:fundamental-theorem} and the fact that $\Delta_d$ is formally self-adjoint
(cf. \Cref{harmonic-forms:lm:laplacians-are-self-adjoint}), we get
\begin{align*}
\mathcal{A}^k_\mathbb{C}(X) = \img(\Delta_d) \oplus \ker(\Delta_d^*) =
\img(\Delta_d) \oplus \ker(\Delta_d) =
\Delta_d(\mathcal{A}^k_\mathbb{C}(X)) \oplus \mathcal H^k(X).
\end{align*}
Using \Cref{harmonic-forms:rem:elliptic-theorem}, the same argument yields
\begin{align*}
\mathcal{A}^{p,q}(X) = \img(\Delta_\opartial) \oplus \ker(\Delta_\opartial^*) =
\img(\Delta_\opartial) \oplus \ker(\Delta_\opartial) =
\Delta_\opartial(\mathcal{A}^{p,q}(X)) \oplus \mathcal H^{p,q}(X).
\end{align*}
\end{proof}
\subsection{De Rahm and Dolbeault cohomologies}\;
In the next section, we aim to prove the Hodge Isomorphism theorems, which state that every class of
closed differential forms (of type $(p,q)$) possesses a unique harmonic representative (of type
$(p,q)$). Although we will not conduct an in-depth discussion of the theory behind cohomology, we
will briefly revisit the fundamental definitions of the de Rahm and Dolbeault cohomologies to
provide the necessary context. For a more detailed insight, see the chapter
\emph{Sheaves and Cohomology} in \cite{Voisin2002}.
\begin{defn}[de Rahm complex]
The \emph{de Rahm complex} is defined as
\begin{align}
\label{harmonic-forms:eq:derahm}
0 \longrightarrow C^\infty(X,\mathbb{C})\cong\mathcal{A}^0_\mathbb{C}(X) \xlongrightarrow{d_0}
\mathcal{A}^1_\mathbb{C}(X) \xlongrightarrow{d_1} \dots \xlongrightarrow{d_{2m-1}}
\mathcal{A}^{2m}_\mathbb{C}(X) \xlongrightarrow{d_{2m}} 0,
\end{align}
with $d_k: \mathcal{A}^k_\mathbb{C}(X) \rightarrow \mathcal{A}^{k+1}_\mathbb{C}(X)$ being the
restriction of the exterior derivative to the differential $k$-forms $\mathcal{A}^k_\mathbb{C}(X)$.
\end{defn}
Note that the property $d^2 = 0$ ensures that this is indeed a complex.
Using this property, we are able to define the de Rahm cohomology.
\begin{defn}[de Rahm cohomology]
Let $Z^k(X,\mathbb{C}):= \ker(d_k)$ be the vector space of \emph{closed} differential $k$-forms and
$B^k(X,\mathbb{C}):= \img(d_{k-1})$ be the vector space of \emph{exact} differential $k$-forms. The
\emph{$k$-th complex de Rahm cohomology group} is defined as
\begin{align*}
H^k_{dR}(X,\mathbb{C}) := Z^k(X,\mathbb{C}) / B^k(X,\mathbb{C}).
\end{align*}
It is $B^k(X,\mathbb{C}) \subset Z^k(X,\mathbb{C})$ because \Cref{harmonic-forms:eq:derahm} is
a complex, so this is well-defined.
\end{defn}
\begin{defn}[Dolbeault complex]
Similar to \Cref{harmonic-forms:eq:derahm}, we define the \emph{$p$-th Dolbeault complex} for all
$0 \leq p \leq m$ as
\begin{align}
\label{harmonic-forms:eq:dolbeault}
0 \longrightarrow \mathcal{A}^{p,0}(X) \xlongrightarrow{\opartial_0}\mathcal{A}^{p,1}(X)
\xlongrightarrow{\opartial_1}\dots\xlongrightarrow{\opartial_{m-1}}\mathcal{A}^{p,m}(X)
\xlongrightarrow{} 0
\end{align}
with $\opartial_k: \mathcal{A}^{p,k}(X) \rightarrow \mathcal{A}^{p,k+1}(X)$ being the restriction
of $\opartial$ to the differential forms of type $(p,k)$.
\end{defn}
It is again obvious that this is indeed a complex because we have $\opartial^2 = 0$. Using this
property, we can define the Dolbeault cohomology.
\begin{defn}[Dolbeault cohomology]
Let $Z_p^k(X,\mathbb{C}):= \ker(\opartial_k)$ be the vector space of $\opartial$-closed
differential forms and $B_p^k(X,\mathbb{C}):= \img(\opartial_{k-1})$ be the vector space of
$\opartial$-exact differential forms of type $(p,k)$. The
\emph{$k$-th Dolbeault cohomology group} of the $p$-th Dolbeault complex is defined as
\begin{align*}
H^{p,k}_\opartial(X,\mathbb{C}) := Z_p^k(X,\mathbb{C}) / B_p^k(X,\mathbb{C}).
\end{align*}
It is $B_p^k(X,\mathbb{C}) \subset Z_p^k(X,\mathbb{C})$ because \Cref{harmonic-forms:eq:dolbeault}
is a complex, so this is well-defined.
\end{defn}
Before we are able to prove the Hodge Isomorphism theorems, we have to show the following propositions.
\begin{prop}
\label{harmonic-forms:lm:harmonic-forms-are-closed}
Let $\alpha\in\mathcal{H}^k(X)$ be harmonic, then $\alpha$ is also closed.
\end{prop}
\begin{proof}
In \Cref{harmonic-forms:lm:kernel-laplacian}, it was shown that $\ker(\Delta_d) = \ker(d) \cap
\ker(d^*)$. Therefore, it is $\mathcal{H}^k(X) \subset \ker(d)$ and the statement is proven.
\end{proof}
\begin{prop}[{\cite[Ch.\,VI §3. Theorem 3.16]{Demailly1997}}]
\label{harmonic-forms:lm:improved-decomposition-of-the-k-forms}
For the differential $k$-forms, there is another orthogonal decomposition given as
\begin{align}
\mathcal{A}^k_\mathbb{C}(X) = \mathcal{H}^k(X) \oplus d(\mathcal A^{k-1}_\mathbb{C}(X)) \oplus
d^*(\mathcal A^{k+1}_\mathbb{C}(X)).
\end{align}
\end{prop}
\begin{proof}
This is a restated version of the proof of the given theorem in \cite{Demailly1997}.
With the definition of the Laplacian $\Delta_d$, it is obvious that
$\Delta_d(\mathcal{A}^k_\mathbb{C}(X))\subset d(\mathcal A^{k-1}_\mathbb{C}(X)) \oplus d^*(\mathcal A^{k+1}_\mathbb{C}(X))$.
Thus, because of \Cref{harmonic-forms:eq:decomposition-of-the-k-forms}, it is enough to prove the
orthogonality of the decomposition. Therefore, let $\alpha\in \mathcal{A}^{k+1}_\mathbb{C}(X)$
and $\beta \in \mathcal{A}_\mathbb{C}^{k-1}(X)$. It \nolinebreak is
\begin{align*}
(d^*\alpha,d\beta)_{L^2} = (\alpha,d^2\beta)_{L^2} = 0.
\end{align*}
Hence, $d(\mathcal A^{k-1}_\mathbb{C}(X))$ and $d^*(\mathcal A^{k+1}_\mathbb{C}(X))$ are orthogonal.
With the previously established equality $\ker(\Delta_d) = \ker(d) \cap \ker(d^*)$
(cf. \Cref{harmonic-forms:lm:kernel-laplacian}), it is also for all $\gamma \in \mathcal{H}^k(X)$
\begin{align*}
(\gamma,d\beta)_{L^2} = (d^*\gamma,\beta)_{L^2} = (0,\beta)_{L^2} = 0 \; \text{ and } \;
(d^*\alpha,\gamma)_{L^2} = (\alpha,d\gamma)_{L^2} = (\alpha, 0)_{L^2} = 0.
\end{align*} Therefore, the statement is proven.
\end{proof}
\begin{cor}
The kernel of the exterior derivative $d$ decomposes orthogonally as
\begin{align}
\label{harmonic-forms:eq:kernel-of-the-exterior-derivative}
\ker(d) = \mathcal{H}^k(X) \oplus d(\mathcal{A}^{k-1}_\mathbb{C}(X)).
\end{align}
\end{cor}
\begin{proof}
With \Cref{harmonic-forms:lm:kernel-laplacian}, it is obvious that
$\mathcal{H}^k(X) \oplus d(\mathcal{A}^{k-1}) \subset \ker(d)$.
Also for all $\alpha \in \mathcal{A}^{k+1}_\mathbb{C}(X)$, it is $d^*\alpha \in \ker(d^*)$
because $(d^*)^2 = 0$. Therefore, if it would be $d^*\alpha \in \ker(d)$, it would already be
$d^*\alpha \in \mathcal{H}^k(X)$. Thus, this statement is a consequence of the last proposition.
\end{proof}
The same argument yields a similar statement for the formal adjoint of the exterior derivative.
\begin{cor}
The kernel of the formal adjoint $d^*$\! decomposes orthogonally as
\begin{align}
\label{harmonic-forms:eq:kernel-of-the-formal-adjoint-of-the-exterior-derivative}
\ker(d^*) = \mathcal{H}^k(X) \oplus d^*(\mathcal{A}_\mathbb{C}^{k+1}(X)).
\end{align}
\end{cor}
\begin{rem}
It has to be mentioned that both of these corollaries have been motivated by an attempt to
separate the proof of the Hodge Isomorphism theorem found in \cite[Theorem 5.23]{Voisin2002}
into multiple components to make it more accessible to the reader.
\end{rem}
At this time, we are able to prove the first of the above-mentioned theorems, which is going to be
essential for the proof of the Hodge Decomposition.
\begin{thm}[Hodge Isomorphism theorem \MakeUppercase{\romannumeral 1} {\cite[Theorem
5.23]{Voisin2002}}]
\label{harmonic-forms:thm:hodge-isomorphism-1}
The natural mapping
\begin{align*}
\mathcal{H}^k(X) \rightarrow H^k_{dR}(X,\mathbb{C}), \;\enspace
\alpha \mapsto [\alpha]
\end{align*}
is an isomorphism. In particular, any class of closed forms in $H^k_{dR}(X,\mathbb{C})$ has a
unique harmonic representative.
\end{thm}
\begin{proof}
We use the same proof as in the given theorem in \cite{Voisin2002} and provide links to our
previously provided statements.
Note that the given mapping is meaningful because harmonic forms are closed
(cf. \Cref{harmonic-forms:lm:harmonic-forms-are-closed}).
Let now $\alpha \in \mathcal{A}^k_\mathbb{C}(X)$ be a closed differential form. With the
decomposition in \Cref{harmonic-forms:eq:decomposition-of-the-k-forms} we can uniquely rewrite
$\alpha =\beta + \Delta_d\gamma = \beta + dd^*\gamma + d^*d\gamma$ with
$\beta, \gamma \in \mathcal{A}^k_\mathbb{C}(X)$ and $\beta$ harmonic. Since $\alpha$ is a closed
form, it is
\begin{align*}
0 = d\alpha = d\beta + d^2d^*\gamma + dd^*d\gamma= d\beta + dd^*d\gamma.
\end{align*}
As harmonic forms are closed, this simplifies to $0 = dd^*d\gamma$. Hence, with the decomposition in
\Cref{harmonic-forms:eq:kernel-of-the-exterior-derivative}, we know that $d^*d\gamma = 0$. Thus, it is
$\alpha = \beta + dd^*\gamma$ and therefore we have $[\alpha] = [\beta]$ because $dd^* \gamma$ is
exact. This already proves the surjectivity.
In order to prove the injectivity, we chose $\alpha' \in \mathcal{A}^k_\mathbb{C}(X)$ to be harmonic and
exact. We can rewrite $\alpha' = d\beta'$ for $\beta'\in \mathcal{A}_\mathbb{C}^{k-1}(X)$.
It is $d\beta' \in \ker(d^*)$ because $d\beta'$ is harmonic.
However, with \Cref{harmonic-forms:eq:kernel-of-the-formal-adjoint-of-the-exterior-derivative}, we
know that $\ker(d^*) \cap d(\mathcal{A}^{k-1}_\mathbb{C}(X)) = \{0\}$. Thus, $\alpha' = d\beta' = 0$, and
therefore, the mapping is injective because $\alpha'$ only maps to an exact differential form if $\alpha' = 0$.
\end{proof}
As a direct result, we get the following corollary.
\begin{cor}[{\cite[Ch.\,VI §3. Theorem 3.17]{Demailly1997}}]
The de Rahm cohomology groups $H_{dR}^k(X,\mathbb{C})$ have finite dimensions.
\end{cor}
\begin{proof}
In \Cref{harmonic-forms:lm:finite-dimensional}, we have already established that $\mathcal{H}^k(X)$
has finite dimension. Thus, the first Hodge Isomorphism theorem immediately proves this result.
\end{proof}
Next, we will prove the same statement but this time for the Dolbeault cohomology groups. It should
not be surprising that the proof will use the same arguments.
\begin{prop}
\label{harmonic-forms:lm:opartial-harmonic-is-opartial-closed}
Let $\alpha \in \mathcal{H}^{p,q}(X)$ be a harmonic differential form, then
$\alpha$ is also $\opartial$-closed.
\end{prop}
\begin{proof} The differential form $\alpha$ is harmonic if and only if it is $\Delta_\opartial$-harmonic
(cf. \Cref{harmonic-forms:cor:harmonic-if-and-only-if-delta-opartial-harmonic}). Therefore, this
statement is a direct consequence of \Cref{harmonic-forms:rem:kernel-laplacian-partial-opartial}.
\end{proof}
\begin{prop}[{\cite[Ch.\,VI §7. Theorem 7.1]{Demailly1997}}]
\label{harmonic-forms:lm:refined-decomposition-of-the-pq-forms}
For the differential forms of type $(p,q)$, there is another orthogonal decomposition given as
\begin{align}
\label{harmonic-forms:eq:another-decomposition-for-k-forms-with-opartial}
\mathcal{A}^{p,q}(X) = \mathcal{H}^{p,q}(X) \oplus \opartial\,(\mathcal{A}^{p,q-1}(X)) \oplus
\opartial^*\!(\mathcal{A}^{p,q+1}(X))
\end{align}
\end{prop}
\begin{proof}
Just repeat the proof of \Cref{harmonic-forms:lm:improved-decomposition-of-the-k-forms} using the
decomposition in \Cref{harmonic-forms:eq:decomposition-of-the-pq-forms} and the properties for the
kernel of $\Delta_\opartial$ established in
\Cref{harmonic-forms:rem:kernel-laplacian-partial-opartial}.
