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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 20: Curvature of line bundles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize NRU HSE, Moscow} \\[15mm] {\small Misha Verbitsky, April 7, 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 Holomorphic bundles (reminder)} \definition {\bf \blue Holomorphic vector bundle} on a complex manifold $M$ is a locally trivial sheaf of $\calo_M$-modules. \definition {\bf \blue The total space} $\Tot(B)$ of a holomorphic bundle $B$ over $M$ is the space of all pairs $\{x\in M, b \in B_x/{\goth m}_x B\}$, where $B_x$ is the stalk of $B$ in $x\in M$ and ${\goth m}_x$ the maximal ideal of $x$. We equip $\Tot(B)$ with the natural topology and holomorphic structure, in such a way that $\Tot(B)$ becomes a locally trivial holomorphic fibration with fiber $\C^r$, $r=\rk B$. \remark {\bf \purple The set of holomorphic sections of a map $\Tot(B)\arrow M$ is naturally identified with the set of sections of the sheaf $B$.} \claim Ler $B$ be a holomorphic bundle. Consider the sheaf $B_{C^\infty}:=B \otimes_{\calo_M} C^\infty M$. {\bf \red Then $B_{C^\infty}$ is a locally trivial sheaf of $C^\infty M$-modules}. \definition $B_{C^\infty}$ is called {\bf \blue smooth vector bundle underlying the holomorphic vector bundle $B$}. \remark The natural map $\Tot(B)\arrow \Tot(B_{C^\infty})$ is a diffeomorphism. \newpage {\bf \blue $\bar\6$-operator on vector bundles (reminder)} \definition Let $B$ be a holomorphic vector bundle on $M$. Consider an operator $\bar\6:\; B_{C^\infty}\arrow B_{C^\infty}\otimes \Lambda^{0,1}(M)$ mapping $b\otimes f$ to $b\otimes \bar\6 f$, where $b$ is a holomorphic section of $B$, and $f$ smooth. This operator is called {\bf\blue a holomorphic structure operator} on $B$. {\bf \red It is well-defined because $\bar\6$ is $\calo_M$-linear}, and $B_{C^\infty}=B \otimes_{\calo_M} C^\infty M$. \definition A {\bf \blue $\bar\6$-operator} on a smooth complex vector bundle $V$ over a complex manifold is a differential operator $V \stackrel {\bar\6}\arrow \Lambda^{0,1}(M)\otimes V$ satisfying $\bar\6(fb) = \bar\6(f)\otimes b + f\bar\6(b)$ for any $f\in C^\infty M, b\in V$. \remark A $\bar\6$-operator {\bf \purple can be extended to $\bar\6:\; \Lambda^{0,i}(M)\otimes V \arrow \Lambda^{0,i+1}(M)\otimes V,$} using the Leibnitz identity $\bar\6 (\eta \otimes b) = \bar\6(\eta)\otimes b + (-1)^{\tilde \eta}\eta\wedge\bar\6(b)$, for all $b\in V$ and $\eta \in \Lambda^{0,i}(M)$. \theorem {\bf \blue (Malgrange)} Let $\bar\6:\; V \arrow \Lambda^{0,1}(M)\otimes V$ be a $\bar\6$-operator on a complex vector bundle, satisfying $\bar\6^2=0$, where $\bar\6$ is extended to \[ V \stackrel{\bar\6}\arrow \Lambda^{0,1}(M)\otimes V \stackrel{\bar\6}\arrow \Lambda^{0,2}(M)\otimes V \stackrel{\bar\6}\arrow \Lambda^{0,3}(M)\otimes V \stackrel{\bar\6}\arrow ... \] as above. {\bf \red Then $B:=\ker \bar\6\subset V$ is a holomorphic bundle of the same rank, and $V=B_{\C^\infty}$.