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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 22: Kodaira-Nakano vanishing theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize NRU HSE, Moscow} \\[15mm] {\small Misha Verbitsky, April 15, 2018 } \end{center} \newpage {\bf \blue Curvature (reminder)} \definition Let $\nabla:\; B \arrow B \otimes \Lambda^1 M$ be a connection on a vector bundle $B$. We extend $\nabla$ to an operator \[ V \stackrel{\nabla}\arrow \Lambda^{1}(M)\otimes V \stackrel{\nabla}\arrow \Lambda^{2}(M)\otimes V \stackrel{\nabla}\arrow \Lambda^{3}(M)\otimes V \stackrel{\nabla}\arrow ... \] using the Leibnitz identity $\nabla(\eta \otimes b) = d\eta + (-1)^{\tilde \eta} \eta \wedge \nabla b$. Then the operator $\nabla^2:\; B \arrow B\otimes \Lambda^{2}(M)$ is called {\bf \blue the curvature} of $\nabla$. \remark {\bf \purple The algebra of differential forms with coefficients in $\End B$ acts on $\Lambda^* M \otimes B$} via $\eta \otimes a (\eta' \otimes b) = \eta \wedge \eta' \otimes a(b)$, where $a\in \End(B)$, $\eta, \eta'\in \Lambda^* M$, and $b\in B$. \remark $\nabla^2(fb) = d^2 f b + df \wedge \nabla b - df \wedge \nabla b + f \nabla^2 b$, hence {\bf \purple the curvature is a $C^\infty M$-linear operator.} {\bf \red We shall consider the curvature $B$ as a 2-form with values in $\End B$}. Then $\nabla^2 := \Theta_B \in \Lambda^2 M \otimes \End B$, where an $\End(B)$-valued form acts on $\Lambda^* M \otimes B$ as above. \newpage {\bf \blue Conventions} Let $B$ be a holomorphic Hermitian bundle, and $\nabla= \bar\6 + \nabla^{1,0}$ its Chern connection. In this lecture, {\bf \red I would use $\6$ instead of $\nabla^{1,0}$.} As usual, we define the sequence \[ V \stackrel{\nabla}\arrow \Lambda^{1}(M)\otimes V \stackrel{\nabla}\arrow \Lambda^{2}(M)\otimes V \stackrel{\nabla}\arrow \Lambda^{3}(M)\otimes V \stackrel{\nabla}\arrow ... \] using the Leibnitz identity $\nabla(\eta \otimes b) = d\eta + (-1)^{\tilde \eta} \eta \wedge \nabla b$. Here the operators $\nabla$ are denoted by $d_\nabla$, their (1,0) and (0,1)-parts $\6_\nabla$ and $\bar\6_\nabla$. {\bf \red To simplify notations, I shall often omit the subscript ${}_\nabla$.} These conventions are sloppy but more or less standard. \newpage {\bf \blue Hodge theory with coefficients in $B$ (reminder)} \proposition Let $B$ be a holomorphic Hermitian bundle on a K\"ahler manifold, and $\nabla= \bar\6 + \6$ its Chern connection. We use the same letters $\6$, $\bar\6$ for the Hodge components of $d_\nabla:\; \Lambda^i(M)\otimes B \arrow \Lambda^{i+1}(M)\otimes B$. Then {\bf \red on $B$-valued differential forms, the following relations hold.} \[ [\Lambda, \6] = \1 \bar\6^*, \ \ \ [\Lambda, \bar\6] = - \1 \6^*, \ \ \ [L, \bar\6^*] = - \1 \6, \ \ \ [L, \6^*] = \1 \bar\6. \] \!\!\!\!\!\!\!\definition Let $\bar\6:\; \Lambda^{p,q}(M)\otimes B\arrow \Lambda^{p,q+1}(M)\otimes B$ be the holomorphic structure operator, extended to differential forms using the Leibniz identity. The anticommutator $\Delta_{\bar\6}:=\{\bar\6, \bar\6^*\}= \bar\6\bar\6^* + \bar\6^* \bar\6$ is called {\bf \blue Dolbeault Laplacian with coefficients in $B$}. It is self-adjoint, positive elliptic operator: $(\Delta_{\bar\6} x, x)= (\bar\6x, \bar\6x) + (\bar\6^*x, \bar\6^*x).$ {\bf \green Hodge theory with coefficients in a bundle:} {\bf \red There is a basis in the Hilbert space $L^2(\Lambda^*(M)\otimes B)$, consisting of eigenvalues of $\Delta_{\bar\6}$, and each eigenspace is finite-dimensional.