\end{proof}
With this decomposition, we are able to find a description for the kernel of the Dolbeault operators
$\opartial$ and $\opartial^*$. It is obvious that
$\mathcal{H}^{p,q}(X) \oplus \opartial\,(\mathcal{A}^{p,q-1}(X)) \subset \ker(\opartial)$. It is
also for all $\alpha \in \mathcal{A}^{p,q+1}(X)$ the image $\opartial^*\!\alpha \in \ker(\opartial^*)$.
Therefore, if it would be $\opartial^*\!\alpha \in \linebreak\ker(\opartial)$, then it would already
be $\opartial^*\!\alpha \in \mathcal{H}^{p,q}(X)$. Hence, the last proposition yields
\begin{align}
\label{harmonic-forms:eq:kernel-of-opartial}
\ker(\opartial) = \mathcal{H}^{p,q}(X) \oplus \opartial\,(\mathcal{A}^{p,q-1}(X)).
\end{align}
Using the same argument again but for the formal adjoint $\opartial^*\!,$ we obtain
\begin{align}
\label{harmonic-forms:eq:kernel-of-the-formal-adjoint-of-opartial}
\ker(\opartial^*) = \mathcal{H}^{p,q}(X) \oplus \opartial^*(\mathcal{A}^{p,q+1}(X)).
\end{align}
This is everything necessary to prove the second Hodge Isomorphism theorem for differential forms
of type $(p,q)$.
\begin{thm}[Hodge Isomorphism theorem \MakeUppercase{\romannumeral 2} {\cite[Ch.\,VI §7. Theorem 7.2]{Demailly1997}}]
\label{harmonic-forms:thm:hodge-iso-2}
The natural mapping
\begin{align*}
\mathcal{H}^{p,q}(X) \rightarrow H^{p,q}_\opartial(X,\mathbb{C}),\; \enspace \alpha \mapsto [\alpha]
\end{align*}
is an isomorphism. In particular, any class of \,$\opartial$-closed forms in
$H^{p,q}_\opartial(X,\mathbb{C})$ has a unique harmonic representative.
\end{thm}
\begin{proof}
Just repeat the proof of \Cref{harmonic-forms:thm:hodge-isomorphism-1} using
\Cref{harmonic-forms:lm:opartial-harmonic-is-opartial-closed},
\Cref{harmonic-forms:lm:refined-decomposition-of-the-pq-forms} and the just established
decompositions \Cref{harmonic-forms:eq:kernel-of-opartial} and
\Cref{harmonic-forms:eq:kernel-of-the-formal-adjoint-of-opartial}.
\end{proof}

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\section{Hodge Decomposition}
In this final chapter, we are left with the proof of the Hodge Decomposition theorem, which is the
primary goal of this thesis. \!With the already established theory from the preceding chapters, the
proof will be straightforward. Since it is apriori unclear whether the Hodge Decomposition depends
on the choice of the Kähler metric, we will also show that this decomposition does not depend on the
metric.
In the second section of this chapter, we are also going to provide a topological application of the
Hodge Decomposition theorem that illustrates only one of the many important consequences in complex
and algebraic geometry.
For the remainder of this chapter, we are going to assume the following setting.
\begin{set}
Let $X$ be a compact Kähler manifold with Riemannian metric $g$ and closed fundamental form
$\omega$. Also, let $h:= g -i\omega$ denote the Kähler metric.
\end{set}
\subsection{Proof of the Hodge Decomposition theorem}\;
\begin{thm}[Hodge Decomposition {\cite[Ch.\,VI §8. Theorem 8.5]{Demailly1997}}]
For the compact Kähler manifold $X$, there does exist an isomorphism
\begin{align}
\label{hodge-decomposition:eq:main-theorem}
H^k_{dR}(X,\mathbb{C}) \cong \bigoplus_{p+q=k} H^{p,q}_{\opartial}(X,\mathbb{C}).
\end{align}
\end{thm}
\begin{proof}
The existence of the decomposition
\begin{align*}
\mathcal{H}^k(X) = \bigoplus_{p+q=k} \mathcal{H}^{p,q}(X)
\end{align*} has been established in \Cref{harmonic-forms:cor:final-harmonic-decomposition}. Using
the two Hodge Isomorphism
\Cref{harmonic-forms:thm:hodge-isomorphism-1,harmonic-forms:thm:hodge-iso-2}, we get
\begin{align*}
H^k_{dR}(X,\mathbb{C}) \cong\mathcal{H}^k(X) = \bigoplus_{p+q=k} \mathcal{H}^{p,q}(X)\cong
\bigoplus_{p+q=k}H^{p,q}_\opartial(X,\mathbb{C}).
\end{align*}
\end{proof}
Note that the definition of the harmonic forms depends on the definition of the Laplacians
$\Delta_d,\Delta_\partial$ and $\Delta_\opartial$. These Laplacians are defined using the formal
adjoint operators $d^*, \partial^*$ and $\opartial^*$ that depend on the $L^2$-metric induced by the
metric $h$. Hence, it is unclear whether the Hodge Decomposition is independent of the choice of
this metric.
\begin{prop}[{\cite[Proposition 6.11]{Voisin2002}}]
\label{hodge-decomposition:prop:independence-of-metric}
The Hodge Decomposition \Cref{hodge-decomposition:eq:main-theorem} does not depend on the choice
of the Kähler metric.
\end{prop}
\begin{proof}
This proof was taken from the stated proposition in \cite{Voisin2002}.
Let $K^{p,q} \subset H^k_{dR}(X,\mathbb{C})$ denote the subspace of cohomology classes that are
representable by a closed form of type $(p,q)$. In \Cref{harmonic-forms:thm:hodge-iso-2}, we have
shown that every class in $H_\opartial^{p,q}(X,\mathbb{C})$ has a harmonic representative, which is in
particular also closed. Therefore, it is already $H_\opartial^{p,q}(X,\mathbb{C}) \subset K^{p,q}$.\\
Now, we are going to show that the other inclusion is also true. Let $\eta \in K^{p,q}$ be a
closed form of type $(p,q)$. With \Cref{harmonic-forms:eq:decomposition-of-the-k-forms}, we can
uniquely rewrite $\eta = \alpha + \Delta_d \beta$ with $\alpha \in \mathcal{H}^k(X)$ and $\beta
\in \mathcal{A}_\mathbb{C}^k(X)$. Since $\Delta_\opartial$ maps forms of type $(p,q)$ to forms of type
$(p,q)$, we know with \Cref{harmonic-forms:eq:thm-laplacian-realtion} that this also holds for
$\Delta_d$. Therefore, we can only consider the components of type $(p,q)$ to get the again unique
equality
\begin{align*}
\eta = \alpha^{p,q} + \Delta_d\beta^{p,q} = \alpha^{p,q} + (dd^*\beta^{p,q} + d^*d\beta^{p,q}).
\end{align*}
With \Cref{harmonic-forms:cor:harmonic-if-and-only-if-components-are-harmonic}, we know that
the component $\alpha^{p,q}$ is also harmonic and particularly closed. As $\eta$ is closed too, we get
\begin{align*}
0 = d\eta = d\alpha^{p,q} + d(dd^* \beta^{p,q} + d^*d\beta^{p,q}) = dd^*d\beta^{p,q}.
\end{align*}
With \Cref{harmonic-forms:eq:kernel-of-the-exterior-derivative}, we already know that
$d^*d\beta^{p,q} = 0$. Thus, we obtain $\eta = \alpha^{p,q} + dd^* \beta^{p,q}$.
Since $\opartial(dd^*\beta^{p,q}) = 0$, we know that
$[\eta] = [\alpha^{p,q}] \in H^{p,q}_\opartial(X,\mathbb{C})$. This proofs that
$H^{p,q}_\opartial(X,\mathbb{C}) = K^{p,q}$. Furthermore, since $K^{p,q}$ and the inclusion of
$K^{p,q}$ into $H^k_{dR}(X,\mathbb{C})$ do not depend on the choice of the metric, this proves
the statement.
\end{proof}
\begin{rem}
After the independence of the Kähler metric has been proven, it is justified to write an equality in
\Cref{hodge-decomposition:eq:main-theorem}.
\end{rem}
\subsection{Application of the Hodge Decomposition}\;
The Hodge Decomposition theorem has numerous applications in complex and algebraic geometry and also
in topology. Since most of these applications need some additional theory that would be beyond the
scope of this thesis, we will only provide one of them, which has historically provided the first
example of a complex manifold that can not be equipped with a Kähler metric.
In the proof of \Cref{hodge-decomposition:prop:independence-of-metric}, we have defined $K^{p,q}
\subset H^k_{dR}(X,\mathbb{C})$ to be the subspace of cohomology classes that are representable by a
closed form of type $(p,q)$. Since the exterior derivative is compatible with complex conjugation
and the conjugation of a closed differential form of type $(p,q)$ is a closed differential form of
type $(q,p)$, we obtain $\overline{K^{p,q}} = K^{q,p}$. Hence, we have the following proposition.
\begin{prop}[{\cite[Corollary 6.12]{Voisin2002}}]
\label{hodge-decomposition:prop:conjugation-dolbeault-cohomologies}
For the Dolbeault cohomology groups, we have the equality \vspace*{-0.1cm}
\begin{align*}
\overline{H^{p,q}_\opartial(X,\mathbb{C})}=H^{q,p}_\opartial(X,\mathbb{C}).
\end{align*}
\end{prop}
\begin{proof}
This proof was taken from the given corollary in \cite{Voisin2002}.
In the proof of \Cref{hodge-decomposition:prop:independence-of-metric}, we have shown that
$K^{p,q} = H_\opartial^{p,q}(X,\mathbb{C})$. Therefore, the statement is a consequence of the
just established similar equality $\overline{K^{p,q}} = K^{q,p}$.
\end{proof}
This proposition can be used to obtain an interesting corollary that uses the Hodge Decomposition
theorem and gives a strong constraint to the topological properties of a Kähler manifold. However,
before we focus on this corollary, we will introduce a new notation inspired by
\cite[Ch. VI §8. Equation 8.11]{Demailly1997} to improve the readability.
\begin{nota}
The $k$-th \emph{Betti number} and the $(p,q)$-th \emph{Hodge number} of the Kähler manifold $X$
are respectively defined as
\begin{align*}
b_k(X) := \dim_\mathbb{C} H^{k}(X,\mathbb{C}), \qquad h_{p,q}(X) := \dim_\mathbb{C}H_\opartial^{p,q}(X,\mathbb{C}).
\end{align*}
\end{nota}
Using this notation, the Hodge Decomposition \Cref{hodge-decomposition:eq:main-theorem} immediately
yields the equality
\begin{align}
\label{hodge-decomposition:eq:betti-and-hodge-numbers}
b_k(X) = \sum_{p+q=k}h_{p,q}(X)
\end{align}
(cf. \cite[Ch.\,VI §8. Equation 8.12]{Demailly1997}).
This equality can now be used to obtain the following corollary.
\begin{cor}[{\cite[Corollary 6.13]{Voisin2002}}]
\label{hodge-decomposition:cor:odd-betti-numbers}
The odd Betti numbers $b_{2k+1}(X)$ are even.
\end{cor}
\begin{proof}
With \Cref{hodge-decomposition:prop:conjugation-dolbeault-cohomologies}, we get
$h_{p,q}(X) = h_{q,p}(X)$. Thus, \Cref{hodge-decomposition:eq:betti-and-hodge-numbers} yields
\begin{align*}
b_{2k+1}(X) = \sum_{p=0}^{2k+1} h_{p,2k+1-p}(X)
= \sum_{p=0}^{k} 2\,h_{p,2k+1-p}(X)
= 2 \sum_{p=0}^{k} h_{p,2k+1-p}(X).
\end{align*}
Therefore, the odd Betti number $b_{2k+1}(X)$ is even.
\end{proof}
This corollary allows us to rule out the existence of a Kähler metric for certain compact complex
manifolds by calculating the odd Betti numbers. One example of such a type of manifold that cannot
be endowed with a Kähler metric are the Hopf surfaces.
\begin{defn}[Hopf surfaces]
Let $\lambda_1,\lambda_2 \in \mathbb{C}$ with $0 < |\lambda_1| \leq |\lambda_2| < 1$. Let
$\mathbb{Z}$ act freely and properly discontinuously on $\mathbb{C}^2\setminus\{0\}$ by the
biholomorphic transformations
\begin{align*}
\mathbb{Z} \times \mathbb{C}^2\!\setminus\!\{0\} \rightarrow \mathbb{C}^2\!\setminus\!\{0\},
\;\quad(l,z_1,z_2) \mapsto (\lambda_1^lz_1,\lambda_2^lz_2).
\end{align*}
The resulting compact complex quotient manifold $(\mathbb{C}^2\setminus\{0\}) / \mathbb{Z}$ is
called a \emph{Hopf surface}.
\end{defn}
\begin{rem}
It has to be mentioned that the definition of the Hopf surfaces differs substantially among
different authors. The provided definition is a combination of the two found in
\cite[p.\,143]{Voisin2002} and \cite[p.\,61]{Huybrechts2004}
\end{rem}
Let now $Y$ be such a Hopf surface. Using the approach mentioned in \cite[p.\,143]{Voisin2002},
one can calculate that $b_1(Y) = 1$. As shown in \Cref{hodge-decomposition:cor:odd-betti-numbers},
the odd Betti numbers must be even for a Kähler manifold. Thus, this calculation proves that $Y$
cannot be a Kähler manifold.
\begin{rem}
According to \cite[p.\,172]{Barth1984}, one of these Hopf surfaces has been the first example of
a compact complex manifold that is not Kähler. It was discovered by Heinz Hopf in 1948. Note,
however, that his definition of the Hopf surfaces has been slightly different than the one
presented in this thesis.
\end{rem}

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\section{Kähler manifolds and formal adjoint operators}
This chapter is dedicated to the study of a distinguished type of manifold known as Kähler manifold.
These manifolds possess a combination of essential structures, including a compatible Riemannian
metric, a holomorphic structure and a symplectic form, i.e. a closed non-degenerate differential
2-form.
The simultaneous presence of these different structures leads to some interesting geometric and
analytic properties. One of them is the existence of the Kähler identities, whose proof will be the
primary goal of this chapter.
