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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf K\"ahler manifolds and holonomy \\[15mm] \small lecture 3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Tel-Aviv University \\[2mm] December 21, 2010, } \end{center} \newpage {\bf \blue K\"ahler manifolds} {\bf\green DEFINITION:} An Riemannian metric $g$ on an almost complex manifiold $M$ is called {\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$. In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$, hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric. {\bf\green DEFINITION:} The differential form $\omega\in \Lambda^{1,1}(M)$ is called {\bf \blue the Hermitian form} of $(M,I,g)$. \remark It is $U(1)$-invariant, hence {\bf \purple of Hodge type (1,1)}. {\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$ is called {\bf \blue K\"ahler} if $d\omega=0$. The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ is called {\bf \blue the K\"ahler class} of $M$, and $\omega$ {\bf \blue the K\"ahler form}. \newpage {\bf \blue Levi-Civita connection and K\"ahler geometry} \definition Let $(M, g)$ be a Riemannian manifold. A connection $\nabla$ is called {\bf \blue orthogonal} if $\nabla(g) =0$. It is called {\bf \blue Levi-Civita} if it is torsion-free. \theorem (``the main theorem of differential geometry'')\\ {\bf \red For any Riemannian manifold, the Levi-Civita connection exists,\\ and it is unique}. {\bf \green THEOREM:} Let $(M,I,g)$ be an almost complex Hermitian manifold. {\bf \purple Then the following conditions are equivalent.} (i) {\bf \red $(M,I,g)$ is K\"ahler} (ii) One has {\red $\nabla(I)=0$,} where $\nabla$ is the Levi-Civita connection. \newpage {\bf \blue Holonomy group} \definition (Cartan, 1923) Let $(B,\nabla)$ be a vector bundle with connection over $M$. For each loop $\gamma$ based in $x\in M$, let $V_{\gamma, \nabla}:\; B\restrict x \arrow B\restrict x$ be the corresponding parallel transport along the connection. The {\bf \blue holonomy group} of $(B,\nabla)$ is a group generated by $V_{\gamma, \nabla}$, for all loops $\gamma$. If one takes all contractible loops instead, $V_{\gamma, \nabla}$ generates {\bf \blue the local holonomy}, or {\bf \blue the restricted holonomy} group. \remark A bundle is {\bf \blue flat} (has vanishing curvature) {\bf\purple if and only if its restricted holonomy vanishes.} \remark If $\nabla(\phi)=0$ for some tensor $\phi\in B^{\otimes i}\otimes (B^*)^{\otimes j}$, {\bf \red the holonomy group preserves $\phi$.} \definition {\bf \blue Holonomy of a Riemannian manifold} is holonomy of its Levi-Civita connection. \example Holonomy of a Riemannian manifold lies in $O(T_x M, g\restrict x)=O(n)$. \example Holonomy of a K\"ahler manifold lies in $U(T_x M, g\restrict x, I \restrict x)=U(n)$. \remark The holonomy group {\bf \red does not depend on the choice of a point $x\in M$.} \newpage {\bf \blue The Berger's list} \theorem (de Rham) A complete, simply connected Riemannian manifold with non-irreducible holonomy {\bf \red splits as a Riemannian product.