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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 2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Tel-Aviv University \\[2mm] December 16, 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 Holomorphic vector bundles} \definition A (smooth) {\bf \blue vector bundle} on a smooth manifold is a locally trivial sheaf of $C^\infty M$-modules. \definition A {\bf \blue holomorphic vector bundle} on a complex manifold is a locally trivial sheaf of $\calo_M$-modules. \remark {\bf \red A section $b$ of a bundle $B$ is often denoted as $b\in B$}. \claim Let $B$ be a holomorphic vector bundle. Consider the sheaf $B_{C^\infty}:=B \otimes_{\calo_M} C^\infty M$. It is clearly locally trivial, hence {\bf \purple $B_{C^\infty}$ is a smooth vector bundle.} \definition $B_{C^\infty}$ is called {\bf \blue a smooth vector bundle underlying $B$}. \newpage {\bf \blue A holomorphic structure operator} \definition Let $d= d^{0,1}+d^{1,0}$ be the Hodge decomposition of the de Rham differential on a complex manifold, $d^{0,1}:\; \Lambda^{p,q}(M) \arrow \Lambda^{p,q+1}(M)$ and $d^{1,0}:\; \Lambda^{p,q}(M) \arrow \Lambda^{p+1,q}(M)$. The operators $d^{0,1}$, $d^{1,0}$ are denoted $\bar\6$ and $\6$ and called {\bf \blue the Dolbeault differentials}. \remark From $d^2=0$, one obtains {\bf \red $\bar\6^2=0$ and $\6^2=0$.} \remark {\bf \purple The operator $\bar\6$ is $\calo_M$-linear.} \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 The kernel of {\bf \purple $\bar\6$ coincides with the set of holomorphic sections} of $B$. \newpage {\bf \blue The $\bar\6$-operator on vector bundles} \definition {\bf\blue A $\bar\6$-operator} on a smooth bundle is a map $V \stackrel {\bar\6}\arrow \Lambda^{0,1}(M)\otimes V$, satisfying $\bar\6(fb) = \bar\6(f)\otimes b + f\bar\6(b)$ for all $f\in C^\infty M, b\in V$. \remark {\bf \purple A $\bar\6$-operator on $B$ can be extended to \[ \bar\6:\; \Lambda^{0,i}(M)\otimes V \arrow \Lambda^{0,i+1}(M)\otimes V,\] } using $\bar\6 (\eta \otimes b) = \bar\6(\eta)\otimes b + (-1)^{\tilde \eta}\eta\wedge\bar\6(b)$, where $b\in V$ and $\eta \in \Lambda^{0,i}(M)$. \remark If $\bar\6$ is a holomorphic structure operator, then $\bar\6^2=0$. \theorem (Atiyah-Bott) Let $\bar\6:\; V \arrow \Lambda^{0,1}(M)\otimes V$ be a $\bar\6$-operator, satisfying $\bar\6^2=0$. {\bf \red Then $B:=\ker \bar\6\subset V$ is a holomorphic vector bundle of the same rank.} \remark This statement is a vector bundle analogue of Newlander-Nirenberg theorem. \definition $\bar\6$-operator $\bar\6:\; V \arrow \Lambda^{0,1}(M)\otimes V$ on a smooth manifold is called {\bf \blue a holomorphic structure operator}, if $\bar\6^2=0$. \newpage {\bf Connections and holomorphic structure operators} \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 a Hodge decomposition of $\nabla$, $\nabla= \nabla^{0,1} + \nabla^{1,0}$, \[ \nabla^{0,1}:\; V \arrow \Lambda^{0,1}(M)\otimes V, \ \ \ \nabla^{1,0}:\; V \arrow \Lambda^{1,0}(M)\otimes V. \] 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 Curvature of a connection} \definition