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