\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig, dutchcal} %\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}} \renewcommand{\H}{{\Bbb H}} \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}{{\it const}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\End}{\operatorname{End}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Sets}{\operatorname{{\cal S}ets}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\Vol}{\operatorname{Vol}} \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{\codim}{\operatorname{codim}} \newcommand{\coim}{\operatorname{coim}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\slope}{\operatorname{slope}} \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 Complex geometry, lecture 6 \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{\proof}{% {\bf \green Proof:\ }} \newcommand{\pstep}{% {\bf \green Proof. Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Complex geometry\\[15mm] \small lecture 6: Real analytic manifolds and Newlander-Nirenberg} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] October 10, 2020 } \end{center} \newpage {\bf \blue Real structures on complex manifolds (reminder)} \definition A smooth map $\Psi:\; M \arrow N$ on an almost complex manifold $(M,I)$ is called {\bf \blue antiholomorphic} if $d\Psi(I)=-I$. A function $f$ is called {\bf\blue antiholomorphic} if $\bar f$ is holomorphic. \exercise Prove that {\bf \purple an antiholomorphic function on $M$ defines an antiholomorphic map from $M$ to $\C$.} \exercise Prove that a map $\Psi:\; M \arrow N$ of almost complex manifolds is antiholomorphic {\bf \purple if and only if $\Psi^*(\Lambda^{0,1}(N))\subset \Lambda^{1,0}(M)$.} \exercise Let $\iota$ be a smooth map from a complex manifold $M$ to itself. Prove that {\bf \purple $\iota$ is antiholomorphic if and only if $\iota^*(f)$ is antiholomorphic for any holomorphic function $f$ on $U\subset M$.} \definition {\bf \blue A real structure} on a complex manifold $M$ is an antiholomorphic involution $\tau:\; M \arrow M$. \example {\bf \purple Complex conjugation defines a real structure on $\C^n$.} \newpage {\bf \blue Fixed points of real structures on manifolds (reminder)} \proposition Let $M$ be a complex manifold and $\iota:\; M\arrow M$ a real structure. Denote by $M^\iota$ the fixed point set of $\iota$. Then, {\bf \red for each $x\in M^\iota$ there exists a $\iota$-invariant coordinate neighbourhood with holomorphic coordinates $z_1, ..., z_n$, such that $\iota^*(z_i)=\bar z_i$.} \pstep For each basis of 1-forms $\nu_1, ..., \nu_n \in \Lambda^{1,0}_x(M)$, there exists a set of holomorphic coordinate functions $u_1, ..., u_n$ such that $du_i \restrict x=\nu_i$. To obtain such a coordinate system, {\bf \purple we chose any coordinate system $v_1, ..., v_n$ and apply a linear transform mapping $dv_i\restrict x$ to $\nu_i$.} {\bf \green Step 2:} The differential $d\iota$ acts on $T_x M$ as a real structure. Using the structure theorem about real structures, we obtain that any real basis $\zeta_1, .., \zeta_n$ of $T_x^* M^\iota$ is a complex basis in the complex vector space $T_x^* M$. Then $\nu_i:=\zeta_i+ \1 I(\zeta_i)\}$ is a basis in $\Lambda^{1,0}_x(M)$. Choose the coordinate system $u_1, ..., u_n$ such that $du_i \restrict x=\nu_i$ (Step 1). {\bf \purple Replacing $u_i$ by $z_i:=u_i+\iota^*(\bar u_i)$, we obtain a holomorphic coordinate system $z_i$ on $M$} (compare with Theorem 1 in Lecture 4) {\bf \purple which satisfies $\iota^*(z_i)=\bar z_i$.} \endproof \definition Let $\{U_i\}$ be an complex atlas on $M$. Assume that any $U_i$ intersecting $M^\iota$ satisfies the conclusion of this proposition. Then $\{U_i\}$ is called {\bf \blue compatible with the real structure}. \newpage {\bf \blue Real analytic manifolds and real structures (reminder)} \proposition Let $M^\iota\subset M$ be a fixed point set of an antiholomorphic involution $\iota$ on a complex manifold $M$, $\{U_i\}$ a complex analytic atlas, and $\Psi_{ij}:\; U_{ij}\arrow U_{ij}$ the gluing functions. Assume that the atlas $U_{i}$ is compatible with the real structure, in the sense of the previous proposition. {\bf \red Then all $\Psi_{ij}$ are real analytic on $M^\iota$, and define a real analytic atlas on the manifold $M^\iota$.