Let $\nabla:\; B \arrow B \otimes \Lambda^1 M$ be a connection on a smooth budnle. Extend it to an operator on $B$-valued forms \[ B \stackrel{\nabla}\arrow \Lambda^{1}(M)\otimes B \stackrel{\nabla}\arrow \Lambda^{2}(M)\otimes B \stackrel{\nabla}\arrow \Lambda^{3}(M)\otimes B \stackrel{\nabla}\arrow ... \] using $\nabla(\eta \otimes b) = d\eta + (-1)^{\tilde \eta} \eta \wedge \nabla b$. The operator $\nabla^2:\; B \arrow B\otimes \Lambda^{2}(M)$ is called {\bf \blue the curvature} of $\nabla$. \remark The algebra of $\End(B)$-valued forms naturally acts on $\Lambda^* M \otimes B$. The curvature satisfies $\nabla^2(fb) = d^2 f b + df \wedge \nabla b - df \wedge \nabla b + f \nabla^2 b= f \nabla^2 b$, hence it is $C^\infty M$-linear. {\bf \purple We consider it as an $\End(B)$-valued 2-form on $M$. } \proposition (Bianchi identity) Using the {\bf \purple graded Jacobi identity}, we obtain $[\nabla, \nabla^2]=[\nabla^2,\nabla]+ [\nabla, \nabla^2]=0$, hence $[\nabla, \nabla^2]=0$. This gives {\bf \blue Bianchi identity:} $\nabla(\Theta_B)=0$. \remark If $B$ is a line bundle, $\End B$ is trivial, and {\bf \red the curvature $\Theta_B$ of $B$ is a closed 2-form.} \definition The cohomology class $c_1(B):=\frac{\1}{2\pi}[\Theta_B]\in H^2(M)$ is called {\bf \blue the real first Chern class of a line bunlde $B$}. {\bf \green An exercise:} Check that $c_1(B)$ is independent from a choice of $\nabla$. \newpage {\bf \blue Curvature of a holomorphic line bundle} \remark When speaking of a {\bf \blue ``curvature of a holomorphic bundle'',} one usually means the curvature of a Chern connection. \remark Let $B$ be a holomorphic Hermitian line bundle, and $b$ its non-degenerate section. Denote by $\eta$ a (1,0)-form which satisfies $\nabla^{1,0} b=\eta\otimes b$. Then $d|b|^2= \Re g(\nabla^{1,0} b, b) = \Re\eta|b|^2$. {\bf \purple This gives $\nabla^{1,0} b= \frac{\6 |b|^2}{|b|^2}b= 2\6\log|b| b.$} \remark Then $\Theta_B(b)= 2\bar\6\6\log|b| b$, {\bf \red that is, $\Theta_B = -2 \6\bar\6\log|b|$.} \corollary If $g' = e^{2f} g$ -- two metrics on a holomorphic line bundle, $\Theta, \Theta'$ their curvatures, {\bf \purple one has $\Theta' - \Theta = -2 \6\bar\6 f$} \claim Let $\eta$ be a closed (1,1)-form in the same cohomology class as $\Theta_{B,h}$. {\bf \red Then $\eta$ is a curvature of a Chern connection} on $B$, for some metric $h'$. {\bf \green Proof:} The difference $\Theta_{B,h}-\eta$ is an exact (1,1)-form, hence {\bf \red belongs to an image of $\6\bar\6$ (``$\6\bar\6$-lemma''): $\Theta_{B,h}-\eta=-2 \6\bar \6 f$.} Then the curvature of a metric $h':=e^{2f}h$ satisfies $\Theta_{B,h} - \Theta_{B,h'} = -2 \6\bar \6 f,$ hence $\eta=\Theta_{B,h'}$. \endproof \remark {\bf \purple Such metric is unique, up to a constant.} \newpage {\bf \blue Calabi-Yau manifolds} \remark Let $B$ be a line bundle on a manifold. Using the long exact sequence of cohomology associated with the exponential sequence \[ 0 \arrow \Z_M \arrow C^\infty M \arrow (C^\infty M)^* \arrow 0, \] {\bf \red we obtain $0 \arrow H^1(M, (C^\infty M)^*) \arrow H^2(M, \Z) \arrow 0$.