} \theorem (Berger's theorem, 1955) Let $G$ be an irreducible holonomy group of a Riemannian manifold which is not locally symmetric. {\bf \red Then $G$ belongs to the Berger's list:} { \begin{tabular}{|l|l|} \hline \multicolumn{2}{|c|}{\bf \color[rgb]{0,0,0.6}Berger's list}\\[1mm] \hline \it Holonomy & \it Geometry\\[1mm] \hline $SO(n)$ acting on $\R^n$ & Riemannian manifolds\\[1mm] \hline $U(n)$ acting on $\R^{2n}$ & K\"ahler manifolds\\[1mm] \hline $SU(n)$ acting on $\R^{2n}$, $n>2$ & Calabi-Yau manifolds\\[1mm] \hline $Sp(n)$ acting on $\R^{4n}$ & hyperk\"ahler manifolds\\[1mm] \hline $Sp(n)\times Sp(1)/\{\pm 1\}$ & quaternionic-K\"ahler\\[1mm] acting on $\R^{4n}$, $n>1$ & manifolds\\[1mm] \hline $G_2$ acting on $\R^7$ & $G_2$-manifolds \\[1mm] \hline $Spin(7)$ acting on $\R^8$ & $Spin(7)$-manifolds\\[1mm] \hline \end{tabular} } \newpage {\bf \blue Chern connection} \definition Let $B$ be a holomorphic vector bundle, and $\bar\6:\; B_{C^\infty}\arrow B_{C^\infty}\otimes \Lambda^{0,1}(M)$ an operator mapping $b \otimes f$ to $b\otimes \bar\6 f$, where $b\in B$ is a holomorphic section, and $f$ a smooth function. This operator is called {\bf \blue a holomorphic structure operator} on $B$. {\bf \red It is correctly defined, because $\bar\6$ is $\calo_M$-linear.} \remark {\bf \purple A section $b\in B$ is holomorphic iff $\bar\6(b)=0$} \definition let $(B, \nabla)$ be a smooth bundle with connection and a holomorphic structure $\bar\6:\; B \arrow \Lambda^{0,1}(M)\otimes B$. Consider the Hodge decomposition of $\nabla$, $\nabla= \nabla^{0,1} + \nabla^{1,0}$. We say that $\nabla$ is {\bf \blue compatible with the holomorphic structure} if $\nabla^{0,1}=\bar\6$. \definition {\bf \blue An Hermitian holomorphic vector bundle} is a smooth complex vector bundle equipped with a Hermitian metric and a holomorphic structure. \definition {\bf\blue A Chern connection} on a holomorphic Hermitian vector bundle is a connection compatible with the holomorphic structure and preserving the metric. \theorem On any holomorphic Hermitian vector bundle, {\bf \red the Chern connection exists, and is unique.} \newpage {\bf \blue Calabi-Yau manifolds} \definition\\ {\bf\blue A Calabi-Yau manifold} is a compact Kaehler manifold with $c_1(M,\Z)=0$. \newcommand{\Ric}{\operatorname{Ric}} \definition Let $(M,I, \omega)$ be a Kaehler $n$-manifold, and $K(M):= \Lambda^{n,0}(M)$ its {\bf \blue canonical bundle.} We consider $K(M)$ as a colomorphic line bundle, $K(M)= \Omega^n M$. The natural Hermitian metric on $K(M)$ is written as \[ (\alpha, \alpha') \arrow \frac{\alpha\wedge \bar \alpha'}{\omega^n}.\] Denote by $\Theta_K$ the curvature of the Chern connection on $K(M)$. The {\bf\blue Ricci curvature} $\Ric$ of $M$ is symmetric 2-form $\Ric(x,y)= \Theta_K(x, Iy)$. \definition A K\"ahler manifold is called {\bf \blue Ricci-flat} if its Ricci curvature vanishes. \theorem (Calabi-Yau) \\ Let $(M, I, g)$ be Calabi-Yau manifold. {\bf \red Then there exists a unique Ricci-flat Kaehler metric in any given Kaehler class.} \remark Converse is also true: {\bf \purple any Ricci-flat K\"ahler manifold has a finite covering which is Calabi-Yau.} This is due to Bogomolov. \newpage {\bf \blue Bochner's vanishing} \theorem (Bochner vanishing theorem) On a compact Ricci-flat Calabi-Yau manifold, {\bf \red any holomorphic $p$-form $\eta$ is parallel} with respect to the Levi-Civita connection: $\nabla(\eta)=0$. \remark Its proof uses spinors (see below). \definition A {\bf \blue holomorphic symplectic manifold} is a manifold admitting a non-degenerate, holomorphic symplectic form. \remark A holomorphic symplectic manifold is Calabi-Yau. The top exterior power of a holomorphic symplectic form {\bf \purple is a non-degenerate section of canonical bundle.