We are mainly going to use the results of the last chapter, but we will also assume that the reader
is familiar with some basic concepts of (almost) complex and Riemannian manifolds. Otherwise, we
would advise the reader to take a look at the second chapter of \cite{Huybrechts2004}. In
particular, sections 2.1, 2.2 and 2.6 are going to be relevant for this thesis.
Because there are many different conventions in notation, we are going to start with a clarification
of the used symbols.
\begin{nota}
For a complex manifold $X$, we are going to use the following notation for the different tangent
bundles:
\begin{itemize}
\item The holomorphic tangent bundle will be written as $T_X := \mathop{\dot{\bigcup}}_{p \in X} T_{X,p}$.
\item The real tangent bundle, i.e. the tangent bundle of the underlying real differentiable manifold, will
be written as $T_{X,\mathbb{R}} := \mathop{\dot{\bigcup}} T_{X,p,\mathbb{R}}$.
\item The complex tangent bundle, i.e the complexification of the real tangent bundle, will
be written as $T_{X,\mathbb{C}} := T_{X,\mathbb{R}} \otimes \mathbb{C} := \mathop{\dot{\bigcup}}_{p\in X}
T_{X,p,\mathbb{R}} \otimes \mathbb{C} =: \mathop{\dot{\bigcup}}_{p \in X} T_{X,p,\mathbb{C}}$.
\end{itemize}
For the associated $k$-multilinear forms, we are going to use the following notation:
\begin{itemize}
\item For the vector bundle of the real $k$-forms, we are going to write $\Omega^k_{X,\mathbb{R}} := \bigwedge^k
T_{X,\mathbb{R}}^*$ and the associated global smooth sections will be written as $\mathcal{A}_{\mathbb{R}}^k(X)$.
\item For the vector bundle of the complex $k$-forms, we are going to write $\Omega^k_{X,\mathbb{C}} := \bigwedge^k
T_{X,\mathbb{C}}^*$ and the associated global smooth sections will be written as
$\mathcal{A}^k_\mathbb{C}(X)$.
\item For the vector bundle of the complex forms of type $(p,q)$, we will write $\Omega^{p,q}_{X} := \bigwedge^{p,q}
T_{X}^*$ and the associated global smooth sections will be written as $\mathcal{A}^{p,q}(X)$.
\end{itemize}
Also, for the global smooth sections of a specific vector bundle $\pi: E \rightarrow X$, we are
going to use the notation $C^\infty(E)$, which should not be confused with the smooth functions
$C^\infty(E,\mathbb{R})$ or $C^\infty(E,\mathbb{C})$.
\end{nota}
\subsection{Hermitian manifolds}\;
In differential geometry, a Riemannian metric is an essential tool to define basic geometric
properties like distance, angle or curvature. In this section, we will introduce hermitian
manifolds, which serve as the complex counterpart of Riemannian manifolds. Our primary use case
right here will be the globalization of our locally defined operators from the last chapter.
For the remainder of this section, we are going to assume the following setting.
\begin{set}
Let $X$ be a complex $m$-dimensional manifold with induced almost complex structure
$I:T_{X,\mathbb{R}} \rightarrow T_{X,\mathbb{R}}$ (cf. \cite[Proposition 2.6.2]{Huybrechts2004}).
Also, assume that there is a compatible Riemannian metric on the underlying $2m$-dimensional real
differentiable manifold that is given as $g: T_{X,\mathbb{R}} \times T_{X,\mathbb{R}} \rightarrow \mathbb{R}$,
such that the induced inner product $g_p : T_{X,p,\mathbb{R}} \times T_{X,p,\mathbb{R}} \rightarrow \mathbb{R}$
is compatible with the almost complex structure $I_p:T_{X,p,\mathbb{R}} \rightarrow T_{X,p,\mathbb{R}}$
for all $p \in X$. Also, let $n:= 2m$ to keep the notation simple.
\end{set}
\begin{defn}[Fundamental form]
Similar to the local case, the \emph{fundamental form} $\omega \in \mathcal{A}_\mathbb{R}^2(X) \cap
\mathcal{A}^{1,1} (X)$ is defined such that for all $p \in X$ and $v,w \in T_{X,p,\mathbb{R}}$, it is
\begin{align*}
\omega_p(v,w) := g_p(I_p(v),w) = -g_p(v,I_p(w)).
\end{align*}
As the Riemannian metric $g$ varies smoothly in $p \in X$, it is obvious that this pointwise
definition indeed defines a global smooth section.
\end{defn}
\begin{rem}
The property of $\omega$ being a real differential 2-form and also of type $(1,1)$ is a direct
consequence of $\omega_p$ possessing this property for every $p \in X$. Also, $\omega_p$ is
non-degenerate because of $g_p$ being positive definite. Hence, $\omega$ is said to be
non-degenerate too.
\end{rem}
\begin{defn}[Hermitian manifold {\cite[Definition 3.1.1]{Huybrechts2004}}]
Our complex manifold $X$, whose underlying real differentiable manifold is also equipped with
a Riemannian metric $g$, which is compatible with the induced almost complex structure $I$,
is called a \emph{hermitian manifold}.
\end{defn}
As we have seen in the local theory chapter, for every $p \in X$, we can use the inclusion
$T_{X,p,\mathbb{R}} \into T_{X,p,\mathbb{C}} \onto T_{X,p}$ to define a positive definite hermitian
form $h_p: T_{X,p} \times T_{X,p} \rightarrow \mathbb{C}$ as
\begin{equation*}
h_p(v,w) := g_p(v,w) -i\omega_p(v,w).
\end{equation*}
Because $h_p$ depends smoothly on $p$, this already induces a global smooth sesquilinear form $h$
that is also positive definite. Such a form is called a \emph{hermitian metric} of the manifold $X$.
With this pointwise definition, it is immediately evident that any hermitian manifold is naturally
equipped with a hermitian metric. Hence, the name is justified.
\begin{rem}
Note that the usual definition of a hermitian manifold is a complex manifold, which is equipped with
a positive definite hermitian metric on every holomorphic tangent space (see e.g. \cite[Section 12]{Schnell2012}).
At first, it may seem that this definition describes a more general object, but these two definitions
are equivalent, which is a direct consequence of \Cref{loc-theory:rem:real-of-hermitian-form}.
Our definition was inspired by \cite[Definition 3.1.1]{Huybrechts2004}.
\end{rem}
\begin{rem}
The same constructions as in \Cref{loc-theory:cor:induced-product-on-exterior-algebra} and
\Cref{loc-theory:rem:hermitian-form-on-exterior-algebra} can be used in a pointwise manner to
obtain Riemannian metrics $g$ and hermitian metrics $h$ on the different exterior algebra
bundles $\Omega^\bullet_{X,\mathbb{R}}, \Omega^{\bullet,\bullet}_{X}$ and
$\Omega^\bullet_{X,\mathbb{C}}$ and the respective section spaces
$\mathcal{A}_\mathbb{R}^\bullet(X),\mathcal{A}^{\bullet,\bullet}(X)$ and
$\mathcal{A}^\bullet_{X,\mathbb{C}}$.
\end{rem}
We know that any real differentiable manifold $M$ can be endowed with a Riemannian metric (cf.
\cite[Proposition 13.3]{Lee2012}) and is therefore also a Riemannian manifold. Also, for a given
almost complex structure $J$, we can choose any Riemannian metric $\hat{g}$ that does not need
to be compatible with $J$ and define another Riemannian metric
\begin{align*}
g'_p(v,w) := \hat{g}_p(v,w) + \hat{g}_p(J_p(v),J_p(w))
\end{align*}
with $p \in M$ and $v,w \in T_pM$. Then, it is
\begin{align*}
g'_p(J_p(v),J_p(w)) &= \hat{g_p}(J_p(v),J_p(w)) +\hat{g}_p(J_p^2(v),J_p^2(w)) \\
&= \hat{g}_p(J_p(v),J_p(w)) + \hat{g}_p(v,w) \\
&= g'_p(v,w).
\end{align*}
Thus, $g'$ defines a compatible Riemannian metric. Combined with the above discussion, this proves
the following proposition.
\begin{prop}
\label{kaehler-manifolds:lm:all-complex-manifolds-are-hermitian}
Every complex manifold is also hermitian and can therefore be endowed with a hermitian metric.
\end{prop}
\begin{rem}
We want to explicitly note that this does not imply that the above-defined real differential
manifold $M$ is a hermitian manifold. This is because $M$ is only a complex manifold if the almost
complex structure $J$ is integrable.
\end{rem}
Additionally, hermitian manifolds are always orientable, i.e. the underlying real differentiable
manifold can be equipped with an orientation. This property is not true for Riemannian manifolds
since there are indeed manifolds that are not orientable. In order to prove this, we have the next
proposition.
\begin{prop}[{\cite[Lemma 3.8]{Voisin2002}}]
\label{kaehler-maifolds:lm:volume-form}
There is a canonical volume form associated with a hermitian manifold that is given as
$\vol=\nobreak \frac{\omega^m}{m!}$. In particular, any hermitian manifold has a natural orientation.
\end{prop}
\begin{proof}
With \Cref{loc-theory:volume-form-locally}, we already know that for every local orthonormal
frame field $v_1,\dots,v_n$ (cf. \cite[Section 4.10]{Warner1983}) of the real tangent bundle,
which is also positively oriented with respect to the natural local orientation, it is
\begin{align}
\label{kaehler-manifolds:eq:vol-form-calc}
\frac{\omega^m}{m!} (v_1,\dots,v_n) = 1.
\end{align}
Therefore, $\frac{\omega^m}{m!}$ defines a nowhere-vanishing global smooth section. Thus, it
induces an orientation on the underlying real differentiable manifold (cf. \cite[Proposition 15.5]{Lee2012}).\\
See also \cite[Lemma 3.8]{Voisin2002} for an explicit calculation of \Cref{kaehler-manifolds:eq:vol-form-calc}.
\end{proof}
In the previous chapter, we have defined three important local operators: the Lefschetz operator,
the dual Lefschetz Operator, and the Hodge star operator. Now, we can also define these operators
in the global context such that they inherit the properties of the local operators.
\begin{defn}
The \emph{global Lefschetz operator} is defined as a vector bundle homomorphism
\begin{align*}
L : \Omega^k_{X,\mathbb{R}} \rightarrow \Omega^{k+2}_{X,\mathbb{R}},\quad \alpha \mapsto \alpha \wedge \omega.
\end{align*}
Because $\omega \in \mathcal A_\mathbb{R}^2(X)$ is a smooth section, this induces a linear
operator on the global smooth sections
$L: \mathcal A_\mathbb{R}^k(X) \rightarrow \mathcal{A}_\mathbb{R}^{k+2}(X)$.
Since $\omega$ is also a smooth differential form of type $(1,1)$, the $\mathbb{C}$-linear
extension of $L$ yields a linear mapping on the complex differential forms
$L_\mathbb{C}: \mathcal{A}^k_{\mathbb{C}}(X) \rightarrow \mathcal{A}_\mathbb{C}^{k+2}(X)$.
\end{defn}
\begin{defn}
The \emph{global Hodge star operator} is defined as a linear mapping
\begin{align*}
\hodgestar: \Omega^k_{X,\mathbb{R}} \rightarrow \Omega^{2m-k}_{X,\mathbb{R}}
\end{align*}
such that for all $\alpha,\beta \in \Omega^k_{X,\mathbb{R}}$, it is
\begin{align*}\alpha \wedge \hodgestar \beta
= g(\alpha,\beta) \cdot \vol.
\end{align*}
This uniquely defines this operator because it is already uniquely defined locally. We obtain
again an induced operator
$\hodgestar: \mathcal{A}_\mathbb{R}^k(X) \rightarrow \mathcal{A}_\mathbb{R}^{2m-k}(X)$
on the global sections and $\mathbb{C}$-linear extension yields
$\hodgestar_\mathbb{C}:\mathcal{A}_\mathbb{C}^k(X) \rightarrow \mathcal{A}_\mathbb{C}^{2m-k}(X)$.
Since the hermitian metric $h$ on $\mathcal{A}^k_\mathbb{C}(X)$ has been obtained by sesquilinearly
extending the Riemannian metric $g$ on $\mathcal{A}_\mathbb{R}^k(X)$, it is for all
$\alpha,\beta \in \mathcal{A}_\mathbb{C}^k(X)$
\begin{align*}
\alpha\wedge\hodgestar_\mathbb{C}\overline{\beta} = h(\alpha,\beta) \cdot \vol.
\end{align*}
\end{defn}
In the local case, we have defined the dual Lefschetz operator $\Lambda$ as the adjoint of the
Lefschetz operator $L$. To define this operator in a global context, we use the same approach as in
\cite[Section 3.1]{Huybrechts2004} and apply the equality established in
\Cref{loc-theory:lm:formula-for-the-dual-lefschetz-operator}.
\begin{defn}
We define the \emph{dual Lefschetz operator} as
\begin{align*}
\Lambda: \Omega_{X,\mathbb{R}}^{k+2} \rightarrow \Omega_{X,\mathbb{R}}^k, \quad \alpha
\mapsto \big((-1)^k \hodgestar \circ\; L \circ \hodgestar \big)(\alpha).
\end{align*}
This again induces an operator on the smooth sections
$\Lambda: \mathcal{A}^{k+2}_\mathbb{R}(X) \rightarrow \mathcal{A}^k_\mathbb{R}(X)$.
Using $\mathbb{C}$-linear extension, we obtain
\begin{align*}
\Lambda_\mathbb{C}: \mathcal{A}_\mathbb{C}^{k+2}(X) \rightarrow \mathcal{A}_{\mathbb{C}}^k(X),
\quad \alpha \mapsto \big((-1)^k \hodgestar_\mathbb{C} \circ\; L_\mathbb{C}
\circ \hodgestar_\mathbb{C} \big)(\alpha).
\end{align*}
A similar calculation as in the proof of \Cref{loc-theory:lm:formula-for-the-dual-lefschetz-operator}
can be used to show that these global operators are indeed adjoint to the global Lefschetz
operators $L$ and $L_\mathbb{C}$, respectively.
\end{defn}
\begin{nota}
Similar to the local case, we will abuse the notation and only write $L, \Lambda$ and $\hodgestar$
instead of $L_\mathbb{C},\Lambda_\mathbb{C}$ and $\hodgestar_\mathbb{C}$ again. We will also not
differentiate between the operators on the different spaces.