} \newpage {\bf \blue Chern connection (reminder)} \definition Let $V$ be a smooth complex vector bundle with connection $\nabla:\; V \arrow \Lambda^1(M)\otimes V$ and holomorphic structure $\bar\6:\; V \arrow \Lambda^{0,1}(M)\otimes V$. Consider the Hodge type decomposition of $\nabla$, $\nabla= \nabla^{0,1} + \nabla^{1,0}$, where \[ \nabla^{0,1}:\; V \arrow \Lambda^{0,1}(M)\otimes V, \ \ \ \nabla^{1,0}:\; V \arrow \Lambda^{1,0}(M)\otimes V. \] We say that {\bf \blue the connection $\nabla$ is compatible with the holomorphic structure} if $\nabla^{0,1}=\bar\6$. \definition {\bf \blue A holomorphic Hermitian vector buncle} is a smooth complex vector bundle equipped with a Hermitian metric and a holomorphic structure. \definition {\bf \blue Chern connection} on a holomorphic Hermitian vector bundle is a unitary connection compatible with the holomorphic structure. \theorem Every holomorphic Hermitian vector bundle {\bf \red admits a Chern connection, which is unique.} \remark When people say about ``curvature of a holomorphic Hermitian line bundle'', {\bf \red they speak about curvature of a Hermitian connection.} \newpage {\bf\blue Bianchi identity} \remark $[\nabla, \{\nabla, \nabla\}]=[\{\nabla, \nabla\},\nabla]+ [\nabla, \{\nabla, \nabla\}]=0$ by the super Jacobi identity. This gives {\bf \blue the Bianchi identity:} $\nabla(\Theta_B\wedge \eta) = \Theta_B \wedge \nabla(\eta)$. \remark When $B$ is a line bundle, $\End(B)$ is trivial, and $\Theta_B$ is a 2-form. \claim {\bf \red A curvature of a line bundle is a closed 2-form.} \proof For any form $\theta\in \Lambda^{i}(M) \otimes \End(B)$, Leibnitz identity gives $\nabla(\theta \wedge \eta) = d\theta \wedge \eta + (-1)^i\theta \wedge \nabla(\eta)$. Bianchi identity gives $\nabla(\Theta_B\wedge \eta) = \Theta_B \wedge \nabla(\eta)$. Therefore $d\Theta_B=0$. \endproof \remark The same argument can be used to show that {\bf \red $\Tr_B \Theta_B^i$ is a closed $2i$-form}, where $\Tr_B$ denotes the trace in $\End(B)$, and $\Theta_B^i$ is the $i$-th power of an $\End(B)$-valued form. \definition Cohomology classes of $\Tr_B \Theta_B^i$ are called {\bf \blue characteristic vlasses} of a bundle $B$ (``Chern-Weil formula''). When $B$ is a line bundle, the cohomology class of $-\frac{\1}{\pi} \Theta_B$ is called {\bf \blue first Chern class of $B$}, denoted $c_1(B)$. \newpage {\bf \blue Luigi Bianchi} \begin{center}{\epsfig{file=Bianchi.jpeg,width=0.43\linewidth}\\[10mm] {\bf \small \blue Luigi Bianchi (1856 - 1928)}} \end{center} \newpage {\bf \blue Exponential sequence} \remark Let $B$ be a line bundle on a manifold, $\{U_\alpha\}$ its cover where $B$ is trivialized, and $\phi_{\alpha\beta}$ the corresponding transition functions defined on $U_\alpha \cap U_\beta$. On each intersection $U_\alpha \cap U_\beta\cap U_\gamma$ we have $\phi_{\alpha\beta}\phi_{\beta\gamma}= \phi_{\alpha\gamma}$, hence {\bf \purple a trivialization of $B$ on $\{U_\alpha\}$ defines a \v Cech 1-cocycle on $B$ with values in $(C^\infty M)^*$.} The following claim is clear from the definitions. \claim {\bf \red Isomorphism classes of vector bundles are in bijective correspondence with $H^1(M, (C^\infty M)^*)$.