\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig} %\usepackage[matrix,arrow]{xy} \newcommand{\green}{\color[rgb]{0,0.4,0}} \newcommand{\purple}{\color[rgb]{0.4,0,0.4}} \newcommand{\red}{\color[rgb]{0.7,0,0}} \newcommand{\blue}{\color{blue}} \def\eqref#1{(\ref{#1})} \newcommand{\goth}{\mathfrak} \newcommand{\g}{{\frak g}} \newcommand{\arrow}{{\:\longrightarrow\:}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\R}{{\Bbb R}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\6}{\partial} \def\1{\sqrt{-1}\:} \newcommand{\restrict}[1]{{\left|_{{#1}}\right.}} \newcommand{\cntrct} % contraction with a vector field {\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}} \def\Bbb#1{\mathbb #1} \newcommand{\calo}{{\cal O}} \newcommand{\cac}{{\cal C}} % Correcting TeX... %\let\oldtilde=\tilde %\renewcommand{\tilde}{\widetilde} \renewcommand{\bar}{\overline} \renewcommand{\phi}{\varphi} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} % Operatornames \newcommand{\even}{{\rm even}} \newcommand{\ev}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\sf const}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\End}{\operatorname{End}} \newcommand{\St}{\operatorname{St}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\symb}{\operatorname{\text{\sf symb}}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Alt}{\operatorname{Alt}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sec}{\operatorname{Sec}} \newcommand{\Can}{\operatorname{Can}} \newcommand{\Sing}{\operatorname{Sing}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\Supp}{\operatorname{Supp}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\dist}{\operatorname{\sf dist}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\ind}{\operatorname{\sf ind}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Tw}{\operatorname{Tw}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Diff}{\operatorname{Diff}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\inbfpare}[1]{{% \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}% }} \newcommand{\comment}[1]{{}} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Hodge theory, lecture 19 \hfil \tiny M. 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 19: Chern connections} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize NRU HSE, Moscow} \\[15mm] {\small Misha Verbitsky, April 4, 2018 } \end{center} \newpage {\bf \blue Curvature} \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 Flat bundles} \definition Let $(B, \nabla)$ be a bundle with connection. {\bf\blue Holonomy group} of $\gamma$ is the group of endomorphisms of the fiber $B_x$ obtained from parallel transports along all paths starting and ending in $x\in M$ \definition A bundle is {\bf \blue flat} if and only if its curvature vanishes. The following theorem easily follows from the Frobenius theorem and interpretation of connection on $B$ as of splitting of the tangent bundle of the total space of the principal $GL(n)$-fibration associated with $B$. \theorem Let $(B, \nabla)$ be a vector bundle with connection over a simply connected manifold. {\bf \red Then $B$ is flat if and only if its holonomy group is trivial}. \newpage {\bf \blue Holomorphic bundles} \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} \remark Let $M$ be a complex manifold. Then {\bf \purple the operator \\ $\bar\6:\; C^\infty M \arrow \Lambda^{0,1}(M)$ is $\calo_M$-linear.