\end{nota}
\begin{rem}
\label{kaehler-manifolds:rem:local-are-global-properties-of-the-hodge-star}
Since the global Hodge star operator is pointwise equivalent to the local Hodge star operator,
it is obvious that the properties established in \Cref{loc-theory:lm:property-hodge-star}
translate to the equivalent global properties.
\end{rem}
\subsection{Formal adjoint operators}\;
In order to establish the Kähler identities, it is essential to introduce the concept of linear
differential and formal adjoint operators. Therefore, the goal of this section will be to define
this type of operator.
In order to do so, we will first define a hermitian $L^2$-metric, which is then used to generalize
the notion of adjoint operators. After this brief introduction, our particular focus will be on the
formal adjoint operators of the exterior derivative $d$ and the associated Dolbeault operators
$\partial$ and $\opartial$, as their comprehensive understanding will be the foundation for the
theory developed later.
For the remainder of this section, we are going to assume the following setting.
\begin{set}
Let $X$ be an $m$-dimensional hermitian manifold with induced hermitian metric $h$. Also, let
$\vol$ denote the canonical volume form on $X$. In order to properly define the $L^2$-metric,
we are also going to assume $X$ is compact.
\end{set}
\begin{defn}[The hermitian $L^2$-metric]
For all $\alpha,\beta \in \mathcal A^k_\mathbb{C}(X)$ the \emph{hermitian $L^2$-metric} is
defined as
\begin{align*}
\left(\alpha,\beta\right)_{L^2} := \int_{X} \alpha \wedge \hodgestar \overline{\beta} = \int_X
h(\alpha,\beta) \cdot \vol.
\end{align*}
Note that by definition of the Hodge star operator, the wedge product $\alpha \wedge
\hodgestar\overline\beta$ is a differential $2m$-form. Thus, it is allowed to integrate this
form. Since $X$ is compact, this integral is always going to be finite.
\end{defn}
\begin{rem}
\label{kaehler-manifolds:rem:l2-metric-properties}
With the linearity of the integral, it is immediate that this $L^2$-metric is indeed sesquilinear.
It is also for all $\alpha,\beta \in \mathcal A^k_\mathbb{C}(X)$
\begin{align*}
\overline{(\alpha,\beta)}_{L^2} = \overline{\int_X h(\alpha,\beta) \cdot \vol} = \int_X
\overline{h(\alpha,\beta)} \cdot \vol = \int_X h(\beta,\alpha) \cdot \vol = (\beta,\alpha)_{L^2}
\end{align*}
because $\vol$ is a real differential form and therefore invariant under complex conjugation. Also,
because $p \mapsto h_p(\alpha_p,\alpha_p)$ is a smooth function and in particular also continuous,
the integral is going to be zero if and only if $h_p(\alpha_p,\alpha_p) = 0$ for all $p \in X$. As
$h_p$ is positive definite, this is only the case if $\alpha_p = 0$ for all $p$. Additionally, it is
$h_p(\alpha_p,\alpha_p) \geq 0$ for all $\alpha_p \neq 0$. Thus, this $L^2$-metric is positive
definite.
\end{rem}
The following definition has been inspired by \cite[Definition 5.15]{Voisin2002}, but we have adapted
it to avoid the discussion of sheaves.
\begin{defn}
\label{kaehler-manifolds:defn:differential-opperators}
Let both $\pi_1: E_1\rightarrow X$ and $\pi_2: E_2\rightarrow X$ be smooth and complex vector
bundles on the complex manifold $X$ with rank $r_1$ and $r_2$, respectively. A
\emph{complex linear differential operator of order $d$} written as
$D: C^\infty(E_1) \rightarrow C^\infty(E_2)$ is a collection of operators
$D_{U_j}: C^\infty(U_j,E_1) \rightarrow C^\infty(U_j,E_2)$ with $(U_j)_{j\in J} = X$ an open
covering and $C^\infty(U_j,E_1)$ the local smooth sections on $U_j$, such that the following
properties hold:
\begin{enumerate}
\item $D$ is compatible with restrictions to smaller open sets, i.e. for $U_k \subset U_j$ an open
subset, it is $D_{U_j\mid_{U_k}} = D_{U_k}$.
\item For every $p \in X$ there is a coordinate neighborhood $U$ with local coordinates
$z_1,\dots,z_m$ and trivializations
\begin{align*}
E_{1 \mid_U} \cong U \times \mathbb{C}^{r_1} \quad \text{and} \quad E_{2 \mid_U} \cong U \times
\mathbb{C}^{r_2},
\end{align*}
such that $D_{U_k \mid_{U}}((\alpha_1,\dots,\alpha_{r_1})) = (\beta_1,\dots,\beta_{r_2})$ with
\begin{align*}
\beta_r= \sum_{S,t}P_{r,S,t} \pdv{\alpha_t}{z_S}.
\end{align*}
The coefficients $P_{r,S,t}$ are complex-valued and smooth, and the sum needs to be finite. Also
$P_{r,S,t} = 0$ for $|S| > d$, and there is at least one such coefficient that is non-zero for
$|S| = d$.
\end{enumerate}
\end{defn}
\begin{exmp}
Suppose an open subset $U \subset X$ is equipped with local coordinates $z_1,\dots,z_m$.
Let $\eta \in \mathcal{A}^k_\mathbb{C}(U)$ be a local differential form given as
\begin{align*}
\eta := \sum_{J_1,J_2} \eta_{J_1,J_2} dz_{J_1} \wedge d\overline{z}_{J_2}.
\end{align*}The image of $\eta$ under the complex exterior derivative
$d: \mathcal{A}_\mathbb{C}^k(U) \rightarrow\mathcal{A}_\mathbb{C}^{k+1}(U)$ is given \nolinebreak as
\begin{align*}
d\eta = \partial \eta + \opartial \eta &= \sum_{j=1}^{m} dz_j \wedge \pdv{\eta}{z_j} +
\sum_{j=1}^{m}d\overline {z}_j \wedge \pdv{\eta}{\overline z_j}\\
&=\sum_{j=1}^{m}\sum_{J_1,J_2}\pdv{\eta_{J_1,J_2}}{z_j} dz_j \wedge dz_{J_1} \wedge d\overline{z}_{J_2} +
\pdv{\eta_{J_1,J_2}}{\overline z_j} d\overline z_j \wedge dz_{J_1} \wedge d\overline{z}_{J_2}.
\end{align*}
This shows that $d, \partial$ and $\opartial$ are linear differential operators of order $1$.
\end{exmp}
After we have defined the $L^2$-metric and linear differential operators, it is possible to
generalize the notion of adjoint operators.
\begin{defn}
\label{kaehler-manifolds:defn:formal-adjoints}
Let $D: \mathcal{A}^k_\mathbb{C}(X) \rightarrow \mathcal{A}^l_\mathbb{C}(X)$ be a linear
differential operator of order $d$. The linear differential operator
$D^*:\mathcal{A}^l_\mathbb{C}(X)\rightarrow \mathcal{A}^k_\mathbb{C}(X)$ of \nolinebreak order
$d$ is called the \emph{formal adjoint operator} of $D$ with respect to the hermitian $L^2$-metric
if the following equality holds for all
$\alpha \in \mathcal{A}_\mathbb{C}^l(X), \beta \in \mathcal{A}_\mathbb{C}^k(X)$:
\begin{align}
\label{kaehler-manifolds:eq:formal-adjunction-property}
\big(D^* \alpha, \beta\big)_{L^2} = \big(\alpha,D\beta\big)_{L^2}.
\end{align}
Also, if it is $D^* = D$, we call $D$ to be \emph{formally self-adjoint}.
\end{defn}
\begin{rem}
Note that the notion of a formal adjoint operator can also be defined for linear differential
operators between any hermitian or euclidean vector bundles whose spaces of sections are endowed
with an $L^2$-metric. Since we will only need these adjoints for differential operators on the
differential forms, our definition will be sufficient. For a more general definition see
\cite[Ch.\,VI §1. Definition 1.5]{Demailly1997} and also the previous discussion there.
\end{rem}
\begin{rem}
\label{kaehler-manifolds:formal-adjoint-of-the-formal-adjoint}
In the setting of the last definition, let $(D^*)^*$ be the formal adjoint of $D^*$. We can
calculate for all $\alpha \in \mathcal A^l_\mathbb{C}(X), \beta \in \mathcal A^k_\mathbb{C}(X)$
\begin{align*}
\big((D^*)^*\beta,\alpha\big)_{L^2} = \big(\beta,D^* \alpha\big)_{L^2} =
\overline{\big(D^*\alpha,\beta\big)}_{L^2} = \overline{\big(\alpha,D\beta\big)}_{L^2} =
\big(D\beta,\alpha\big)_{L^2}.
\end{align*}
Since this is true for all $\alpha$ and $\beta$, it is also $D$, the formal adjoint of
$D^*$.
\end{rem}
As this calculation illustrates, we could use the relation between complex conjugation and the hermitian
$L^2$-metric to equivalently require for all
$\alpha \in \mathcal{A}^k_\mathbb{C}(X),\beta\in\mathcal{A}^l_\mathbb{C}(X)$
\begin{align}
\label{kaehler-manifolds:eq:formal-adjunction-property-2}
\big(\alpha,D^*\beta\big)_{L^2} = \big(D\alpha,\beta\big)_{L^2}
\end{align}
instead of the formal adjunction property in \Cref{kaehler-manifolds:eq:formal-adjunction-property}.
In order to calculate the formal adjoints of the exterior derivative and the two associated
Dolbeault operators $\partial$ and $\opartial$, we have the following lemma that has been inspired
by \cite[Sections 5.1.2 and 5.1.3]{Voisin2002}.
\begin{lm}
Let $\alpha \in \mathcal{A}^k_\mathbb{C}(X)\cap\mathcal{A}^{p,q}(X)$ and
$\beta \in \mathcal{A}^{k+1}_\mathbb{C}(X)$. Then, it is
\begin{align*}
(d\alpha,\beta)_{L^2} = \big(\alpha,-\hodgestar d\hodgestar\beta\big)_{L^2}.
\end{align*}
Additionally, if $\beta$ is of type $(p+1,q)$, it is
\begin{align*}
(\partial \alpha, \beta)_{L^2} = (\alpha,-\hodgestar\partial\hodgestar\beta)
\end{align*}
and if $\beta$ is of type $(p,q+1)$, it holds to be
\begin{align*}
(\opartial\alpha,\beta) = (\alpha,-\hodgestar\opartial\hodgestar \beta).
\end{align*}
\end{lm}
\begin{proof}
This proof extends the argument in \cite[Ch.\,VI §3. Theorem 3.9]{Demailly1997}.
Let $\alpha \in \mathcal A^k_\mathbb{C}(X)$ and $\beta \in \mathcal A^{k+1}_\mathbb{C}(X)$.
With Leibniz's rule, we get the following expression
\begin{align*}
d(\alpha\wedge\hodgestar\overline\beta) = d\alpha \wedge \hodgestar\overline\beta +
(-1)^k \alpha \wedge d(\hodgestar\overline\beta).
\end{align*}
This can be used to calculate
\begin{align*}
\label{kaehler-manifolds:eq:rhs-calc-adjoint-op}
(d\alpha,\beta)_{L^2} = \int_Xd\alpha \wedge \hodgestar \overline{\beta} = \int_X d(\alpha \wedge
\hodgestar\overline\beta) - (-1)^k \int_X\alpha \wedge d(\hodgestar\overline\beta).
\end{align*}
With Stokes theorem and the results from \Cref{loc-theory:lm:property-hodge-star}, which according
to \Cref{kaehler-manifolds:rem:local-are-global-properties-of-the-hodge-star} also apply to the
global Hodge star operator, we are able to calculate further
\begin{align*}
(d\alpha,\beta)_{L^2}
&= (-1)^{k+1} (-1)^{k} \int_X\alpha \wedge \hodgestar \hodgestar d (\hodgestar \overline \beta)\\
&= -\int_X\alpha \wedge \hodgestar(\overline{\hodgestar d \hodgestar \beta})\\
&= \big(\alpha, -\hodgestar d\hodgestar\beta\big)_{L^2}.
\end{align*}
This already proves the first equality. In order to prove the other two equalities, we can use
almost the same calculation with the exception of Stokes Theorem. Therefore, let
$\eta\in\mathcal{A}_\mathbb{C}^{2m-1}(X) \cap \mathcal{A}^{m,m-1}(X)$. It is $\partial\eta = 0$
and thus $\opartial\eta = d\eta$. We get
\begin{align*}
\int_X \partial\eta = 0
\end{align*}
and with Stokes theorem, we also obtain
\begin{align*}
\int_X\opartial\eta = \int_Xd\eta = 0.
\end{align*}
If it would be $\eta\in\mathcal{A}^{m-1,m}(X)$, we would get the same results. Since it is
\begin{align*}
\mathcal{A}_{\mathbb{C}}^{2m-1}(X) = \mathcal{A}^{m,m-1}(X)\oplus\mathcal{A}^{m-1,m}(X),
\end{align*}
(cf. \cite[Corollary 2.6.8]{Huybrechts2004}) our calculation shows that
\begin{align*}
\int_X \partial\eta = \int_X \opartial \eta = 0
\end{align*}
for all $\eta \in \mathcal{A}_\mathbb{C}^{2m-1}(X)$.
Since $\alpha \wedge \hodgestar\beta \in \mathcal{A}_{\mathbb{C}}^{2m-1}(X)$, it is
\begin{align*}
\int_X\partial(\alpha\wedge\hodgestar\overline\beta) = 0 \enspace\;\text{and} \enspace\;
\int_X\opartial(\alpha\wedge\hodgestar\overline\beta) = 0.
\end{align*}
Thus, the same calculation as above can be used to prove the other two statements.
\end{proof}
As an immediate result of this lemma, we obtain an explicit definition for the formal adjoint operators
$d^*, \partial^*$ and $\opartial^*$.
\begin{cor}[{\cite[Lemma 5.7, Lemma 5.8]{Voisin2002}}]
The formal adjoint operators of $d,\partial$ and $\opartial$ are explicitly given as
\begin{align*}
d^* := -\hodgestar d \hodgestar, \quad\; \partial^* := - \hodgestar \partial \hodgestar,\quad\;
\opartial^* := -\hodgestar \opartial \hodgestar.
\end{align*}
\end{cor}
\begin{rem}[{\cite[Lemma 3.1.4]{Huybrechts2004}}]
\label{kaehler-manifolds:rem:splitting-of-the-formal-adjoint-of-the-exterior-derivative}
We know that the exterior derivative splits as $d = \partial + \opartial$. Therefore, the explicit
expressions of the formal adjoint operators can be used to verify the existence of a similar
splitting $d^*=\partial^* + \opartial^*$.