}\\ {\bf \green Elliptic regularity:} {\bf \red Each eigenvector of $\Delta_{\bar\6}$ on $L^2(\Lambda^*(M)\otimes B)$ is a smooth form.} \theorem {\bf \red The space of $\Delta_{\bar\6}$-harmonic forms is identified with} the {\bf \blue Dolbeault cohomology with coefficients in $B$}, that is, with $\frac{\ker \bar \6}{\im\bar\6}$. We denote the $(p,q)$-part of Dolbeault cohomology by $H^{p, q}(M, B)$. \newpage {\bf \blue Serre's duality (reminder)} \remark The operator $*:\;\Lambda^{p,i}(M)\otimes_{\calo_M} B\arrow \Lambda^{n-p,n-i}(M)\otimes_{\calo_M} B^*$ exchanges $\bar\6$ and $\pm\bar\6^*$. {\bf \purple Therefore, it preserves the space $\ker \Delta_{\bar\6}$} of $\bar\6$-harmonic forms. \remark By definition, $H^n(\Omega^n M)= \frac{\Lambda^{n,n}(M)}{\im \bar\6}$. Stokes formula implies that $\int_M \bar\6(\alpha)=0$ for all $\alpha$. This gives a natural map $H^n(\Omega^n M)\stackrel{\int_M}\arrow \C$. \theorem {\bf \blue (Serre's duality)}\\ Let $M$ be an $n$-dimensional, compact complex manifold, and $B$ an Hermitian holomorphic bundle. Then {\bf \red the multiplication \[ H^i(\Omega^p M\otimes B) \times H^{n-i}(\Omega^{n-p} M\otimes B^*)\arrow H^n(\Omega^n M) \stackrel{\int_M}\arrow \C \] defines a non-degenerate pairing.} \proof Indeed, for each $\eta \in \ker \Delta_{\bar\6}$, the form $*\eta$ also belongs to $\ker \Delta_{\bar\6}$, but $\int_M \eta\wedge *\eta>0$, hence this pairing is non-degenerate. \endproof \corollary {\bf \blue (Serre's duality for $p=n$)} Let $M$ be a compact complex manifold, $\dim_\C M =n$. {\bf \red Then $H^i(B)\cong H^{n-i}(B^*\otimes K_M)^*$, where $K_M= \Omega^n M$.} \definition $K_M= \Omega^n M$ is called {\bf \blue canonical bundle}. \newpage {\bf \blue Schedule for April and May } \vfil {\bf \purple Lectures: Misha Verbitsky (April 14, May 16) \\ Ekaterina Amerik (April 18, 21, 25)\\[10mm] \red \Large Exam: Saturday, May 19.} \vfil \newpage {\bf \blue Laplacians and curvature } \remark Curvature of Chern connection: $\Theta_B=\{\nabla^{1,0}, \bar\6\}$, in the present notation it's $\Theta_B=\{\6, \bar\6\}$. \theorem {\bf \blue (Bochner-Kodaira-Nakano identity)}\\ {\bf \red $\Delta_{\bar\6} = \Delta_{\6} + H_B$, where $H_B:=-\1 [\Lambda, \Theta_B]$.} \proof Super-Jacobi identity gives \begin{multline*} [\Lambda, \Theta_B]= [\Lambda, \{\6, \bar\6\}] = \{[\Lambda, \6], \bar\6\} + \{\6, [\Lambda, \bar\6]\}\\ = \1 \{\bar\6^*, \bar\6\} - \1\{\6^*, \6\}= \1 \Delta_{\bar\6} - \1 \Delta_{\6}. \end{multline*} \endproof \remark The operators $\Delta_{\bar\6}$ and $\Delta_{\6}$ are positive. Therefore, if $(H_Bx, x) >0$, one has $(\Delta_{\bar\6}x,x)>0$. {\bf \purple If $(H_Bx, x) >0$ for all $x$, the operator $\Delta_{\bar\6}$ has no kernel, and the corresponding cohomology group vanishes.} \newpage {\bf \blue Positive line bundles} \definition A holomorphic line bundle is called {\bf \blue positive} if its first Chern class is cohomologous to a K\"ahler form. {\bf \green Theorem 1:} {\bf \blue (Kodaira-Nakano)} Let $L$ be a positive line bundle on a compact K\"ahler manifold. {\bf \red Then for any bundle $B$ there exists $N>0$ such that $H^i(B\otimes L^N)=0$ for all $i>0$}. \proof We deduce Theorem 1 from Theorem 2 below. {\bf \green Theorem 2:} {\bf \blue (Kodaira-Nakano)} Let $B$ be a holomorphic Hermitian line bundle on $n$-dimensional K\"ahler manifold, $\Theta_B$ its curvature, and $L_{\Theta_B}:\; \Lambda^*(M) \arrow \Lambda^*(M)$ the operator of multiplication by $\Theta_B$. Suppose that the self-adjoint operator $H_B:=\1[L_{\Theta_B}, \Lambda]$ satisfies $(H_B(x), x)>0$ for any non-zero $k$-form $x$, $k>n$. {\bf \red Then $H^p(B\otimes \Omega^q M)=0$ for all $p+q >n$.