hodge/BAThesis/kaehler-manifolds.tex

\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}