} \definition {\bf \blue Exponential exact sequence} is the following exact sequence of sheaves: \[ 0 \arrow \Z_M \arrow C^\infty M \arrow (C^\infty M)^* \arrow 0, \] Since $H^i(C^\infty M)=0$ for $i>0$, {\bf \purple the corresponding long exact sequence gives $0 \arrow H^1(M, (C^\infty M)^*) \tilde \arrow H^2(M, \Z) \arrow 0$.} \newpage {\bf \blue Line bundles on $\C P^\infty$} \exercise Let $B$ be a vector bundle on $M$. {\bf \purple Prove that $B\oplus B'$ is trivial for some bundle $B'$.} \claim For any complex line bundle $L$ on $M$, {\bf \red there exists a map $\phi:\; M \arrow \C P^n$ such that $L= \pi^* \calo(-1)$. } \proof Indeed, suppose that $L\oplus B'$ is trivial, $B_1 =L \oplus B' = V \otimes_\C C^\infty M$. Then each point $x\in M$ defines a line $\phi(x)\in {\Bbb P}V$ such that $L\restrict x = \phi(x) \subset B_1 \restrict x= V$. In this situation, {\bf \purple $L$ is obtained as a pullback of the tautological vector bundle.} \endproof \newpage {\bf \blue First Chern class} \definition The isomorphism $H^1(M, (C^\infty M)^*) \tilde \arrow H^2(M, \Z)$ maps a line bundle $L\in H^1(M, (C^\infty M)^*)$ to its {\bf \blue integer Chern class} $c_1^\Z(B)\in H^2(M, \Z)$ \theorem {\bf \blue (Gauss-Bonnet)} \\ Let $L$ be a line bundle on $M$. Then the natural map $H^2(M, \Z)\arrow H^2(M, \R)$ {\bf \red maps $c_1^\Z(L)$ to the class $c_1(L)=-\frac{\1}{\pi} [\Theta_L]$ defined above.} \proof Let $\calo(-1)$ be the tautological bundle on $\C P^n$. As shown above, any line bundle $L$ on $M$ can be obtained as $\phi^*(\calo(-1))$, hence it suffices to prove $[\Theta_L]= \1\pi c_1^\Z(L)$ for $L=\calo(-1)$ on $\C P^n$. Since a cohomology class in $H^2(\C P^n)$ is determined by its restriction to $\C P^1$, it would suffice to prove this formula for $L=\calo(-1)$ on $\C P^1$. In this case, $c_1^\Z(L)$ is the Euler characteristic of $L$, and $[\Theta_L]= \1\pi c_1^\Z(L)$ is the usual Gauss-Bonnet formula on a 2-sphere, which can be obtained by computing the volume of $S^2$. \endproof \newpage {\bf \blue Real structures on a complex vector space} \definition {\bf \blue Real structure on a complex vector space} is anticomplex involution. \exercise For any complex vector space $V$ and a real structure $\iota$, denote by $V_\R$ the fixed point set of $\iota$. {\bf \purple Prove that $V= V_\R \otimes_\R \C$.} \example Let $V$ be a Hermitian vector space, and$\End_\C V$ its endomorphism space. Consider the real structure $\phi \stackrel \iota \arrow -\phi^*$, where $\phi^*$ denotes the Hermitian conjugate. {\bf \purple Prove that the fixed point set of $\iota$ is the space of anti-Hermitian matrices $\goth u(V)$.} \newpage {\bf \blue Curvature of the Chern connection} \proposition {\bf \red Curvaure $\Theta_B$ of a Chern connection on $B$ is a (1,1)-form: $\Theta_B \in \Lambda^{1,1}(M)\otimes \End(B)$.