} \definition Let $B$ be a complex line bundle, and $\xi_B$ its defining element in $H^1(M, (C^\infty M)^*)$. Its image in $H^2(M, \Z)$ is called {\bf\blue the integer first Chern class} of $B$. \remark {\bf \purple A complex line bundle $B$ is (topologically) trivial if and only if $c_1(B)=0$.} \theorem (Gauss-Bonnet) A real Chern class of a vector bundle {\bf \purple is an image of the integer Chern class $c_1(B,\Z)$} under the natural homomorphism $H^2(M, \Z)\arrow H^2(M, \R)$. \definition {\bf\blue A first Chern class} of a complex $n$-manifold is $c_1(\Lambda^{n,0}(M))$. \definition\\ {\bf\blue A Calabi-Yau manifold} is a compact Kaehler manifold with $c_1(M,\Z)=0$. \newpage {\bf \blue Calabi-Yau theorem} \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.} \newpage {\bf \blue Calabi-Yau theorem and Monge-Amp\`ere equation} \remark Let $(M, \omega)$ be a K\"ahler $n$-fold, and $\Omega$ a non-degenerate section of $K(M)$, Then $|\Omega|^2 = \frac{\Omega\wedge \bar \Omega}{\omega^n}$ If $\omega_1$ is a new Kaehler metric on $(M,I)$, $h, h_1$ the associated metrics on $K(M)$, then $\frac {h} {h_1}= \frac {\omega_1^n}{\omega^n}$ \corollary A metric $\omega_1= \omega+\6\bar\6\phi$ {\bf \red is Ricci-flat if and only if $(\omega+\6\bar\6\phi)^n = \omega^n e^f$,} where $-2\6\bar\6 f= \Theta_{K,\omega}$. {\bf\green Proof:} For such $f$, $\phi$, one has $\log\frac {h} {h_1}= - f$. This gives \[ \Theta_{K,\omega_1}= \Theta_{K,\omega}+ \6\bar\6\frac {h} {h_1} = \Theta_{K,\omega}-2\6\bar\6 f = 0. \] \endproof \theorem (Calabi-Yau) Let $(M, \omega)$ be a compact Kaehler $n$-manifold, and $f$ any smooth function. {\bf \red Then there exists a unique up to a constant function $\phi$} such that $(\omega+ dd^c \phi)^n = A e^f \omega^n,$ where $A$ is a positive constant obtained from the formula $\int_M A e^f \omega^n= \int_M \omega^n$. \remark \[ (\omega+ dd^c \phi)^n = A e^f \omega^n, \] is called {\bf\blue the Monge-Ampere equation.} \newpage {\bf \blue Uniqueness of solutions of complex Monge-Ampere equation} \proposition (Calabi) {\bf \red A complex Monge-Ampere equation has at most one solution,} up to a constant. {\bf \green Proof. Step 1:} Let $\omega_1, \omega_2$ be solutions of Monge-Ampere equation. Then $\omega_1^n = \omega_2^n$. By $dd^c$-lemma, one has $\omega_2= \omega_1 + dd^c \psi$. {\bf \purple We need to show $\psi=const$.} {\bf \green Step 2:} This gives \[ 0 = (\omega_1 + dd^c \psi)^n - \omega_1^n= dd^c \psi\wedge \sum_{i=0}^{n-1}\omega_1^i \wedge\omega_2^{n-1-i}. \] {\bf \green Step 3:} Let $P:=\sum_{i=0}^{n-1}\omega_1^i \wedge\omega_2^{n-1-i}$. This is a positive $(n-1, n-1)$-form. {\bf \purple There exists a Hermitian form $\omega_3$ on $M$ such that $\omega_3^{n-1}=P$.} {\bf \green Step 4:} Since $dd^c \psi\wedge P =0$, this gives $\psi dd^c \psi\wedge P=0$. Stokes' formula implies \[ 0 = \int_M \psi \wedge\6 \bar\6\psi\wedge P= - \int_M \6\psi \wedge\bar\6 \psi\wedge P = - \int_M |\6\psi|_3^2\omega_3^n. \] where $|\cdot|_3$ is the metric associated to $\omega_3$. {\bf \red Therefore $\bar\6 \psi=0$.} \endproof \end{document}