Furthermore, we can use them to verify $(\partial^*)^2 = (\opartial^*)^2 = 0$.
It should be mentioned, however, that these results would have already been possible with only
the abstract \Cref{kaehler-manifolds:defn:formal-adjoints}.
\end{rem}
\subsection{Kähler identities}\;
To conclude this chapter, we will introduce a special kind of hermitian manifold known as Kähler
manifold. Kähler manifolds are characterized by the fact that their associated fundamental form is
closed, and therefore, they are additionally equipped with a symplectic structure.
Although we will not take a closer look at this additional structure, we are interested in another
property. For Kähler manifolds, there exist some interesting relations between the dual Lefschetz
operator $\Lambda$ and the Dolbeault operators $\partial$ and $\opartial$. These relations are
called Kähler identities, and they will be an essential property for the proof of the Hodge
Decomposition theorem.
\begin{defn}
A \emph{Kähler manifold} is a hermitian manifold whose associated fundamental form $\omega$ is
closed, i.e. $d \omega = 0$. In this case, the hermitian metric $h$ is called \emph{Kähler}.
\end{defn}
\begin{exmp}
Let $Y$ be a Riemannian surface, i.e. a $1$-dimensional complex manifold. With
\Cref{kaehler-manifolds:lm:all-complex-manifolds-are-hermitian}, we know that $Y$ is also hermitian.
Also any $2$-form $\omega \in \mathcal{A}_\mathbb{R}^2(X)$ is closed because $Y$ has real
dimension $2$. Thus, $Y$ is also a Kähler manifold.
\end{exmp}
For additional examples see \cite[Examples 3.1.9]{Huybrechts2004}. Furthermore, with the theory
established in the previous sections of this chapter, we can finally provide the Kähler identities.
\begin{thm}[{\cite[Proposition 6.5]{Voisin2002}}]
\label{kaehler-manifolds:thm:kaehler-identities}
On a compact Kähler manifold, we have the identities
\begin{align*}
[\Lambda,\opartial] = -i\partial^*, \quad [\Lambda,\partial] = i\opartial^*,
\end{align*}
with the Lie bracket being defined as the commutator.
\end{thm}
We are not going to prove this statement but rather refer to the proofs in
\cite[Proposition 6.5, Lemma 6.6]{Voisin2002} and \cite[Proposition 3.1.12]{Huybrechts2004}.
\begin{rem}
Typically, the Kähler identities consist of additional equations. However, for the purposes of this
thesis, only the two presented in this theorem will be relevant. A more complete list can be found
in the stated proposition in \cite{Huybrechts2004}.
\end{rem}

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\section{Local Theory}
This chapter aims to provide the reader with some of the essential notions and tools for the local
theory needed to form a comprehensive understanding of the concept of hermitian manifolds and
in particular, Kähler manifolds. The main focus will be on complex vector spaces and hermitian
forms on those spaces and also the tools linear algebra provides.
Later, we are going to focus on the different tangent bundles of complex manifolds, which are
collections of vector spaces that vary in a geometric way on the manifold. Therefore, most of the
notions and tools we are going to introduce in this chapter will be translated into the global
context later.
The main goal of this chapter will be the definition of the local Hodge star operator \nolinebreak
$\hodgestar$ and the definition of the local Lefschetz and dual Lefschetz operator $L$ and
$\Lambda$. For this purpose, we will start by providing a brief overview of the topic of
complexification of vector spaces. Afterwards, we will focus on euclidean and hermitian vector
spaces. Finally, we will conclude with the definition of the operators mentioned above.
The reader should be aware that this chapter is based on the similarly named chapter in Daniel
Huybrechts' \emph{Complex Geometry} \cite{Huybrechts2004}, although the level of detail found there
is not to be expected.
\subsection{Complexification of vector spaces}\;
To simplify the notation, we are going to assume the following setting for the remainder of this
section.
\begin{set}
Let $V$ denote a real $n$-dimensional vector space. Also, assume that $V$ is an almost complex
vector space, i.e. $V$ is equipped with an
endomorphism $I: V\rightarrow V$ such that $I^2 = -\id$. This endomorphism is called the almost
complex structure of $V$.
\end{set}
Using this almost complex structure, we can also think of $V$ as a complex vector space with the
$\mathbb{C}$-module structure defined as $(a+ib)\cdot v = a v + b I(v)$ for all $a,b \in
\mathbb{R}$. For this complex vector space, we will write $(V,I)$. With the product rule for the
determinant, we can calculate
\begin{align*}
\det(I)^2 = \det(I^2) = \det(-\id) = (-1)^n.
\end{align*}
Since $\det(I)$ is real, we conclude that $n = 2m$ for some $m \in \mathbb{N}$.
Furthermore, $V$ and $(V,I)$ are equal as sets and if $(v_1,\dots,v_d)$ is a complex basis of
$(V,I)$, it is immediate that $(v_1,I(v_1),\dots,v_d,I(v_d))$ is a real basis of $V$. Therefore,
their dimensions relate as
\begin{align*}
\dim_\mathbb{C} (V,I) = d = \frac{1}{2} \dim_\mathbb{R} V = \frac{1}{2}n = m.
\end{align*}
Additionally, as an almost complex vector space, $V$ is endowed with a natural orientation. This
boils down to the fact that the real space $\mathbb{C}^m$ has a natural orientation given by the
basis $(e_1, ie_1, \dots e_m, ie_m)$, with the $e_1,\dots,e_m$ being the standard basis vectors (cf.
\cite[Corollary 1.2.3]{Huybrechts2004}).
At the same time, it is possible to construct a different complex vector space using $V$.
\begin{defn}
The \emph{complexification} $V_\mathbb{C}$ of $V$ is defined as
$V_\mathbb{C} := V \otimes_\mathbb{R} \mathbb{C}$.
\end{defn}
Let $\left(v_1, \dots, v_n\right)$ be a real basis of $V$. With the properties of the tensor
product, it is $\left(v_1 \otimes 1, \dots , v_n \otimes 1\right)$ a complex basis of
$V_\mathbb{C}$. This shows that there exists an inclusion $V \into V_\mathbb{C}$ and for the
dimension of $V_\mathbb{C}$, we get
\begin{align*}
\dim_\mathbb{C} V_\mathbb{C} = n = \dim_\mathbb{R} V.
\end{align*}
Now, the almost complex structure $I$ can be linearly extended to an almost complex structure
$I_\mathbb{C}$ on $V_\mathbb{C}$. This is defined as $I_\mathbb{C} (v\otimes 1 + w \otimes i) :=
I(v) \otimes 1 + I(w)\otimes i$, and it is evident that this linear extension also has the property
$I_\mathbb{C}^2 = -\id$. Thus, we also call this to be an almost complex structure.
\begin{nota}
Note that for a vector $v\otimes \lambda \in V_\mathbb{C}$, it is a common practice to sometimes
omit the tensor product in the notation, just noting $\lambda v$ instead of $v \otimes \lambda$. If
it is possible without confusion, we will also write $I$ instead of $I_\mathbb{C}$ for the complex
extension of the almost complex structure.
\end{nota}
The following proposition shows how the two $\mathbb{C}$-module structures on $V_\mathbb{C}$,
defined by the almost complex structure $I$ and by multiplication with $i$, compare to each other.
\begin{prop}[Decomposition of $V_\mathbb{C}$ {\cite[Lemma 1.2.5]{Huybrechts2004}}]
\label{loc-theory:lm:decomposition-of-vc}
For the complexification $V_\mathbb{C}$ we have the decomposition
$V_\mathbb{C} = V^{1,0} \oplus V^{0,1}$ with
\begin{align*}
V^{1,0} := \left\{v \in V_\mathbb{C} \mid I(v) = iv\right\} \enspace
\text{and} \enspace V^{0,1} := \left\{v \in V_\mathbb{C} \mid I(v) =- iv\right\}.
\end{align*}
\end{prop}
\begin{proof}We extend the proof of the stated lemma in \cite{Huybrechts2004}.
Let $v \in V_\mathbb{C}$. It is $v = \frac{1}{2} (v- iI(v)) + \frac{1}{2} (v + iI(v))$. A simple
calculation shows
\begin{align*}
I(v-iI(v)) = I(v) -I(iI(v)) = I(v) - iI^2(v) = I(v) + iv = i(-iI(v) + v)
\end{align*}
and thus $\frac{1}{2} (v -iI(v)) \in V^{1,0}$. With a similar calculation we obtain
$\frac{1}{2} (v+iI(v)) \in V^{0,1}$. At the same time, it holds to be $V^{1,0} \cap V^{0,1} = \{0\}$.
Thus, the inclusion $V^{1,0} \oplus V^{0,1} \into V_\mathbb{C}$ is injective and with the above
calculation, it is also surjective. Hence, it is a canonical isomorphism, so the decomposition is
proven.
\end{proof}
\begin{rem} We expand the argument in the proof of \cite[Lemma 1.2.5]{Huybrechts2004}.
The proof of the last proposition shows that a vector $w \in V^{1,0}$ can be written as
$w = v-iI(v)$ for some $v \in V_\mathbb{C}$. At the same time, we can split $v = x + iy$
with $x,y \in V$. Then it is
\begin{align*}
\overline{w} = \overline{v -iI_\mathbb{C}(v)} &= \overline{x + iy - i (I(x) + iI(y))}\\
&= x -iy+iI(x)+I(y) = \overline{v}+ i(I(x) + -iI(y)) = \overline{v} + iI_\mathbb{C}(\overline{v}).
\end{align*}
Hence it is $\overline{w} \in V^{0,1}$. Similar calculations show that for $w \in V^{0,1}$, it is
$\overline{w} \in V^{1,0}$ and $\overline{\widebar{w}} = w$. Since complex conjugation is
$\mathbb{R}$-linear, this already proves that $V^{1,0}$ and $V^{0,1}$ are isomorphic as real vector
spaces.
\end{rem}
\begin{rem}
\label{loc-theory:rem:c-linear-c-antilinear}
Using the proof of the last proposition and the natural inclusion \linebreak$V \into V_\mathbb{C},\,
v \mapsto v \otimes 1$, we can define an $\mathbb{R}$-linear isomorphism
\begin{align*}
\varphi_1: (V,I) \rightarrow V^{1,0},\;\enspace v \mapsto \big(v\otimes 1-iI_\mathbb{C}(v\otimes 1)\big).
\end{align*}
However, we are able to calculate
\begin{align*}
\varphi_1(I(v)) = I(v) \otimes 1 - iI_\mathbb{C}(I(v) \otimes 1) = I_\mathbb{C}(v \otimes 1)
-iI_\mathbb{C}^2(v \otimes 1) = I_\mathbb{C}\big(v\otimes 1 -iI_\mathbb{C}(v\otimes 1)\big).
\end{align*}
Hence, we obtain $\varphi_1(I(v)) = I_\mathbb{C}(\varphi_1(v)) = i\varphi_1(v)$ because
$\varphi_1(v) \in V^{1,0}$. Since the $\mathbb{C}$-module structure on $(V,I)$ is defined using $I$,
we know that $\varphi_1$ is also a $\mathbb{C}$-linear isomorphism. At the same time, we are able to
define a similar $\mathbb{R}$-linear isomorphism
\begin{align*}
\varphi_2: (V,I) \rightarrow V^{0,1} \;\enspace
v \mapsto (v \otimes 1 + iI_\mathbb{C}(v \otimes 1)).
\end{align*}
The same calculation yields $\varphi_2(I(v)) = -i\varphi_2(v)$. Thus, $\varphi_2$ is a
$\mathbb{C}$-antilinear isomorphism.
\end{rem}
Next, we are going to define an induced almost complex structure on the dual space $V^*$. Because of
its induced nature, this almost complex structure is also called $I$ and it is defined as a mapping
$I: V^* \rightarrow V^*$, such that $I(f)(v) = f(I(v))$ for all $f \in V^*$ and $v \in V$.
Now, the following lemma ensures the compatibility of the complexification with the dual space of $V$.
\begin{lm}[{\cite[Lemma 1.2.6]{Huybrechts2004}}]
\label{loc-theory:lm:compatibility-of-dual-and-complexification}
It is $(V_\mathbb{C})^* = \Hom_\mathbb{R}(V,\mathbb{C}) = (V^*)_\mathbb{C}$ and it also holds to be
\begin{alignat*}{2}
(V^{1,0})^* &= \left\{f \in \Hom_\mathbb{R}(V,\mathbb{C}) \mid f(I(v)) = if(v)\; \forall v \in V\right\} = (V^*)^{1,0},\\
(V^{0,1})^* &= \left\{f \in \Hom_\mathbb{R}(V,\mathbb{C}) \mid f(I(v)) = -if(v)\; \forall v \in V\right\} = (V^*)^{0,1}.
\end{alignat*}
\end{lm}
\begin{proof}
It is $(V_\mathbb{C})^* = \Hom_\mathbb{C} (V_\mathbb{C}, \mathbb{C})$ and
$(V^*)_\mathbb{C} = \Hom_\mathbb{R} (V, \mathbb{C})$. In order to prove that these two spaces
are equal, we have to prove the existence of a canonical isomorphism
$\Hom_\mathbb{R}(V,\mathbb{C}) \cong \Hom_\mathbb{C} (V_\mathbb{C}, \mathbb{C})$.
Let $f \in \Hom_\mathbb{R}(V,\mathbb{C})$ and extend it to an $\mathbb{R}$-linear mapping
$\tilde f: V_\mathbb{C} \rightarrow \mathbb{C}$ by setting $\tilde f(v \otimes \lambda): = \lambda
f(v)$ for all $v \in V$ and $\lambda \in \mathbb{C}$. This mapping is also $\mathbb{C}$-linear
because we can show for all $\mu \in \mathbb{C}$
\begin{align*}
\tilde f(\mu\cdot(v \otimes \lambda)) = \tilde f (v \otimes \mu\lambda) = \mu\lambda f(v) = \mu
\cdot \tilde f (v \otimes \lambda).
\end{align*}
This shows that for every $f$, we can find a unique $\tilde f \in \Hom_\mathbb{C}(V_\mathbb{C},
\mathbb{C})$.\\
Let now $g \in \Hom_\mathbb{C}(V_\mathbb{C}, \mathbb{C})$. Using the inclusion
$V \into V_\mathbb{C}$, we can restrict $g$ to obtain a mapping $h: V \rightarrow \mathbb{C}$
that is defined as $h(v):= g(v \otimes 1)$.