} \remark Due to Bochner's vanishing, {\bf \red holonomy of Ricci-flat Calabi-Yau manifold lies in $SU(n)$}, and {\bf \red holonomy of Ricci-flat holomorphically symplectic manifold lies in $Sp(n)$.} \definition A holomorphically symplectic Ricci-flat Kaehler manifold is called {\bf \blue hyperk\"ahler}. \remark Since $Sp(n)=SU({\Bbb H}, n)$, a {\bf \purple hyperk\"ahler manifold admits quaternionic action in its tangent bundle.} \newpage {\bf \blue Bogomolov's decomposition theorem} \theorem {\bf \blue (Cheeger-Gromoll)} Let $M$ be a compact Ricci-flat Riemannian manifold with $\pi_1(M)$ infinite. {\bf \red Then a universal covering of $M$ is a product of $\R$ and a Ricci-flat manifold.} \corollary A fundamental group of a compact Ricci-flat Riemannian manifold is {\bf \blue ``virtually polycyclic'':} {\bf \purple it is projected to a free abelian subgroup with finite kernel.} \remark This is equivalent to any compact Ricci-flat manifold having a finite covering which has free abelian fundamental group. \remark This statement contains the Bieberbach's solution of Hilbert's 18-th problem on classification of crystallographic groups. \theorem {\bf \blue (Bogomolov's decomposition)} Let $M$ be a compact, Ricci-flat Kaehler manifold. {\bf \red Then there exists a finite covering $\tilde M$ of $M$ which is a product of Kaehler manifolds of the following form:} \[ \tilde M = T \times M_1 \times ... \times M_i \times K_1 \times ... \times K_j, \] with all $M_i$, $K_i$ simply connected, $T$ a torus, and $\Hol(M_l) = Sp(n_l)$, $\Hol(K_l)=SU(m_l)$ \newpage {\bf \blue Harmonic forms} Let $V$ be a vector space. {\bf \blue A metric $g$ on $V$ induces a natural metric on each of its tensor spaces:} $g(x_1\otimes x_2 \otimes ... \otimes x_k, x_1'\otimes x_2' \otimes ... \otimes x_k') = g(x_1, x'_1)g(x_2, x'_2) ... g(x_k, x'_k)$. {\bf \purple This gives a natural positive definite scalar product on differential forms over a Riemannian manifold $(M,g)$:} $g(\alpha, \beta) := \int_M g(\alpha, \beta) \Vol_M$. The topology induced by this metric is called {\bf \blue $L^2$-topology.} \definition Let $d$ be the de Rham differential and $d^*$ denote the adjoint operator. The {\bf \blue Laplace operator} is defined as $\Delta:= dd^*+d^*d$. A form is called {\bf \blue harmonic} if it lies in $\ker \Delta$. \theorem {\bf \red The image of $\Delta$ is closed} in $L^2$-topology on differential forms. \remark This is a very difficult theorem! \remark On a compact manifold, the form $\eta$ is {\bf \purple harmonic iff $d\eta = d^*\eta =0$.} Indeed, $(\Delta x, x)= (dx, dx) + (d^*x, d^*x)$. \corollary This defines a map ${\cal H}^i(M) \stackrel \tau \arrow H^i(M)$ from harmonic forms to cohomology. \newpage {\bf \blue Hodge theory} \theorem (Hodge theory for Riemannian manifolds)\\ {\bf \purple On a compact Riemannian manifold,} the map ${\cal H}^i(M) \stackrel \tau \arrow H^i(M)$ to cohomology {\bf \red is an isomorphism.} {\bf \green Proof. Step 1:} $\ker d\; \bot\; \im d^*$ and $\im d \;\bot\; \ker d^*$. Therefore, {\bf \purple a harmonic form is orthogonal to $\im d$.} This implies that {\bf \red $\tau$ is injective}. {\bf \green Step 2:} {\bf \purple $\eta \bot \im \Delta$ if and only if $\eta$ is harmonic.