} \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 $B_{C^\infty}=B \otimes_{\calo_M} C^\infty M$. \remark The kernel of $\bar\6:\; B_{C^\infty}\arrow B_{C^\infty}\otimes \Lambda^{0,1}(M)$ {\bf \red coincides with the image of $B$} under the natural sheaf embedding $B\hookrightarrow B_{C^\infty}$, with $b \arrow b \otimes 1$. \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 $\eta \in \Lambda^{0,i}(M)$. \newpage {\bf \blue Complexification (reminder)} \definition Let $M_\R$ be a real analytic manifold, and $M_\C$ a complex analytic manifold equipped with an antiholomorphic involution, such that $M_\R$ is the set of its fixed points. Then $M_\C$ is called {\bf \blue complexification} of $M_\R$. \definition A tensor on a real analytic manifold is called {\bf\blue real analytic} if it is expressed locally by a sum (infinite, generally speaking) of coordinate monomials with real analytic coefficients. \claim Let $M_\R$ be a real analytic manifold, $M_\C$ its complexification, and $\Phi$ a tensor on $M_\R$. {\bf \red Then $\Phi$ is real analytic if and only if $\Phi$ can be extended to a holomorpic tensor $\Phi_\C$ in some neighbourhood of $M_\R$ inside $M_\C$.} \proof The ``if'' part is clear, because every complex analytic tensor on $M_\C$ is by definition real analytic on $M_\R$. Conversely, suppose that $\Phi$ is expressed by a sum of coordinate monomials with real analytic coefficients $f_i$. Let $\{U_i\}$ be a cover of $M$, and $\tilde U_i:=U_i\times B_\epsilon$ the corresponding cover of a neighbourhood of $M_\R$ in $M_\C$ constructed above. Chosing $\epsilon$ sufficiently small, we can assume that the Taylor series giving coefficients of $\Phi$ converges on each $\tilde U_i$. {\bf \purple We define $\Phi_\C$ as the sum of these series.} \endproof \newpage {\bf \blue Holomorphic structure operator} \remark {\bf \purple For any holomorphic bundle, one has $\bar\6^2=0$}. Indeed, a holomorphic bundle admits a local trivialization. \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 $V=B_{\C^\infty}$.} \proof We prove this theorem in additional assumption that $V$ and $\bar \6$ is real analytic. This assumption can be justified using the Newlander-Nirenberg theorem. Let $M_\C\subset M\times \bar M$ be a small neighbourhood of diagonal, considered as a complexification of $M$, and $V_\C$ a holomorphic vector bundle on $M_\C$ obtained from the real vector bundle $V$. We extend $\bar\6$ to a differential operator $\bar\6_\C$ on $V_\C$. Then $\bar\6_\C$ is a connection in the restriction of $V_\C$ to the fibers of the natural projection $\pi:\; M_\C \arrow M$, and the condition $\bar\6^2=0$ implies that it is flat. {\bf \purple The sheaf $\ker \bar\6$ constant on fibers of $\pi$,} hence $B=\pi_*(\ker \bar\6)$ is a holomorphic bundle of the same rank as $V$. \endproof \newpage {\bf \blue Connections and holomorphic structures} \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. \newpage {\bf \blue Chern connection} \theorem Every holomorphic Hermitian vector bundle {\bf \red admits a Chern connection, which is unique.} \pstep Given a complex vector bundle $B$, define {\bf \blue complex conjugate bundle} $\bar B$ as the same $\R$-bundle with complex conjugate $\C$-action. Then {\bf \purple a connection $\nabla$ on $B$ defines a connection $\bar\nabla$ on $\bar B$, with $\bar\nabla^{1,0}=\overline{\nabla^{0,1}}$ and $\bar\nabla^{0,1}=\overline{\nabla^{1,0}}$.} {\bf \green Step 2:} Define {\bf \blue $\nabla^{1,0}$-operator} on a complex vector bundle $B$ as a map $B \stackrel {\nabla^{1,0}}\arrow \Lambda^{1,0}(M)\otimes B$, satisfying $\Lambda^{1,0}(fb) = \6(f)\otimes b + f\nabla^{1,0}(b)$ for any $f\in C^\infty M, b\in B$. {\bf \purple A $\bar\6$-operator on $B$ defines an $\nabla^{1,0}$-operator on $\bar B$, and vice versa}. {\bf \green Step 3:} Hermitian form defines an isomorphism of complex vector bundles $B \stackrel g \arrow \bar B^*$. Holomorphic structure on $B$ defines a $\bar\6$-operator on $\bar B=B^*$, which is the same as $\nabla^{1,0}$-operator $\nabla^{1,0}_g$ on $B$. {\bf \purple This gives a connection operator $\nabla:= \bar\6 + \nabla^{1,0}_g$ on $B$,} which is Hermitian by construction. \endproof \end{document}