Since $g$ was $\mathbb{C}$-linear, $h$ is already an $\mathbb{R}$-linear mapping.
This shows that $h \in \Hom_\mathbb{R}(V,\mathbb{C})$, and since those two constructions are
obviously inverse to each other, this completes the proof of the first statement.\footnote{This
first part of the proof was created using a Math Stack Exchange post of the user \emph{Mark}
(\url{https://math.stackexchange.com/users/470733/mark}) that can be found on
\url{https://math.stackexchange.com/q/4718935} and was last checked on the 25th of August, 2023.}\\
For the second statement, we use \Cref{loc-theory:rem:c-linear-c-antilinear} to get
\begin{align*}
(V^{1,0})^* = \Hom_\mathbb{C}(V^{1,0},\mathbb{C})&= \Hom_\mathbb{C}((V,I),\mathbb{C}) \\&=
\left\{f \in \Hom_\mathbb{R}(V,\mathbb{C}) \mid f(I(v)) = if(v)\; \forall v \in V\right\}.
\end{align*}
Additionally, for the other subspace, we can use the same remark to obtain
\begin{align*}
(V^{0,1})^* = \Hom_\mathbb{C}(V^{0,1},\mathbb{C}) &= \Hom_{\overline{\mathbb{C}}}((V,I),
\mathbb{C}) \\&= \left\{f \in \Hom_\mathbb{R}(V,\mathbb{C})\mid f(I(v)) = -if(v)\; \forall v \in
V\right\}.
\end{align*}
\end{proof}
\begin{nota}
Because of the last lemma, we will only write $V^*_\mathbb{C}$, omitting the brackets from now on.
\end{nota}
\subsection{Euclidian and hermitian vector spaces}\;
Later, we are going to define the notion of a hermitian manifold, i.e. a complex manifold whose
holomorphic tangent space in every point is equipped with a hermitian form. In order to do so, this
section will cover some fundamental statements about those forms on complex vector spaces.
For the remainder of this section, we are going to assume the following setting.
\begin{set}
Let $(V,g)$ be a real $n$-dimensional euclidean vector space, i.e. $g$ is a positive definite
symmetric bilinear form on the real space $V$. Also, assume that $V$ is equipped with an almost
complex structure $I$.
\end{set}
\begin{defn}
The inner product $g$ is said to be \emph{compatible with the almost complex structure} $I$ if it
holds to be $g(I(v),I(w)) = g(v,w)$ for all $v,w \in V$.
\end{defn}
\begin{nota}
If the inner product $g$ on $V$ is compatible with the almost complex structure $I$, we usually
only write $(V,g,I)$.
\end{nota}
The just-established notion of a compatible inner product gives rise to an additional notion.
\begin{defn}
The \emph{fundamental form} associated to $(V,g,I)$ is defined as the form $\omega \in
\bigwedge^2V^* \cap \bigwedge^{1,1} V^*$, such that for all $v,w \in V$ it is
\begin{align*}
\omega(v,w) := g(I(v),w) = - g(v, I(w)).
\end{align*}
Note that the second equality is equivalent to $g$ being compatible with the almost complex
structure $I$. Also, note that this immediately yields $\omega(I(v),I(w)) = \omega(v,w)$.
\end{defn}
\begin{rem}
The expression $ \bigwedge^2V^* \cap \bigwedge^{1,1} V^*$ has to be explained. With
\Cref{loc-theory:lm:compatibility-of-dual-and-complexification}, we know that
$\bigwedge^2 V^*\subset \bigwedge^2 V_\mathbb{C}^*$. At the same time, it is
\begin{align}
\label{loc-theory:eq:decomps-of-complex-2-forms}
\bigwedge\nolimits^2 V_\mathbb{C}^* = \bigwedge\nolimits^{2,0} V^* \oplus \bigwedge\nolimits^{1,1}
V^* \oplus \bigwedge\nolimits^{0,2} V^*
\end{align}
(cf. \cite[Proposition 1.2.8 (ii), Example 1.2.34]{Huybrechts2004}).
and for these reasons, the intersection is meaningful as it happens in $\bigwedge^2V^*_\mathbb{C}$.
This expression describes all the alternating real 2-forms of type $(1,1)$, i.e. alternating real
2-forms on $V$, that are also $\mathbb{C}$-linear in it's first argument and $\mathbb{C}$-antilinear
in its second argument if viewed as forms on $V_\mathbb{C}$.
\end{rem}
Now, for the fundamental form $\omega$ from the last definition to be well-defined, we have
to check whether $\omega$ is indeed an alternating real 2-form on $V$ and is also of type $(1,1)$.
For the first statement, real bilinearity follows directly with the bilinearity of $g$. Also using the symmetry
of $g$, we calculate for all $v,w \in V$
\begin{align*}
\omega(v,w) = g(I(v),w) = g(I^2(v),I(w)) = - g(v,I(w)) = - g(I(w),v) = -\omega(w,v).
\end{align*}
Hence, $\omega$ is alternating and therefore a real 2-form. With
\Cref{loc-theory:eq:decomps-of-complex-2-forms}, it suffices to show that the $\mathbb{C}$-bilinear
extension of $\omega$ vanishes on all pairs of vectors $v,w$ in $V^{1,0}$ or $V^{0,1}$ to
prove that it is of type $(1,1)$. In the first case, i.e. $v,w \in V^{1,0}$, we calculate
\begin{align*}
\omega(v,w) = \omega(I(v),I(w)) = \omega(iv,iw) = i^2 \omega(v,w) = -\omega(v,w).
\end{align*}
Hence $\omega(v,w) = 0$. The first equation holds because the complex bilinear extension
inherits this property from the real form $\omega$. The calculation for the other case can be carried out
analogously. Hence, $\omega$ is indeed of type $(1,1)$ and this establishes the well-definedness of
$\omega$.
For $(V,g,I)$, we can also define a positive definite hermitian form on the complex space $(V,I)$.
This form is defined as
\begin{align*}
h: (V,I) \times (V,I) \rightarrow \mathbb{C},\;\enspace
(v,w) \mapsto g(v,w) - i\omega(v,w).
\end{align*}
Additionally, the inner product $g$ on $V$ can be extended sesquilinearly to a positive definite
hermitian form on $V_\mathbb{C}$. This extension is defined as
\begin{align*}
h_\mathbb{C} : V_\mathbb{C} \times V_\mathbb{C} \rightarrow \mathbb{C},\; \enspace
(v \otimes \lambda, w \otimes \mu) \mapsto (\lambda \overline{\mu}) \cdot g(v,w).
\end{align*}
See also \cite[p.\,30]{Huybrechts2004} for the similar definitions.
However, it has to be checked whether these two positive definite hermitian forms are well-defined.
\begin{prop}[{\cite[Lemma 1.2.15]{Huybrechts2004}}]
For $(V,g,I)$, the form $h:= g- i\omega$ is indeed a positive definite
hermitian form on $(V,I)$. Also, the extension $h_\mathbb{C}$ of the inner product $g$ defines a
positive definite hermitian form on $V_\mathbb{C}$.
\end{prop}
\begin{proof}
We expand the calculation in the proof of the stated lemma in \cite{Huybrechts2004} and add the
argument for our second statement. Let
$v,w \in (V,I)$. It is $h(v,v) = g(v,v) - i \omega(v,v) =g(v,v)$ because $\omega$ is alternating and
therefore $\omega(v,v) = 0$. Since $g$ is positive definite, this proves that $h$ is positive definite
as well. Furthermore, since $g$ is symmetric, it is also
\begin{align*}
h(v,w) = g(v,w) - i\omega(v,w) = g(w,v) + i\omega(w,v) = \overline{h(w,v)},
\end{align*}
and it also holds to be
\begin{align*}
h(I(v),w) &= g(I(v),w) - i\omega(I(v),w) \\&= g(I^2(v),I(w)) -i (g(I^2(v),w)) \\&= -g(v,I(w)) +
ig(v,w) \\&= i\big(i g(v,I(w)) + g(v,w)\big) \\&= i\big(- i\omega(v,w) + g(v,w)\big) = i h(v,w).
\end{align*}
On $(V,I)$, the image under $I$ corresponds to multiplication with $i$ because the
$\mathbb{C}$-module structure is defined using the almost complex structure. This proves the
$\mathbb{C}$-linearity in the first argument, as the $\mathbb{R}$-linearity is already inherited from
$g$ and $\omega$.
For the $\mathbb{C}$-antilinearity in the second argument, we combine the last two statements to
get
\begin{align*}
h(v,I(w)) = \overline{h(I(w),v)} = \overline{ih(w,v)} = -ih(v,w).
\end{align*} This completes the proof of the first statement.
To prove the second statement, it is already clear by definition that $h_\mathbb{C}$ is
$\mathbb{C}$-linear in its first argument and $\mathbb{C}$-antilinear in its second argument. Let
$(v_1,\dots,v_n)$ be an orthonormal basis of $V$ with respect to the inner product $g$. With the
properties of the tensor product, it is again $(v_1 \otimes 1, \dots, v_n \otimes 1)$ a basis of
$V_\mathbb{C}$. Therefore, we can write every element $u \in V_\mathbb{C}$ as
$u = \sum_{j=1}^{n} \lambda_j(v_j \otimes 1)= \sum_{j=1}^{n} v_j \otimes \lambda_j$. We are then
able to calculate
\begin{align*}
h_\mathbb{C}(u,u) = h_\mathbb{C}\Big(\sum_{j=1}^n v_j \otimes \lambda_j, \sum_{k=1}^n v_k \otimes
\lambda_k\Big) = \sum_{j,k=1}^{n} \lambda_j\overline\lambda_k g(v_j,v_k) = \sum_{j=1}^n|\lambda_j|^2
\geq 0,
\end{align*}
and $h_\mathbb{C}(u,u) = 0$ if and only if $u = 0$. Hence, $h_\mathbb{C}$ is positive definite.
Furthermore, it holds to be
\begin{align*}
h_\mathbb{C}(v \otimes \lambda,w \otimes \mu) = \lambda \overline{\mu} \cdot g(v,w) = \overline
{\overline \lambda \mu \cdot g(v,w)} = \overline{\overline \lambda \mu \cdot g(w,v)} =
\overline{h_\mathbb{C} (w \otimes \mu, v \otimes \lambda)}.
\end{align*}
Thus, $h$ and $h_\mathbb{C}$ are both positive definite hermitian forms.
\end{proof}
\begin{nota} In \cite[Lemma 1.2.17]{Huybrechts2004}, it is shown that these two hermitian forms only
differ by a factor of $\frac{1}{2}$ under the natural inclusion $(V,I) \into V^{1,0}$. This may be a
reason for the common practice not to differentiate between $h$ and $h_\mathbb{C}$ in the notation,
which we will also adhere to.
\end{nota}
\begin{rem}
\label{loc-theory:rem:real-of-hermitian-form}
In the last proposition, it has been proven that the compatible inner product $g$ already defines a
positive definite hermitian form on $(V,I$). Let now $\tilde h$ be an arbitrary positive definite hermitian
form on $(V,I)$. We can define the \emph{real part} of $h$ as follows.
\begin{align*}
\Re(\tilde h)(v,w) := \frac{1}{2} \big(\tilde h(v,w) + \overline{\tilde h(v,w)}\big) =
\frac{1}{2}\big(\tilde h(v,w) + \tilde h(w,v)\big)
\end{align*}
With the second equality, it is obvious that this defines a real positive definite and symmetric
bilinear form on $V$. It is also
\begin{align*}
\Re(\tilde h)(I(v),I(w)) &= \frac{1}{2} \big(\tilde h(I(v),I(w)) + \tilde h(I(w),I(v))\big)\\&=
\frac{1}{2} \big(\tilde h(v,w) + \tilde h(w,v)\big) \\&= \Re(\tilde h)(v,w)
\end{align*}
and thus $\Re(\tilde h)$ defines a compatible inner product on $V$. Because these constructions are
inverse to each other, there is a one-to-one relation between positive definite hermitian forms on
$(V,I)$ and inner products on $V$ that are compatible with $I$. See also
\cite[Section. 3.1.1]{Voisin2002} for further information about this relation.
\end{rem}
Later, we will also need inner products and hermitian forms on the exterior algebra spaces
$\bigwedge^kV^*$ and $\bigwedge^k V^*_\mathbb{C}$. In the following two lemmas, we are going
to construct inner products on $V^*$ and $\bigwedge^kV$ using the existing inner product $g$ on $V$,
and we will eventually use those constructions to define an induced inner product on
$\bigwedge\nolimits^kV^*$. This construction is inspired by \cite[Section 11]{Schnell2012}, but no
proofs are provided there.
Before we begin, we need to take a look at the natural linear mapping
\begin{align*}
g^\flat: V \rightarrow V^*, \;v\mapsto g(v,-).
\end{align*}
With respect to the inner product $g$, we can choose an orthonormal basis $(v_1,\dots,v_n)$ of $V$.
For all $r,s \in \mathbb{N}$ with $1 \leq r,s \leq n$, we get
\begin{align*}
g^\flat(v_r)(v_s) = g(v_r,v_s) = \delta_{rs}.
\end{align*}
Hence, $g^\flat(v_r) = v^r$ with $v^r$ being the dual basis vector of $v_r$. This already shows that
$g^\flat$ is an isomorphism. Let now $g^\sharp: V^* \rightarrow V$ denote the inverse mapping of
$g^\flat$. Using this mapping, we get the following lemma.
\begin{lm}
\label{loc-theory:lm:product-on-dual-space}
The inner product $g$ on $V$ induces an inner product on $V^*$. It is defined \nolinebreak as
\begin{align*}
\tilde{g}: V^* \times V^* \rightarrow \mathbb{R},\; \enspace
(v,w) \mapsto g\big(g^\sharp(v),g^\sharp(w)\big).
\end{align*}
\end{lm}
\begin{proof}
It is obvious that $\tilde{g}$ defines a bilinear mapping. Let now $(v_1,\dots,v_n)$ be an
orthonormal basis of $V$ again. Also let $(v^1,\dots,v^n)$ denote the corresponding dual basis of
$V^*$. If only evaluated on those dual basis vectors, $\tilde g$ simplifies as
\begin{align*}
\tilde{g}(v^r,v^s) = g(g^\sharp(v^r), g^\sharp(v^s)) = g(v_r,v_s).