} \proof All gluing functions from one coordinate system compatible with the real structure to another {\bf \purple commute with $\iota$, acting on coordinate functions as the complex conjugation.} This gives $\Psi_{ij}(\bar z_i)= \overline {\Psi_{ij}(z_i)}$. Therefore, $\Psi_{ij}$ preserve $M^\iota$, and are expressed by real-valued functions on $M^\iota$. \endproof \newpage {\bf \blue Real analytic manifolds and real structures 2 (reminder)} \proposition {\bf \red Any real analytic manifold can be obtained from this construction.} \pstep Let $\{U_i\}$ be a locally finite atlas of a real analytic manifold $M$, and $\Psi_{ij} :\; U_{ij}\arrow U_{ij}$ the gluing maps. We realize $U_i$ as an open ball with compact closure in $\Re(\C^n)=\R^n$. By local finiteness, there are only finitely many such $\Psi_{ij}$ for any given $U_i$. Denote by $B_\epsilon$ an open ball of radius $\epsilon$ in the $n$-dimensional real space $\im(\C^n)$. {\bf \green Step 2:} Let $\epsilon>0$ be a sufficiently small real number such that all $\Psi_{ij}$ can be extended to gluing functions $\tilde \Psi_{ij}$ on the open sets $\tilde U_i:=U_i\times B_\epsilon\subset \C^n$. {\bf \purple Then $(\tilde U_i, \Psi_{ij})$ is an atlas for a complex manifold $M_\C$.} Since all $\Psi_{ij}$ are real, they are preserved by the natural involution acting on $B_\epsilon$ as $-1$ and on $U_i$ as identity. This involution defines a real structure on $M_\C$. Clearly, $M$ is the set of its fixed points. \endproof \newpage {\bf \blue Complexification} \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 of coordinate monomials with real analytic coefficients. \claim Let $M_\R$ be a real analytic manifold, $(M_\C, \iota)$ 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 $M_\C$.} Moreover, {\bf \red $\Phi$ is real on $M_\R$ if $\iota^*\Phi_\C =\bar\Phi_\C$.} \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 in coordinates by a sum of tensorial 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 Categories} \newcommand{\Ob}{\operatorname{\mathcal {Ob}}} \newcommand{\Mor}{\operatorname{\mathcal{Mor}}} \definition {\bf \blue A category} $\cac$ is a collection of data called ``objects'' and ``morphisms between objects'' which satisfies the axioms below. {\bf \green DATA.}\\ \phantom{XX} {\bf \red Objects:} A class $\Ob(\cac)$ of {\bf \blue objects} of $\cac$. \\ \phantom{XX} {\bf \red Morphisms:} For each $X, Y\in \Ob(\cac)$, one has a set $\Mor(X,Y)$ of {\bf \blue morphisms from $X$ to $Y$}.\\ \phantom{XX} {\bf \red Composition of morphisms:} For each $\phi\in \Mor(X,Y), \psi \in \Mor(Y,Z)$ there exists {\bf \blue the composition} $\phi\circ \psi \in \Mor(X, Z)$\\ \phantom{XX} {\bf \red Identity morphism:} For each $A\in \Ob(\cac)$ there exists a morphism $\Id_A \in \Mor(A,A)$. {\bf \green AXIOMS.}\\ \phantom{XX} {\bf \red Associativity of composition:} $\phi_1\circ(\phi_2\circ\phi_3)=(\phi_1\circ\phi_2)\circ\phi_3$.\\ \phantom{XX} {\bf \red Properties of identity morphism:} For each $\phi\in \Mor(X,Y)$, one has $\Id_x\circ \phi = \phi = \phi\circ \Id_Y$\\ \newpage {\bf \blue Categories (2)} \definition Let $X, Y\in \Ob(\cac)$ -- objects of $\cac$. A morphism $\phi\in \Mor(X,Y)$ is called {\bf \blue an isomorphism} if there exists $\psi\in \Mor(Y,X)$ such that $\phi \circ \psi = \Id_X$ and $\psi\circ\phi = \Id_Y$. In this case, the objects $X$ and $Y$ are called {\bf \blue isomorphic}. {\bf \green Examples of categories:}\\ {\bf \purple Category of sets:} its morphisms are arbitrary maps.\\ {\bf \purple Category of vector spaces:} its morphisms are linear maps.\\ {\bf \purple Categories of rings, groups, fields:} morphisms are homomorphisms.\\ {\bf \purple Category of topological spaces:} morphisms are continuous maps.\\ {\bf \purple Category of smooth manifolds:} morphisms are smooth maps. \newpage {\bf \blue Functors} \definition Let $\cac_1, \cac_2$ be two categories. A {\bf \blue covariant functor} from $\cac_1$ to $\cac_2$ is the following set of data. 1. {\bf \purple A map $F:\; \Ob(\cac_1)\arrow \Ob(\cac_2)$}. 2. {\bf \purple A map $F:\; \Mor(X,Y) \arrow \Mor(F(X), F(Y))$ defined for any pair of objects $X, Y\in \Ob(\cac_1)$.