} Indeed, $(\eta, \Delta x)= (\Delta x, x)$. {\bf \green Step 3:} Since $\im \Delta$ is closed, {\bf \red every closed form $\eta$ is decomposed as $\eta = \eta_h + \eta'$,} where $\eta_h$ is harmonic, and $\eta'= \Delta\alpha$. {\bf \green Step 4:} When $\eta$ is closed, $\eta'$ is also closed. Then $0 =(d\eta, d \alpha) = (\eta, d^* d \alpha)= (\Delta\alpha, d^* d \alpha)= (dd^* \alpha, d^* d \alpha) + (d^* d\alpha, d^* d \alpha)$. The term $(dd^* \alpha, d^* d \alpha)$ vanishes, because $d^2=0$, hence $(d^* d\alpha, d^* d \alpha)=0$. This gives $d^* d\alpha=0$, and $(d^* d\alpha, \alpha) =(d\alpha, d\alpha)=0$. We have shown that {\bf \purple for any closed $\eta$ decomposing as $\eta = \eta_h + \eta'$, with $\eta'= \Delta\alpha$, $\alpha$ is closed} {\bf \green Step 5:} This gives $\eta'= d d^*\alpha$, hence {\bf \red $\eta$ is a sum of an exact form and a harmonic form.} \endproof {\small \remark This gives a way of obtaining the Poincare duality via PDE.} \newpage {\bf \blue Hodge decomposition on cohomology} \theorem {\it (this theorem will be proven in the next lecture)}\\ On a compact Kaehler manifold $M$, {\bf \red the Hodge decomposition is compatible with the Laplace operator. } This gives a decomposition of cohomology, $H^i(M) = \bigoplus_{p+q=i}H^{p,q}(M)$, with $\overline{H^{p,q}(M)} = H^{q,p}(M)$. \corollary {\bf \red $H^p(M)$ is even-dimensional for odd $p$.} {\bf \blue The Hodge diamond:} {\small \[ \begin{array}{ccccccccc} &&&&H^{n,n}&&&& \\[3mm] &&&H^{n,n-1}&&H^{n-1,n}&&& \\[3mm] \ \ \ \ &&H^{n,n-2}&&H^{n-1,n-1}&&H^{n-2,n}&& \ \ \ \ \\[3mm] \vdots& &\vdots &&\vdots &&\vdots&&\vdots\\[3mm] &&H^{2,0}&&H^{1,1}&&H^{0,2}&& \\[3mm] &&&H^{1,0}&&H^{0,1}&& &\\[3mm] &&&&H^{0,0}&&&& \\[3mm] \end{array} \]} \remark {\bf \purple $H^{p,0}(M)$ is the space of holomorphic $p$-forms.} Indeed, $dd^* + d^* d = 2 (\bar\6\bar\6^* + \bar\6^*\bar\6)$ (next lecture), hence {\bf \red a holomorphic form on a compact K\"ahler manifold is closed.} \newpage {\bf \blue Holomorphic Euler characteristic} \definition {\bf\blue A holomorphic Euler characteristic} $\chi(M)$ of a K\"ahler manifold is a sum $\sum(-1)^p\dim H^{p,0}(M)$. \theorem (Riemann-Roch-Hirzebruch) For an $n$-fold, {\bf \red $\chi(M)$ can be expressed as a polynomial expressions of the Chern classes,} $\chi(M)=td_{n}$ where $td_n$ is an $n$-th component of the Todd polynomial, {\small \[ td(M) = 1 + \frac1 {2}c_1 + \frac{1}{12}(c_1^2+c_2) + \frac{1}{24}c_1c_2 + \frac1{720}(-c_1^4 + 4c_1^2c_2 + c_1c_3 + 3c_2^22 - c_4) + ... \]} \vspace{-10mm} \remark The Chern classes are obtained as polynomial expression of the curvature (Gauss-Bonnet). Therefore {\bf \purple $\chi(\tilde M)= p\chi(M)$ for any unramified $p$-fold covering $\tilde M \arrow M$.} \remark Bochner's vanishing and the classical invariants theory imply: 1. When $\Hol(M) = SU(n)$, we have {\bf \red $\dim H^{p,0}(M)= 1$ for $p=1,n$, and 0 otherwise. } In this case, $\chi(M)=2$ for even $n$ and 0 for odd. 2. When $\Hol(M) = Sp(n)$,we have {\bf \red $\dim H^{p,0}(M)= 1$ for even $p$ $0\leq p\leq 2n$, and 0 otherwise.