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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:\ }} \newcommand{\exercise}{{\bf \green EXERCISE:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Hodge theory} \\[15mm] \small Lecture 23: Calabi-Yau theorem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize NRU HSE, Moscow} \\[15mm] {\small Misha Verbitsky, May 16, 2018 } \end{center} \newpage {\bf \blue REMINDER: Holomorphic 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 $\eta \in \Lambda^{0,i}(M)$. \definition {\bf \blue A holomorphic vector bundle} on a complex manifold $(M,I)$ is a vector bundle equipped with a $\bar \6$-operator which satisfies $\bar\6^2=0$. In this case, $\bar\6$ is called {\bf \blue a holomorphic structure operator}. \exercise Consider the Dolbeault differential $\bar\6:\; \Lambda^{p,0}(M)\arrow \Lambda^{p,1}(M)= \Lambda^{p,0}(M)\otimes\Lambda^{0,1}(M)$. {\bf \red Prove that it is a holomorphic structure operator on $\Lambda^{p,0}(M)$.} \definition The corresponding holomorphic vector bundle $(\Lambda^{p,0}(M),\bar\6)$ is called {\bf \blue the bundle of holomorphic $p$-forms}, denoted by $\Omega^p(M)$. \newpage {\bf \blue REMINDER: Chern connection} \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 operator $\bar \6$. \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 REMINDER: 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) Clearly, $[\nabla, \nabla^2]=[\nabla^2,\nabla]+ [\nabla, \nabla^2]=0$, hence $[\nabla, \nabla^2]=0$. This gives {\bf \blue Bianchi identity:} $\nabla(\Theta_B)=0$, where $\Theta$ is considered as a section of $\Lambda^2(M)\otimes \End(B)$, and $\nabla:\; \Lambda^2(M)\otimes \End(B)\arrow\Lambda^3(M)\otimes \End(B)$. the operator defined above \newpage {\bf \blue REMINDER: Curvature of a holomorphic line bundle} \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 Let $\nabla$ be a unitary connection in a line bundle. 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 $\nabla$. \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 holomorphic 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 $\Theta' - \Theta = -2 \6\bar\6 f$} \newpage {\bf \blue $\6\bar\6$-lemma} \theorem {\bf \blue (``$\6\bar\6$-lemma'')}\\ Let $M$ be a compact Kaehler manifold, and $\eta \Lambda^{p,q}(M)$ an exact form. Then $\eta = \6\bar\6\alpha$, for some $\alpha \in \Lambda^{p-1,q-1}(M)$. Its proof uses Hodge theory. \corollary Let $(L,h)$ be a holomorphic line bundle on a compact complex manifold, $\Theta$ its curvature, and $\eta$ a (1,1)-form in the same cohomology class as $[\Theta]$. {\bf \red Then there exists a Hermitian metric $h'$ on $L$ such that its curvature is equal to $\eta$}. {\bf \green Proof:} Let $\Theta'$ be the curvature of the Chern connection associated with $h'$. Then $\Theta' - \Theta = -2\6\bar\6 f$, wgere $f = \log(h'h^{-1})$. Then {\bf \purple $\Theta' - \Theta=\eta- \Theta=-2\6\bar\6 f$ has a solution $f$ by $\6\bar\6$-lemma,} because $\eta- \Theta$ is exact. \endproof \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 $B$, denoted by $c_1(B,\Z)$ or $c_1(B)$. \remark {\bf \purple A complex line bundle $B$ is (topologically) trivial if and only if $c_1(B,\Z)=0$.} \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 Ricci form of a K\"ahler manifold} \theorem {\bf \blue(Bogomolov)} Let $M$ be a compact K\"ahler $n$-manifold with $c_1(M,\Z)=0$. {\bf \red Then the canonical bundle $K_M:=\Omega^n(M)$ is trivial.} \proof Follows from the Calabi-Yau theorem (later today). \endproof In other words, a manifold is Calabi-Yau if and only if its canonical bundle is trivial. \definition Let $(M,\omega)$ be a K\"ahler manifold. The metric on $K_M$ can be written as $|\Omega|^2 = \frac{\Omega\wedge \bar \Omega}{\omega^n}$. The {\bf \blue Ricci form} on $M$ is the curvature of the Chern connection on $K_M$. The manifold $M$ is {\bf \blue Ricci-flat} if its Ricci form vanishes. \remark Since a canonical bundle $K_M$ of a Calabi-Yau manifold is trivial, it admits a metric with trivial connection. Calabi conjectured that {\bf \purple this metric on $K_M$ is induced by a K\"ahler metric $\omega$ on $M$} and proved that such a metric is unique for any cohomology class $[\omega]\in H^{1,1}(M,\R)$. Yau proved that it always exists. \definition A Ricci-flat K\"ahler metric is called {\bf \blue Calabi-Yau metric}. \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 $\frac {h} {h_1}= \frac {\omega_1^n}{\omega^n}$. \remark For two metrics $\omega_1, \omega$ in the same K\"ahler class, one has {\bf \purple $\omega_1-\omega=dd^c\phi$, for some function $\phi$} ($dd^c$-lemma). \corollary A metric $\omega_1= \omega+\6\bar\6\phi$ {\bf \red is Ricci-flat if and only if $(\omega+dd^c\phi)^n = \omega^n e^f$,} where $-2\6\bar\6 f= \Theta_{K,\omega}$ {\bf \purple (such $f$ exists by $\6\bar\6$-lemma).