} \proof $\Delta_{\bar\6}= \Delta_\6 -H_B$, which gives $(\Delta_{\bar\6}x, x)= (\Delta_{\6}x, x) - (H_Bx,x)>0$. \endproof \newpage {\bf \blue Kunihiko Kodaira} \begin{center}{\epsfig{file=Kodaira_Kunihiko.jpg,width=0.33\linewidth}\\[10mm] {\bf \small \blue Kunihiko Kodaira (1915-1997)}} \end{center} \newpage {\bf \blue Negative line bundles and cohomology} \remark For any vector bundles $E, F$ with connections, their curvatures are related as $\Theta_{E\otimes F} = \Theta_E + \Theta_F$. Also, if $L$ is a line bundle, one has $\Theta_{L^*}=- \Theta_L$. \definition Let $L$ be a line bundle such that $L^*$ is positive. Then $L$ is called {\bf \blue negative}. \remark Suppose that $L$ is a negative line bundle on $(M,\omega)$, and $\1\omega$ the curvature of $L$. Then $H_{L}:=\1[L_{\Theta_L}, \Lambda]= H$. {\bf \red On $k$-forms $H_L$ acts as multiplication by $k-n$.} {\bf \purple Therefore, $L$ satisfies conditions of Theorem 2, and $H^p(L\otimes \Omega^q M)=0$ for all $p+q >n$.} \newpage {\bf \blue Kodaira-Nakano theorem: the proof} \remark Suppose $L$ is a negative vector bundle, with $H_L= -\1[L_{\Theta_L}, \Lambda]= H$. The operator $H_{E\otimes F}$ is expressed as $H_{E\otimes F}= H_E + H_F$. This gives $H_{B\otimes L^N} = H_B- N H$. For $N > \alpha$, where $\alpha$ is the biggest eigenvalue of $H_B$, we have \[ (H_{B\otimes L^N}x, x)= (H_B x, x)+ N(n-k)|x|^2> 0. \] {\bf \purple Then Theorem 2 gives $H^p(L^N\otimes B \otimes \Omega^q M)=0$ for all $p+q >n$.} \corollary Let $L$ be a negative line bundle. Then {\bf \red for all vector bundles $B$ with Chern connection, there exists $N>0$ such that $H^i(L^{-N} \otimes B)=0$ for all $i>0$.} \proof Apply the previous corollary to $B^*\otimes K_M$, and use the Serre's duality \[ 0 = H^{n-i} (L^N\otimes B^* \otimes \Omega^m M)= H^i(B \otimes L^{-N})^*. \] \endproof \remark {\bf \purple We have proved Theorem 1.} \endproof \newpage {\bf \blue Forms realized as a curvature (reminder)} {\bf \green Proposition 1:} Let $\omega$ be an integer (1,1)-form with integer cohomology class on a compact K\"ahler manifold. {\bf \red Then $\omega$ is a curvature of a holomorphic line bundle}. \pstep Exponential exact sequence $0 \arrow \Z_M \arrow \calo_M \arrow \calo^*_M\arrow 0$ gives \[ H^1(\calo^*_M) \stackrel c \arrow H^2(M, \Z) \stackrel p \arrow H^2(M, \calo_M), \] where $H^1(\calo^*_M)= \Pic(M)$ is the group of holomorphic line bundles, $c$ maps a bundle to its first Chern class, and $p$ projects $H^2(M)$ to its Hodge component $H^2(M, \calo_M)= H^{0,2}(M)$. Then {\bf \purple for any integer class $[\omega]\in H^{1,1}(M)\cap H^2(M, \Z)$, there exists a line bundle $L$ such that $[\omega]= c_1(L)$.} {\bf \green Step 2:} Take any metric $h$ on $L$. Its curvature $\omega_h$ is a closed (1,1)-form, cohomologous to $\omega$. By $dd^c$-lemma, $\omega_h -\omega = -2 \6\bar\6 f$ for some $f\in C^\infty M$. {\bf \purple By Corollary 1, curvature of $h':=e^{2f}h$ satisfies $\omega_h -\omega_{h'}= -2 \6\bar \6 f$,} giving $\omega_{h'}= \omega$. \endproof \newpage {\bf \blue Base points of line bundles} \definition Let $L$ be a holomorphic line bundle on $M$. A point $x\in M$ is called {\bf \blue base point} for $M$ if any global section of $L$ {\bf \blue vanishes in $x$,} that is, if it does not generate the stalk $L_x$ over the ring of germs $\calo_{M, x}$. A bundle is called {\bf \blue base point free} if it has no base points. \example Let $\C^{n+1}\backslash 0 \stackrel \pi \arrow \C P^n$ be the natural projection. For each point $x\in \C P^n$, the fiber $\pi^{-1}(x)$ is $\C^*\subset l_x$, where $l_x$ is the line associated with $x$. Denote by $\calo(-1)$ the bundle with the fiber $l_x$ in each $x\in \C P^n$. Each line functional on $\C^{n+1}$ defines a line functional on each $l_x$, which holomorphically depends on $x$. {\bf \red This means that the dual line bundle, denoted $\calo(1)$, is base point free.