\end{align*}
Thus, $\tilde{g}$ directly inherits the inner product properties from \nolinebreak$g$.
\end{proof}
\begin{lm}
\label{loc-theory:lm:product-on-exterior-algebra}
The inner product $g$ on $V$ induces an inner product on $\bigwedge\nolimits^kV$, which is defined as
\begin{align*}
g_k: \bigwedge\nolimits^kV \times \bigwedge\nolimits^kV &\rightarrow \mathbb R\\
\big(v_1 \wedge \dots \wedge v_k, w_1\wedge\dots \wedge w_k\big) &\mapsto
\det\Big(\big(g(v_r,w_s)\big)_{rs}\Big).
\end{align*}
\end{lm}
\begin{proof}
With the multilinearity of the determinant, it is again obvious that $g_k$ is a bilinear mapping.
Since the determinant is invariant under transposition, $g_k$ is also symmetric. Let now
$(v_1,\dots,v_n)$ be an orthonormal basis of $V$ again. We know that this induces a basis
$(v_J)_J$ of $\bigwedge^k V$. (cf. \cite[Proposition 14.8]{Lee2012}.) If we only evaluate $g_k$
on these basis vectors again, we get
\begin{align*}
g_k(v_J,v_K) = \det\Big(\big(g(v_{j_r}, v_{k_s})\big)_{rs}\Big) =
\delta_{JK}.
\end{align*}
This is because if $J \neq K$, there is at least one $j_l \in J$ such that $j_l \not\in K$. Thus we
obtain $g(v_{j_l}, v_{\tilde{k}}) = 0$ for all $\tilde{k} \in K$. Hence, the $l$-th column of the
matrix $G:=\big(g(v_{j_r},v_{k_s})\big)_{rs}$ is everywhere zero and therefore
$\det(G) = 0$. If $J$ and $K$ are equal however, then it is $G = \id_k$ and therefore
$\det(G) = 1$. With this, we have for all $\alpha := \sum_{J} \alpha_J v_J\in \bigwedge^k V$
\begin{align*}
g_k(\alpha,\alpha) = \sum_{J} \alpha_J^2\, g_k(v_J,v_J) = \sum_{J} \alpha_J^2 \geq 0.
\end{align*}
This calculation also implies that $g_k(\alpha,\alpha) = 0$ if and only if $\alpha = 0$. Thus, the
statement is proven.
\end{proof}
The combination of the last two lemmas finally gives us an inner product on
$\bigwedge\nolimits^kV^*$.
\begin{cor}
\label{loc-theory:cor:induced-product-on-exterior-algebra}
The induced inner product on $\bigwedge^kV^*$ is given as
\begin{align*}
\tilde{g}_k: \bigwedge^kV^* \times \bigwedge^kV^* &\rightarrow \mathbb{R}\\
\big(v^1 \wedge \dots \wedge v^k, w^1\wedge\dots \wedge w^k\big) &\mapsto
\det\Big(\big(g(g^\sharp(v^j),g^\sharp(w^k))\big)_{jk}\Big).
\end{align*}
\end{cor}
\begin{rem}
\label{loc-theory:rem:preserving-orthonormality}
The two proofs of
\Cref{loc-theory:lm:product-on-dual-space,loc-theory:lm:product-on-exterior-algebra}
also show that the induced inner products preserve orthonormality, i.e. for an orthonormal basis
$(v_1,\dots,v_n)$ of $V$, the induced bases $(v^1,\dots,v^n)$ of $V^*$ and $(v_J)_{J}$ of
$\bigwedge^k V$ are also orthonormal with respect to the induced inner products. Since \Cref{loc-theory:cor:induced-product-on-exterior-algebra} just combines these inner products,
this property also holds for the induced inner product $\tilde{g}_k$ on $\bigwedge\nolimits^kV^*$.
\end{rem}
\begin{rem}
\label{loc-theory:rem:hermitian-form-on-exterior-algebra}
Similar to the construction of the positive definite hermitian form on $V_\mathbb{C}$, we are able
to obtain positive definite hermitian forms $\tilde{h}_k$ on the exterior algebra spaces
$\bigwedge\nolimits^kV_\mathbb{C}^*$, by extending the inner products $\tilde{g}_k$ sesquilinearly
(cf. \cite[p.\,33]{Huybrechts2004}).
\end{rem}
\begin{nota}
It is common practice to only write $g$ and $h$ for all these inner products and hermitian forms,
respectively. This is because they have similar properties and since they are all defined on different
spaces, it should always be clear from the context which form is meant.
\end{nota}
\subsection{Local operators}\;
The next section will focus on the definition of some essential operators. These are initially
defined as local operators on vector spaces but are going to be used on vector bundles later.
However, most of their properties can already be shown locally.
Therefore, we will assume the following setting for the remainder of this section.
\begin{set}
Let $(V,g, I)$ be an euclidean vector space of dimension $n$ with a compatible almost complex
structure. Also, let $\omega$ denote the fundamental form associated with $g$ and let $h$ denote
the induced hermitian forms on $(V,I)$ and $V_\mathbb{C}$. The existence of the almost complex
structure $I$ ensures $n = 2m$ fore some $m \in \mathbb{N}$.
\end{set}
\begin{defn}[Lefschetz operator]
With $\omega$ the associated fundamental form to $(V,g,I)$, the real \emph{Lefschetz operator} is
defined as the linear mapping
\begin{align*}
L: \bigwedge\nolimits^k V^* \rightarrow \bigwedge\nolimits^{k+2} V^*,\;\enspace
\alpha \mapsto \omega \wedge \alpha.
\end{align*}
The complex \emph{Lefschetz operator} on $\bigwedge\nolimits^kV^*_\mathbb{C}$ is then
defined as the $\mathbb{C}$-linear extension $L_\mathbb{C}: \bigwedge\nolimits^k V^*_\mathbb{C}
\rightarrow \bigwedge\nolimits^{k+2} V^*_\mathbb{C}$, i.e. $L_\mathbb{C}(\beta) = \omega \wedge
\beta$ for all $\beta \in \bigwedge\nolimits^{k+2} V^*_\mathbb{C}$.
\end{defn}
\begin{rem}
Note that $\omega \in \bigwedge\nolimits^2 V^* \cap \bigwedge\nolimits^{1,1}
V^*\subset\bigwedge\nolimits^2V_\mathbb{C}^*$ and this is the reason for these two wedge products
to be both meaningful. Also, we have to keep in mind that the definition of this operator depends on the
fundamental form $\omega$, which itself depends on the choice of the inner product $g$.
Furthermore, due to the fundamental form $\omega$ being of type $(1,1)$, it is apparent that the
restriction of $L_\mathbb{C}$ to forms of type $(p,q)$ behaves like
$L: \bigwedge\nolimits^{p,q} V^* \rightarrow \bigwedge\nolimits^{p+1,q+1}V^*$
and thus the complex Lefschetz operator preserves the revised structure of
$\bigwedge\nolimits^k V_\mathbb{C}^* = \bigoplus_{p+q=k}\bigwedge\nolimits^{p,q}V^*$
(cf. \cite[Proposition 1.2.8 (ii)]{Huybrechts2004}).
\end{rem}
In linear algebra, we have the notion of an adjoint operator with respect to an inner product on a
vector space. Therefore, we can use our induced inner product on the exterior algebra spaces
$\bigwedge\nolimits^kV^*$ to define the dual of the Lefschetz operator.
\begin{defn}
The \emph{dual Lefschetz operator} $\Lambda$ is defined as the adjoint operator of $L$ with respect
to the inner product $g$, i.e. the uniquely defined mapping
\begin{align*}
\Lambda: \bigwedge\nolimits^{k+2} V^* \rightarrow \bigwedge\nolimits^{k} V^*,
\end{align*}
such that for all $\alpha \in \bigwedge\nolimits^{k+2} V^*$ and $\beta \in \bigwedge\nolimits^{k}V^*$,
it is $g(\Lambda(\alpha), \beta) = g(\alpha,L(\beta))$.
\end{defn}
\begin{rem}
Note that this operator is indeed uniquely defined because of the non-degeneracy of the inner
product $g$. This is because if we let $\alpha \in \bigwedge\nolimits^{k+2} V^*$ and assume there
would be a second operator $\widetilde{\Lambda}$ admitting the same adjunction property, such
that $\Lambda(\alpha) \neq \widetilde{\Lambda}(\alpha)$, it would be for all
$\beta \in \bigwedge\nolimits^k V^*$
\begin{align*}
g(\Lambda(\alpha) - \widetilde{\Lambda}(\alpha), \beta) = g\left(\Lambda (\alpha), \beta\right) -
g(\widetilde{\Lambda}(\alpha),\beta) = g(\alpha, L(\beta)) - g(\alpha, L(\beta)) = 0.
\end{align*}
Thus, the non-degeneracy of $g$ can be used to obtain $\Lambda (\alpha)-\widetilde{\Lambda}(\alpha) = 0$,
which is a contradiction to our assumption.
\end{rem}
To properly define the next operator now, it is necessary to discuss the existence of a volume
element in $\bigwedge^n V^*$ first.
\begin{rem}
\label{loc-theory:volume-form-locally}
We already know that $V$ has a natural orientation, which we are going to call $\sigma$ for now.
As mentioned in \cite[p.\,83 below Theorem 4-6]{Spivak1965}, there exists a unique volume form
$\vol \in \bigwedge^nV^*$, such that $\vol(v_1, \dots, v_n)= 1$ whenever $(v_1, \dots, v_n)$ is
an orthonormal basis of $V$ that is positively orientated with respect to $\sigma$. This volume form can
be given as $\vol = v^1 \wedge \dots \wedge v^n$.
Now, we want to show that $\vol = \frac{1}{m!} \omega^m$. %with $\omega \in \bigwedge^2 V^*$
%being the fundamental form associated to $(V,g,I)$.
Therefore, let $(w_1,\dots,w_m)$ be a complex orthonormal basis of $(V,I)$ with respect to the
hermitian form $h$. A simple calculation proves that the induced real basis $(w_1,I(w_1),\dots,w_m,I(w_m))$
is orthonormal with respect to the inner product $g$ on $V$. We can calculate for all $w_j$ and $w_k$
\begin{align*}
\omega(w_j,I(w_k)) = g(I(w_j),I(w_k)) = g(w_j,w_k) = \delta_{jk}
\end{align*}
and also
\begin{align*}
\omega(w_j,w_k) = g(I(w_j),w_k) = 0.
\end{align*}
Let $I(w_j)^*$ denote the dual basis vector of $I(w_j)$. With the above calculation, we conclude
that $\omega = \sum_{j=1}^{m} w^j \wedge I(w_j)^*$ and therefore it is
\begin{align*}
\omega^m = \Big(\sum_{j=1}^{m} w^j \wedge I(w_j)^*\Big)^m= m! \cdot \big(w^1 \wedge I(w_1)^*\big)
\wedge \dots \wedge \big(w^m\wedge I(w_m)^*\big)
= m! \cdot \vol.
\end{align*}
\end{rem}
This concludes our collection of the necessary elements to define the Hodge star operator.
\begin{defn}[Hodge star operator]
The \emph{Hodge star operator} on $\bigwedge^k V^*$ is defined as a linear mapping
$\hodgestar: \bigwedge^k V^* \rightarrow \bigwedge^{n-k}V^*$, such that for all
$\alpha,\beta \in\bigwedge^kV^*$, it is
\begin{align}
\label{loc-theory:eq:hodge-star}
\alpha \wedge \hodgestar\beta = g(\alpha,\beta) \cdot \vol.
\end{align}
The $(n-k)$ form $\hodgestar \beta$ is called the \emph{Hodge dual} of $\beta$.
\end{defn}
Note that \Cref{loc-theory:eq:hodge-star} uniquely defines the Hodge dual because for $r,s \in
\mathbb{N}$ with $r + s = n$, the exterior product defines a non-degenerate pairing
\begin{align*}
\bigwedge\nolimits^r V^* \times \bigwedge\nolimits^s V^* \rightarrow \bigwedge\nolimits^{n} V^*.
\end{align*}
To show that such an operator exists, we choose an orthonormal basis $(v_1,\dots,v_n)$ of
$V$ that is positively oriented with respect to the natural orientation of $V$. With
\Cref{loc-theory:rem:preserving-orthonormality}, we already know that the induced basis $(v^J)_{J}$ of
$\bigwedge^kV^*$ is orthonormal as well. For an arbitrary permutation $\tau \in S_n$, we set the Hodge star
operator to map as follows:
\begin{align}
\label{loc-theory:eq:explicit-hodge-star}
\hodgestar\big(v^{\tau(1)}\wedge\dots\wedge v^{\tau(k)}\big) = \sign(\tau) \cdot \;v^{\tau(k+1)}
\wedge\dots\wedge v^{\tau(n)}.
\end{align}
With this definition, it is
\begin{align*}
v^{\tau(1)}\wedge\dots\wedge v^{\tau(k)} \wedge \hodgestar\big(v^{\tau(1)}\wedge\dots\wedge v^{\tau(k)}\big) &= \sign(\tau) \cdot v^{\tau(1)} \wedge\dots\wedge v^{\tau(n)}\\
&= \sign(\tau)^2\cdot v^1\wedge \dots \wedge v^n \\&= 1 \cdot \vol.
\end{align*}
At the same time, it is for all $v^{j_1}\wedge\dots\wedge v^{j_k} \neq \pm v^{\tau(1)}\wedge\dots\wedge v^{\tau(k)}$
\begin{align*}
v^{j_1}\wedge\dots\wedge v^{j_k} \wedge \hodgestar\big(v^{\tau(1)}\wedge\dots\wedge
v^{\tau(k)}\big) = 0.
\end{align*} This is because if we assume without loss of generality that $j_1\neq\dots\neq j_k$,
then there exists at least one $s\in\mathbb{N}$ with $(k+1) \leq s \leq n$, such that $\tau(s) \in \{j_1,\dots,j_k\}$.
This shows that the above mapping indeed explicitly defines the Hodge star. See also \cite[p.\,56f]{Schnell2012},
where we have later discovered a very similar calculation.
\begin{prop}[Properties of the Hodge star {\cite[Proposition 1.2.20]{Huybrechts2004}}]\;\\
\label{loc-theory:lm:property-hodge-star}
Among others, the Hodge star operator has the following properties:
\begin{enumerate}
\item The Hodge star operator on $\bigwedge^kV^*$ is an isometric isomorphism, i.e. it is
bijective and for all $\alpha, \beta \in \bigwedge^kV^*$, it is $g(\alpha,\beta) = g(\hodgestar\alpha,\hodgestar\beta)$.