} In this case, $\chi(M)=n+1$. \corollary $\pi_1(M)=0$ if $\Hol(M) = Sp(n)$, or $\Hol(M) = SU(2n)$. If $\Hol(M) = SU(2n+1)$, $\pi_1(M)$ is finite. \newpage {\bf \blue Spinors and Clifford algebras} \definition {\bf \blue A Clifford algebra} of a vector space $V$ with a scalar product $q$ is an algebra generated by $V$ with a relation $xy+yx = q(x,y) 1$. \newcommand{\Mat}{\operatorname{Mat}} \remark A Clifford algebra of a complex vector space with $V=\C^n$ with $q$ non-degenerate {\bf \red is isomorphic to $\Mat\left(\C^{n/2}\right)$ ($n$ even) and $\Mat\left(\C^{\frac{n-1}{2}}\right)\oplus \Mat\left(\C^{\frac{n-1}{2}}\right)$ ($n$ odd).} \definition {\bf \blue The space of spinors} of a complex vector space $V, q$ is a fundamental representation of $Cl(V)$ ($n$ even) and one of two fundamental representations of the components of $\Mat\left(\C^{\frac{n-1}{2}}\right)\oplus \Mat\left(\C^{\frac{n-1}{2}}\right)$ ($n$ odd). \remark A 2-sheeted covering $\Spin(V) \arrow SO(V)$ naturally acts on the spinor space, which is called {\bf\blue the spin representation of $\Spin(V)$.} \definition Let $\Gamma$ be a principal $SO(n)$-bundle of a Riemannian oriented manifold $M$. We say that $M$ is a spin-manifold, {\bf \blue if $\Gamma$ can be reduced to a $Spin(n)$-bundle.} \remark {\bf \purple This happens precisely when the second Stiefel-Whitney class $w_2(M)$ vanishes.} \newpage {\bf \blue Spinor bundles and Dirac operator} \definition {\bf \blue A bundle of spinors} on a spin-manifold $M$ is a vector bundle associated to the principal $Spin(n)$-bundle and a spin representation. \definition Consider the map $TM \otimes \Spin \arrow \Spin$ induced by the Clifford multiplication. One defines {\bf \blue the Dirac operator} $D:\; \Spin \arrow \Spin$ as a composition of $\nabla:\; \Spin \arrow\Lambda^1 M\otimes\Spin = TM \otimes\Spin$ and the multiplication. \definition A {\bf \blue harmonic spinor} is a spinor $\psi$ such that $D(\psi)=0$. \theorem (Bochner's vanishing) A harmonic spinor $\psi$ on a compact manifold with vanishing scalar curvature $Sc= Tr(\Ric)$ {\bf \red satisfies $\nabla\psi=0$.} {\bf \green Proof:} The {\bf \blue coarse Laplacian} $\nabla^* \nabla$ is expressed through the Dirac operator using the {\bf \blue Lichnerowitz formula} $\nabla^* \nabla - D^2 = - \frac 1 4 Sc$. When these two operators are equal, {\bf \red any harmonic spinor $\psi$ lies in $\ker\nabla^* \nabla$, giving $(\psi, \nabla^* \nabla\psi) = (\nabla\psi, \nabla\psi)=0$.} \endproof \newpage {\bf \blue Bochner's vanishing on Kaehler manifolds} \remark {\bf \purple A Kaehler manifold is spin if and only if $c_1(M)$ is even,} or, equivalently, if there exists a square root of a canonical bundle $K^{1/2}$. \remark On a Kaehler manifold of complex dimension $n$, {\bf \red one has a natural isomorphism between the spinor bundle and $\Lambda^{*,0}(M)\otimes K^{1/2}$} (for $n$ even) and $\Lambda^{2*,0}(M)\otimes K^{1/2}$ (for $n$ odd). \remark On a K\"ahler manifold, the Dirac operator corresponds to $\6 + \6^*$. \corollary {\bf \purple On a Ricci-flat K\"ahler manifold, all $\alpha \in \ker (\6 + \6^*)\restrict{\Lambda^{*,0}(M)}$ ara parallel.} \remark $\ker \6 + \6^*= \ker \{\6, \6^*\}$, where $\{\cdot, \cdot\}$ denotes the anticommutator. However, $\{\6, \6^*\}= \{\bar\6, \bar\6^*\}$ as K\"ahler identities imply. Therefore, {\bf \purple on a K\"ahler manifold, harmonic spinors are holomorphic forms}. \theorem {\bf \blue (Bochner's vanishing)} Let $M$ be a Ricci-flat Kaehler manifold, and $\Omega\in \Lambda^{p,0}(M)$ a holomorphic differential form. {\bf \red Then $\nabla \Omega=0$.} \end{document}