} \pstep Let $B$ be a Hermitan bundle. Consider the operator $\phi \stackrel \iota \arrow -\phi^*$ acting on $\End(B)$, where $\phi \arrow \phi^*$ denotes the Hermitian conjugation. Since $\iota^2=\Id$, and this is an anticomplex operator, it defines the real structure, and its fixed point set is ${\goth u}_B$. {\bf \green Step 2:} Since the Chern connection preserves the Hermitian structure $g$, one has $\nabla(g)=0$, which gives $\nabla^2(g)=0$. This means that $\Theta_B\in \Lambda^2 M \otimes {\goth u}_B$, {\bf \purple and this for is real with respect to the real structure defined by $\iota$}. {\bf \green Step 3:} The $(0,2)$-part of the curvature vanishes, because $\bar\6^2=0$. The $(2,0)$-part of the curvature vanishes, because $\iota(\Theta_B) = \Theta_B$, and {\bf \purple any real structure on $\End(B)$ exchanges $\Lambda^{2,0}(M)\otimes \End(B)$ and $\Lambda^{0,2}(M)\otimes \End(B)$}. \endproof \corollary {\bf \red For the Chern connection $\nabla=\bar\6+\nabla^{1,0}$ on $B$, one has $\Theta_B =\{\nabla^{1,0}, \bar\6\}$.} \corollary {\bf \red A curvature of a holomorphic Hermitian line bundle is a closed (1,1)-form.} \newpage {\bf \blue Curvature of the Chern connection on a line bundle} \remark Let $B$ he a Hermitian holomorphic line bundle, and $b\in \Gamma(B)$ a nowhere vanishing holomorphic section. Then \[ d|b|^2= (\nabla^{1,0} b, b)+ (b, \nabla^{1,0} b)= 2 \Re (\nabla^{1,0} b, b), \] which gives \[ \nabla^{1,0} b= \frac{\6 |b|^2}{|b|^2}b= 2\6\log|b| b. \] {\bf \purple We obtaim that $\Theta_B(b)= 2\bar\6\6\log|b| b$, hence $\Theta_B = -2 \6\bar\6\log|b|$.} {\bf \green Corollary 1:} Let $g' = e^{2f} g$ be Hermitian metrics on a holomorphic line bundle, and $\Theta, \Theta'$ the corresponding curvatures. {\bf \red Then $\Theta' - \Theta = -2 \6\bar\6 f$.} \endproof \definition Tensor multiplication {\bf \purple defines the structure of abelian group on the set $\Pic(M)$ of equivalence classes of holomorphic line bundles on a complex manifod $M$}. This group is called {\bf \blue the Picard group} of $M$. \newpage {\bf \blue $dd^c$-lemma (reminder)} \theorem Let $\eta$ be a form on a compact K\"ahler manifold, satisfying one of the following conditions.\\ (1). $\eta$ is an exact $(p,q)$-form. (2). $\eta$ is $d$-exact, $d^c$-closed. \\ (3). $\eta$ is $\6$-exact, $\bar\6$-closed. \\ {\bf \red Then $\eta\in \im dd^c=\im \6\bar\6$.} \proof Notice immediately that in all three cases {\bf \purple $\eta$ is closed and orthogonal to the kernel of $\Delta$}, hence its cohomology class vanishes. Since $\eta$ is exact, it lies in the image of $\Delta$. Operator $G_\Delta:=\Delta^{-1}$ is defined on $\im \Delta= \ker \Delta^\bot$ and commutes with $d, d^c$. In case (1), $\eta$ is $d$-exact, and $I(\eta)= \bar\eta$ is $d$-closed, hence $\eta$ is $d$-exact, $d^c$-closed like in (2). Then $\eta= d \alpha$, where $\alpha:=G_\Delta d^*\eta$. Since $G_\Delta$ and $d^*$ commute with $d^c$, the form $\alpha$ is $d^c$-closed; since it belongs to $\im \Delta=\im G_\Delta$, it is $d^c$-exact, $\alpha=d^c \beta$ which gives $\eta = dd^c \beta$. In case (3), we have $\eta= \6 \alpha$, where $\alpha:=G_\Delta \6^*\eta$. Since $G_\Delta$ and $\6^*$ commute with $\bar\6$, the form $\alpha$ is $\bar\6$-closed; since it belongs to $\im \Delta$, it is $\bar\6$-exact, $\alpha=\bar\6 \beta$ which gives $\eta = \6\bar\6 \beta$. \endproof \newpage {\bf \blue Forms realized as a curvature of a line bundle} \corollary 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 \end{document}