} These data define a functor if they are {\bf \purple compatible with compositions,} that is, satisfy $F(\phi) \circ F(\psi) = F(\phi\circ\psi)$ for any $\phi\in \Mor(X,Y)$ and $\psi\in \Mor(Y,Z)$, and {\bf \purple map identity morphism to identity} morphism. \newpage {\bf \blue Small categories} \remark This way, one could speak of {\bf \blue category of all categories}, with categories as objects and functors as morphisms. {\bf \red A caution} To avoid set-theoretic complications, Grothendieck added another axiom to set theory, ``universum axiom'', postulating existence of ``universum'', a very big set, and worked with ``small categories'' -- categories where the set of all objects and sets of morphisms belong to the universum. In this sense, ``category of all categories'' is not a ``small category'', because the set of its object (being comparable to the set of all subsets of the universum) is too big to fit in the universum. In practice, mathematicians say ``category'' when they mean ``small category'', tacitly assuming that any given category is ``small''. This is why not many people call ``category of all categories'' a category: nobody wants to deal with set-theoretic complications. \newpage {\bf \blue Example of functors} {\bf \purple A ``natural operation'' on mathematical objects is usually a functor.} Examples: 1. A map $X \arrow 2^X$ from the set $X$ to the set of all subsets of $X$ is a functor from the category $\Sets$ of sets to itself. 2. A map $M \arrow M^2$ mapping a topological space to its product with itself is a functor on topological spaces. 3. A map $V \arrow V \oplus V$ is a functor on vector spaces; same for a map $V \arrow V\otimes V$ or $V \arrow (V \oplus V)\otimes V$. 4. {\bf \blue Identity functor} from any category to itself. 5. A map from topological spaces to $\Sets$, putting a topological space to the set of its connected components. \exercise {\bf \purple Prove that it is a functor.} \newpage {\bf \blue Equivalence of functors} \definition Let $X, Y\in \Ob(\cac)$ be objects of a category $\cac$. A mprphism $\phi\in \Mor(X,Y)$ is called {\bf \blue an isomorphism} if there exists $\psi\in \Mor(Y,X)$ such that $\phi \circ \psi = \Id_X$ and $\psi\circ\phi = \Id_Y$. In this case $X$ and $Y$ are called {\bf \blue isomorphic}. \definition Two functors $F, G:\;\cac_1\arrow \cac_2$ are called {\bf\blue equivalent} if for any $X \in \Ob(\cac_1)$ we are given an isomorphism $\Psi_X:\; F(X) \arrow G(X)$, in such a way that for any $\phi\in \Mor(X,Y)$, one has $ F(\phi) \circ \Psi_Y= \Psi_X\circ G(\phi)$. \remark Such commutation relations are usually expressed by {\bf \blue commutative diagrams}. For example, the condition $F(\phi) \circ \Psi_Y= \Psi_X\circ G(\phi)$ is expressed by a commutative diagram \begin{equation*} \begin{CD} F(X) @>{F(\phi)}>> F(Y)\\ @V{\Psi_X}VV @VV{\Psi_Y}V\\ G(X) @>{G(\phi)}>> G(Y) \end{CD} \end{equation*} \newpage {\bf \blue Equivalence of categories} \definition A functor $F:\; \cac_1 \arrow \cac_2$ is called {\bf \blue equivalence of categories} if there exists a functor $G:\;\cac_2 \arrow \cac_1$ such that the compositions $G\circ F$ and $G\circ F$ are equivaleent to the identity functors $\Id_{\cac_1}$, $\Id_{\cac_2}$. \remark It is possible to show that this is equivalent to the following conditions: {\bf \purple $F$ defines a bijection on the set of isomorphism classes of objects of $\cac_1$ and $\cac_2$, and a bijection \[ \Mor(X,Y) \arrow \Mor(F(X), F(Y)).\] for each $X, Y \in \Ob(\cac_1)$}. \remark From the point of view of category theory, {\bf \red equivalent categories are two instances of the same category} (even if the cardinality of corresponding sets of objects is different). \newpage {\bf \blue Germ of a complex manifold} \definition Let $K\subset M$ be a closed subset of a complex manifold, homeomorphic to $K_1\subset M_1$, where $M_1$ is also a complex manifold. Fixing the homeomorphism $K\cong K_1$, we may identify these sets and consider $K$ as a subset $M_1$. We say that $M$ and $M_1$ {\bf\blue have the same germ in $K$} if there exist biholomorphic open subsets $U_1 \subset M_1$ and $U\subset M$ containing $K$, with the biholomorphism $\phi:\; U \arrow U_1$ identity on $K$. \definition {\bf \blue Germ of a manifold $M$ in $K\subset M$} is an equivalence class of open subsets $U \subset M$ containing $K$, with this equivalence relation. \definition Consider category $\cac_\iota$, with objects complex manifolds $(M, \iota)$ equipped with a real structure, and morphisms holomorphic maps commuting with $\iota$. \theorem {\bf \blue (Grauert)} {\bf \red Category of real analytic manifolds is equivalent to the category of germs of $M\in \cac_\iota$ in $M^\iota\subset M$.