} \remark Let $L$ be a base point free line bundle, and ${\goth m}_x\subset \calo_{M, x}$ the maximal ideal of a point $x\in M$. Consider {\bf \blue the fiber} $L\restrict x:= L_x /{\goth m}_x$. It is a vector space over $\C$ of rank 1. A map $H^0(M, L)\arrow L_x$ taking $f$ to $f\restrict x$ defines a map $\phi_x\in \Hom_\C(H^0(M, L), L\restrict x)\cong H^0(M, L)^*$. {\bf \purple The isomorphism $\Hom_\C(H^0(M, L), L\restrict x)\cong H^0(M, L)^*$ is defined only after we fix an isomorphism $L\restrict x \cong \C$,} that is, up to a constant multiplier. Therefore, {\bf \red the map $x\arrow \psi_x$ defines a holomorphic map $\phi:\; M \arrow {\Bbb P}H^0(M,L)^*$}. {\bf \purple This map is well-defined only if $L$ is base point free.} \newpage {\bf \blue Very ample line bundles} \claim Let $L$ be a base point free holomorhic line bundle on $M$, and $\phi:\; M \arrow {\Bbb P}H^0(M,L)^*$ the natural map defined above. {\bf \red Then $L\cong \phi^*(\calo(1))$.} \pstep This statement is essentially a tautology: global sections of $\calo(1)$ on ${\Bbb P} V$ are identified with $V^*$, hence global sections of $\phi^*(\calo(1))$ are identified with $H^0(M,L)$. {\bf\green Step 2:} To see that this identification is carried over to the corresponding sheaves, we consider the tautological line bundle $\calo(-1)$ with the fiber $l_x$ at each $x\in {\Bbb P}H^0(M,L)^*$. By construction, $l_x$ is dual to $L\restrict x$, and this duality is holomorphic on $x$, hence $\phi^*(\calo(-1))=L^*$. \endproof \definition A line bundle $L$ is called {\bf \blue very ample} if it is base point free, and the corresponding holomorphic map $\phi:\; M \arrow {\Bbb P}H^0(M,L)^*$ is an embedding. It is called {\bf \blue ample} of $L^{\otimes N}$ is very ample for some $N>0$. \definition A complex manifold is called {\bf \blue projective} if it admits a holomorphic embedding to $\C P^n$. \newpage {\bf \blue Kodaira embedding theorem} \corollary Let $(M, \omega)$ be a compact K\"ahler manifold such that the cohomology class of $\omega$ is rational. {\bf \red Then $M$ admits a positive line bundle.} \proof Let $n\omega$ be a K\"ahler form proportional to $\omega$ with integer K\"ahler class. {\bf \purple Then $n\omega$ is realized as a curvature of a holomorphic line bundle (Proposition 1).} This line bundle is positive. \endproof \theorem {\bf \blue (Kodaira)} {\bf \red A positive bundle is ample}.\\ \proof {\em (will be proven later in April)} \endproof \corollary {\bf \blue (Kodaira embedding theorem)}\\ {\bf \red A compact K\"ahler manifold $M$ is projective if and only if it admits a K\"ahler form with rational cohomology class.} \proof Using the Corollary above, we obtain that $M$ admits a positive holomorphic line bundle $L$, and the theorem of Kodaira above implies that $L$ is ample, hence $L^{\otimes N}$ defines a projective embedding. Conversely, if $M\subset \C P^n$ is a projective manifold, the cohomology class of Fubini-Study form is proportional to a rational class, hence $M$ admits a K\"ahler form with a rational cohomology class. \endproof \end{document}