\item For all $\alpha \in \bigwedge^kV^*$, it is $\hodgestar^2\alpha = (-1)^k \alpha$. In particular, it is
$\hodgestar^{-1} = (-1)^k \hodgestar$.
\label{loc-theory:lm:property-hodge-star-one}
\end{enumerate}
\end{prop}
\begin{proof}
With the explicit definition in \Cref{loc-theory:eq:explicit-hodge-star}, it is quite easy to determine
that the Hodge star operator maps any orthonormal basis to an orthonormal basis, and this proves
property (1).
For the proof of property (2), we use a local version of the calculation in \cite[Lemma 5.5]{Voisin2002}.
We can apply property (1) and calculate for all $\alpha,\beta \in\bigwedge^kV^*$
\begin{align*}
\alpha\wedge\hodgestar\beta = g(\alpha, \beta) \cdot \vol = g(\hodgestar\alpha,\hodgestar\beta)
\cdot \vol = g(\hodgestar\beta,\hodgestar\alpha) \cdot \vol = \hodgestar\beta \wedge \hodgestar^2
\alpha.
\end{align*}
With $n-(n-k) = k$, we know $\hodgestar^2 \alpha$ is a $k$-form again. As $\hodgestar\beta$ is a
$(n-k)$ form, we obtain
\begin{align*}
\hodgestar\beta \wedge \hodgestar^2\alpha &= (-1)^{k(n-k)} \hodgestar^2 \alpha \wedge \hodgestar
\beta \\&= (-1)^{k(2m-k)} \hodgestar^2\alpha\wedge\hodgestar\beta\\ &= (-1)^k \hodgestar^2 \alpha
\wedge \hodgestar\beta.
\end{align*}
Given that this holds for all $\beta \in \bigwedge^kV^*$, the second property is already proven.
\end{proof}
Additionally, there is an interesting relation between the Hodge star operator and the Lefschetz and
dual Lefschetz operators, which is going to be useful later.
\begin{lm}[{\cite[Lemma 1.2.23]{Huybrechts2004}}]
\label{loc-theory:lm:formula-for-the-dual-lefschetz-operator}
For $\alpha \in \bigwedge^{k+2} V^*$ the image under the dual Lefschetz operator $\Lambda$ can be
explicitly calculated as
\begin{align*}
\Lambda (\alpha) = \big(\hodgestar^{-1} \circ \,L \circ \hodgestar\big)(\alpha) = \big( (-1)^k
\hodgestar \circ \,L \circ \hodgestar \big)(\alpha).
\end{align*}
\end{lm}
\begin{proof} We expand the proof of the given lemma in \cite{Huybrechts2004}.
Let $\beta \in \bigwedge^kV^*$. Using the definition of the Hodge star and the definition of the
Lefschetz operator, we can calculate
\begin{align*}
g(\alpha, L\beta)\cdot \vol = g(L\beta,\alpha) \cdot \vol = L\beta \wedge \hodgestar \alpha =
\omega \wedge \beta \wedge \hodgestar \alpha.
\end{align*}
As $\omega$ is a $2$-form and the wedge product is associative, we have
\begin{align*}
g(\alpha, L\beta)\cdot \vol = (-1)^{2k} \beta \wedge \omega \wedge \hodgestar\alpha = \beta
\wedge (\omega \wedge \hodgestar\alpha).
\end{align*}
Applying the definition of the Lefschetz operator again and using the definition of the Hodge star
operator yields
\begin{align*}
g(\alpha, L\beta)\cdot \vol &=\beta \wedge L (\hodgestar \alpha) \\
&= \beta \wedge (\hodgestar \hodgestar^{-1}) L (\hodgestar \alpha) \\
&= \beta \wedge \hodgestar (\hodgestar^{-1} (L (\hodgestar \alpha))) \\
&= g\big(\beta, (\hodgestar^{-1}( L (\hodgestar \alpha)))\big) \cdot \vol \\
&=g(\hodgestar^{-1} (L (\hodgestar \alpha)), \beta)\cdot\vol.
\end{align*}
Thus, we have shown the equality
$g(\Lambda \alpha, \beta) = g(\alpha,L\beta) = g(\hodgestar^{-1}(L(\hodgestar \alpha)), \beta)$.
Since this holds for all $k$-forms $\beta$, the non-degeneracy of $g$ proves the first equality
of the statement. The second equality then follows directly with
\creflmpart{loc-theory:lm:property-hodge-star}{loc-theory:lm:property-hodge-star-one}. % see custom command in main_thesis
\end{proof}
Similar to the $\mathbb{C}$-linear extension of the Lefschetz operator, we will also need the
$\mathbb{C}$-linear extension of the Hodge star operator.
\begin{defn}
The $\mathbb{C}$-linear extension of the Hodge star
$\hodgestar_\mathbb{C} :\bigwedge^kV_\mathbb{C}^* \rightarrow \bigwedge^{n-k} V_\mathbb{C}^*$
is defined such that for all $\alpha,\beta \in \bigwedge^kV^*_\mathbb{C}$, we have
\begin{align*}
\alpha \wedge \hodgestar_\mathbb{C}\overline{\beta} = h(\alpha, \beta) \cdot \vol.
\end{align*}
Note that this expression is meaningful because
$\vol = \frac{1}{m!} \omega^m \in \bigwedge^nV^*\cap \bigwedge^{m,m}V^*$.
Since the hermitian form $h$ on $\bigwedge^kV_\mathbb{C}^*$ was previously defined as the
sesquilinear extension of the inner product $g$ on $\bigwedge^kV^*$, it is immediate that this is indeed
the $\mathbb{C}$-linear extension of the real Hodge star operator.
\end{defn}
There is also the $\mathbb{C}$-linear extension
$\Lambda_\mathbb{C}: \bigwedge^{k+2}V_\mathbb{C}^*\rightarrow \bigwedge^{k} V_\mathbb{C}^*$
of the dual Lefschetz operator. Using the explicit formula in
\Cref{loc-theory:lm:formula-for-the-dual-lefschetz-operator}, this extension is given as
\begin{align*}
\Lambda_\mathbb{C} = (-1)^k \hodgestar_\mathbb{C} \circ L_\mathbb{C} \circ \hodgestar_\mathbb{C}.
\end{align*}
We can use the same calculation as in the proof of \Cref{loc-theory:lm:formula-for-the-dual-lefschetz-operator}
to show that this is indeed the adjoint operator to $L$ with respect to the hermitian form $h$.
\begin{nota}
It is common practice to abuse the notation, denoting the complex extensions
$L_\mathbb{C},\hodgestar_\mathbb{C}$ and $\Lambda_\mathbb{C}$ as $L, \hodgestar$
and $\Lambda$, respectively.
\end{nota}

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\section{Introduction}
\stoptocentries % Personal macro: Following sections shouldn't appear in toc
\subsection*{Contents of the thesis}\;
The Hodge Decomposition theorem for compact Kähler manifolds is a fundamental theorem of the Hodge
Theory. It provides a decomposition of the de Rahm cohomology groups into suitable Dolbeault
cohomology groups, thus yielding a connection between the topology and the complex structure of a
compact Kähler manifold.
\begin{thm}[Hodge Decomposition]
For a compact Kähler manifold $X$, there is a direct sum decomposition
\begin{align*}
H^k_{dR}(X,\mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}_{\opartial}(X,\mathbb{C}).
\end{align*}
\end{thm}
The primary objective of this thesis will be the elaboration of the proof of this fundamental
theorem. In order to achieve this, we will have to introduce the needed theory first. We are going
to start by presenting the consequences of the existence of an almost complex structure and a
compatible euclidean inner product on a real vector space. For this purpose, we will mainly use the
tools of Linear Algebra.
With this, we will be able to define the local versions of the \emph{Hodge star operator}
$\hodgestar$ and the \emph{Lefschetz} and dual \emph{Lefschetz operators} $L$ and $\Lambda$.
Afterwards, our focus is going to shift to complex manifolds and their different tangent bundles.
Although it is assumed that the reader is already familiar with the definition and basic properties
of complex manifolds, we will begin with the definition and also the elaboration of the properties
of hermitian manifolds, which are the complex counterparts of Riemannian manifolds.
After we have used our local findings for the operators mentioned above to define similarly named
global operators for hermitian manifolds, we will also introduce an \emph{$L^2$-metric} that will be
used to generalize the idea of adjoint operators to \emph{formal adjoint opertors}. We will be
particularly interested in the formal adjoint operators of the exterior derivative $d$ and the
Dolbeault operators $\partial$ and $\opartial$, which will be noted as $d^*,\partial^*$ and
$\opartial^*$.
Those formal adjoint operators will particularly interest us because they appear in the \emph{Kähler
identities}. These identities relate the Dolbeault operators and their formal adjoints to each other
using the dual Lefschetz operator.
\begin{thm}
On a compact Kähler manifold, we have the identities
\begin{align*}
[\Lambda,\opartial] = -i\partial^*, \quad [\Lambda,\partial] = i\opartial^*,
\end{align*}
with the Lie bracket being defined as the commutator.
\end{thm}
Next, we are going to introduce the theory of \emph{harmonic differential forms}. In order to do so,
we will define the \emph{Laplacians} $\Delta_d, \Delta_\partial$ and $\Delta_\opartial$ and work out
their properties. We will then use the \emph{Kähler identities} to prove the next important theorem.
\begin{thm}
For the Laplacians $\Delta_d, \Delta_\partial$ and $\Delta_\opartial$ on a compact Kähler manifold,
we have the following relation
\begin{align*}
\frac{1}{2}\Delta_d = \Delta_\partial = \Delta_\opartial.
\end{align*}
\end{thm}
Since harmonic and $\Delta_\opartial$-harmonic forms will be defined as forms annihilated by
$\Delta_d$ and $\Delta_\opartial$, respectively, this theorem shows that those two notions are
equivalent for Kähler manifolds. Furthermore, we will use this theorem to establish the following
corollary, which will be crucial for proving the \emph{Hodge Decomposition} theorem.
\begin{cor}
For the compact Kähler manifold $X$, the complex harmonic differential $k$-forms $\mathcal{H}^k(X)$
decompose as
\begin{align*}
\mathcal{H}^k(X) = \bigoplus_{p+q=k}\mathcal{H}^{p,q}(X),
\end{align*}
with $\mathcal{H}^{p,q}(X)$ being the harmonic differential forms of type $(p,q)$.
\end{cor}
The final statements needed for our proof of the \emph{Hodge Decomposition} theorem will be the
\emph{Hodge Isomorphism theorems}, which enable us to apply the findings of the harmonic forms
theory to the de Rahm and Dolbeault cohomologies by providing two isomorphisms.
\begin{thm}[Hodge Isomorphism theorem \MakeUppercase{\romannumeral 1}]
The natural mapping
\begin{align*}
\mathcal{H}^k(X) \rightarrow H^k_{dR}(X,\mathbb{C}), \;\enspace
\alpha \mapsto [\alpha]
\end{align*}
is an isomorphism. In particular, any class of closed forms in $H^k_{dR}(X,\mathbb{C})$ has a
unique harmonic
representative.
\end{thm}
\begin{thm}[Hodge Isomorphism theorem \MakeUppercase{\romannumeral 2}]
The natural mapping
\begin{align*}
\mathcal{H}^{p,q}(X) \rightarrow H^{p,q}_\opartial(X,\mathbb{C}),\; \enspace \alpha \mapsto
[\alpha]
\end{align*}
is an isomorphism. In particular, any class of $\opartial$-closed forms in
$H^{p,q}_\opartial(X,\mathbb{C})$
has a unique harmonic representative.
\end{thm}
After these two isomorphism theorems are proven, we already have the \emph{Hodge Decomposition}
given as an isomorphism and in order to get the above \emph{Hodge Decomposition} theorem, we only
need to prove the independence of the Kähler metric.
To conclude this thesis, we will then provide an application of the \emph{Hodge Decomposition}. We
are going to show that the \emph{Hopf surfaces}, which are compact 2-dimensional complex manifolds,
can not be equipped with a Kähler metric.
\subsection*{Remarks on the implementation}\;
For the \emph{Hodge Decomposition}, several different proofs are already known. Therefore, the idea
of this thesis is the collection and the coherent presentation of one of these possible proofs from
the perspective of an undergraduate student who is already familiar with some of the basic concepts
of complex and algebraic geometry.
In this context, this thesis broadly follows the proof presented in \emph{Claire Voisin's} book
\emph{Hodge Theory and Complex Algebraic Geometry I}. However, other popular sources have also
influenced this thesis. Therefore, we will reference similar or equal statements in this literature
whenever possible. This is done to allow for the possibility of verification and to encourage the
reader to engage more deeply with the content.
Also, since multiple different notation conventions exist in complex geometry, we will try to stick
to the notation suggested by \emph{Voisin} in her book. However, we will also provide the reader
with explanations for the used notation throughout the thesis so that even the unfamiliar reader
will be able to follow.
\subsection*{Conventions}\;
Throughout the thesis, we are always going to limit our discussion to differentiable manifolds
without border. In order to have a Riemannian metric on every differentiable manifold, we are also
only going to allow paracompact manifolds.
Additionally, we are going to adhere to the following meaning of the used symbols.
\begin{table}[ht]
\centering
\begin{tabular}{cp{0.6\textwidth}}
$\mathbb{N}$ & Natural numbers, including 0\\
$\subset$ & Not necessarily proper subset \\
$\subsetneq$ & Proper subset\\
$\into$ & Injection or monomorphism\\
$\onto$ & Surjection or epimorphism\\
\end{tabular}
\end{table}
Also, the uppercase letters $J$ and $K$ will usually denote multi-indices except for one instance,
when $J$ denotes an almost complex structure. Additionally, if there are uppercase letters in the
index, they also denote multi-indices. For a basis vector $v_j$, we will write $v^j$ for the dual
basis vector.
\subsection*{Acknowledgments}\;
I want to thank Andreas Demleitner for his advice and for his constant willingness and patience in
answering my questions. I want to thank Alina Vogler and especially Jonathan Stahlmann for their
advice regarding the rules of the English language. I am also very thankful for the opportunity to
write this thesis under the supervision of Stefan Kebekus, whom I want to thank for his help and
guidance.
\vspace*{1.3cm}
\begin{flushright}
\emph{Daniel Rath}\\
\emph{Freiburg im Breisgau}\\
\emph{August 30, 2023}
\end{flushright}
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