} {\bf\green Proof. Step 1:} For such $f$, $\phi$, one has $\log\frac {h} {h_1}= -\log e^f=- f$. As shown above, the corresponding curvatures are related as $\Theta_{K,\omega_1}-\Theta_{K,\omega}=-2\6\bar\6\log(h/h_1)$. This gives \[ \Theta_{K,\omega_1}= \Theta_{K,\omega} -2\6\bar\6\log(h/h_1) = \Theta_{K,\omega}-2\6\bar\6 f. \] {\bf\green Proof. Step 2:} {\bf \red Therefore, $\omega_1$ is Ricci-flat if and only if $\Theta_{K,\omega}-2\6\bar\6 f$.} \endproof To find a Ricci-flat metric {\bf \purple it remains to solve an equation $(\omega+dd^c\phi)^n = \omega^n e^f$ for a given $f$.} \newpage {\bf \blue The complex Monge-Amp\`ere equation} To find a Ricci-flat metric {\bf \purple it remains to solve an equation $(\omega+dd^c\phi)^n = \omega^n e^f$ for a given $f$.} \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 $(\omega+ \1\6\bar\6 \phi)^n = A e^f \omega^n,$ where $A$ is a positive constant obtained from the formula $\int_M A e^f \omega^n= \int_M \omega^n$. \definition \[ (\omega+ \1\6\bar\6 \phi)^n = A e^f \omega^n, \] 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 construction, one has $\omega_2= \omega_1 + \1\6\bar\6 \psi$. {\bf \purple We need to show $\psi=const$.} {\bf \green Step 2:} $\omega_2= \omega_1 + \1\6\bar\6 \psi$ gives \[ 0 = (\omega_1 + \1\6\bar\6 \psi)^n - \omega_1^n= \1\6\bar\6 \psi\wedge \sum_{i=0}^{n-1}\omega_1^i \wedge\omega_2^{n-1-i}. \] {\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 $\1\6\bar\6 \psi\wedge P =0$, this gives $\psi \6\bar\6 \psi\wedge P=0$. 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 \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 on a complex manifold, 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 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 holomorphic 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 a 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 is based on spinors: $\eta$ gives a harmonic spinor, and {\bf \purple on a Ricci-flat Riemannian spin manifold, any harmonic spinor is parallel.} \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.} \newpage {\bf \blue Hyperk\"ahler manifold} \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 $Sp(n)$} (a group of complex unitary matrices preserving a complex-linear symplectic form). \definition A holomorphically symplectic K\"ahler manifold with holonomy in $Sp(n)$ 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 EXAMPLES.} \example An even-dimensional complex vector space. \example An even-dimensional complex torus. \example {\bf \purple A non-compact example:} $T^* \C P^n$ (Calabi). \remark $T^*\C P^1$ {\bf \blue is a resolution of a singularity $\C^2/{\pm1}$.} \remark Let $M$ be a 2-dimensional complex manifold with holomorphic symplectic form outside of singularities, which are all of form $\C^2/{\pm1}$. Then {\bf \purple its resolution is also holomorphically symplectic.} \example Take a 2-dimensional complex torus $T$, then all the singularities of $T/{\pm1}$ are of this form. Its resolution $\widetilde {T/{\pm1}}$ is called {\bf \green a Kummer surface}. {\bf \red It is holomorphically symplectic}. \remark Take a symmetric square $\Sym^2 T$, with a natural action of $T$, and let $T^{[2]}$ be a blow-up of a singular divisor. {\bf \purple Then $T^{[2]}$ is naturally isomorphic to the Kummer surface $\widetilde {T/{\pm1}}$.} \newpage {\bf \blue K3 surfaces} \definition {\bf \blue A K3-surface} is a deformation of a Kummer surface. {\bf \red ``K3: Kummer, K\"ahler, Kodaira''} (a name is due to A. Weil). \begin{center}\epsfig{file=Broad_Peak8051m.jpg,width=0.5\linewidth} {\it\color{blue} ``Faichan Kangri (K3) is the 12th highest mountain on Earth.''} \end{center} \theorem Any complex compact surface with $c_1(M)=1$ and $H^1(M)=0$ {\bf \purple is isomorphic to K3.} Moreover, {\bf \blue it is hyperk\"ahler.} \newpage {\bf \blue Hilbert schemes} \remark {\bf\blue A complex surface} is a 2-dimensional complex manifold. \definition A {\bf\blue Hilbert scheme} $M^{[n]}$ of a complex surface $M$ is a classifying space of all ideal sheaves $I\subset \calo_M$ for which the quotient $\calo_M/I$ has dimension $n$ over $\C$. \remark A Hilbert scheme {\bf \purple is obtained as a resolution of singularities} of the symmetric power $\Sym^n M$. \theorem (Fujiki, Beauville) {\bf \red A Hilbert scheme of a hyperk\"ahler surface is hyperk\"ahler.} \example {\bf\blue A Hilbert scheme of K3}. \example Let $T$ is a torus. Then it acts on its Hilbert scheme freely and properly by translations. For $n=2$, the quotient $T^{[n]}/T$ is a Kummer K3-surface. For $n>2$, it is called {\bf \blue a generalized Kummer variety}. \remark There are 2 more ``sporadic'' examples of compact hyperk\"ahler manifolds, constructed by K. O'Grady. {\bf \purple All known compact hyperkaehler manifolds are these 2 and the three series:} tori, Hilbert schemes of K3, and generalized Kummer. \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, $dd^* + d^* d = 2 (\bar\6\bar\6^* + \bar\6^*\bar\6)$, 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. \end{document}