} \exercise {\bf \purple Prove this theorem.} \newpage {\bf \blue Hans Grauert} \begin{center} \epsfig{file=Grauert.png,width=0.40\linewidth}\\[10mm] { \it \small\green Hans Grauert in Bonn, 2000\\ (8.02.1930 - 4.09.2011)} \end{center} \newpage {\bf \blue Extension of tensors to a complexification} {\bf \green Lemma 1:} Let $X$ be an open ball in $\C^n$ equipped with the standard anticomplex involution, $X_\R=X \cap \R^n$ its fixed point set, and $\alpha$ a holomorphic tensor on $X$ vanishing in $X_\R$. {\bf \red Then $\alpha=0$.} \proof {\bf \purple Any holomorphic function which vanishes on $\R^n$ has all its derivatives vanishing.} Therefore its Taylor serie vanish. Such a function vanishes on $\C^n$ by analytic continuation principle. This argument can be applied to all coefficients of $\alpha$. \endproof \definition An almost complex structure $I$ on a real analytic manifold is {\bf \blue real analytic} if $I$ is a real analytic tensor. \corollary Let $(M,I)$ be a real analytic almost complex manifold, $M_\C$ its complexification, and $I_\C:\; TM_\C \arrow TM_\C$ the holomorphic extension of $I$ to $M_\C$. {\bf \red Then $I_\C^2 = -\Id$.} {\bf \green Proof:} {\bf \purple The tensor $I_\C^2 +\Id$ is holomorphic and vanishes on $M_\R$,} hence the previous lemma can be applied. \endproof \newpage {\bf \blue Underlying real analytic manifold} \remark {\bf \red A complex analytic map $\Phi:\; \C^n \arrow \C^n$ is real analytic as a map $\R^{2n} \arrow \R^{2n}$.} Indeed, the coefficients of $\Phi$ are real and imaginary parts of holomorphic functions, and real and imaginary parts of holomorphic functions can be expressed as Taylor series of the real variables. \definition Let $M$ be a complex manifold. The {\bf \blue underlying real analytic manifold} $M_\R$ is the same manifold, with the same gluing functions, considered as real analytic maps. \remark The sheaf of real analytic functions on $M_\R$ can be defined as {\bf \purple the sheaf of converging power series generated by holomorphic and antiholomorphic functions.} Indeed, such functions are real analytic in any of the real analytic map; conversely, {\bf \purple any real analytic function on $M_\R$ is a converging power serie on $\Re z_i, \Im z_i$, where $z_i$ are holomorphic coordinates on $M$.} \newpage {\bf \blue Complexification of the underlying real analytic manifold} \definition Let $M$ be a complex manifold. The {\bf \blue complex conjugate manifold} is the same manifold with almost complex structure $-I$ and antiholomorphic functions on $M$ holomorphic on $\bar M$. \claim Let $M$ be an integrable almost complex manifold. Denote by $M_\R$ its underlying real analytic manifold. {\bf \red Then a complexification of $M_\R$ can be given as $M_\C:=M \times \bar M$,} with the anticomplex involution $\tau(x,y)=(y,x)$. \proof Clearly, the fixed point set of $\tau$ is the diagonal, identified with $M_\R =M$ as usual. Both holomorphic and antiholomorphic functions on $M_\R$ are obtained as restrictions of holomorphic functions from $M_\C$, hence the sheaf of real analytic functions on $M_\R$ is a subsheaf of $\calo_{M_\C}$ of holomorphic functions on $M_\C$ restricted to $M_\R$. \endproof \newpage {\bf \blue Integrability of almost complex structures (reminder)} \definition An almost complex structure $I$ on a manifold is called {\bf\blue integrable} if any point of $M$ has a neighbourhood $U$ diffeomorphic to an open subset of $\C^n$, in such a way that the almost complex structure $I$ is induced by the standard one on $U\subset \C^n$. \claim {\bf \red Complex structure on a manifold $M$ uniquely determines an integrable almost complex structure, and is determined by it.} \proof Complex structure on a manifold $M$ is determined by the sheaf of holomorphic functions $\calo_M$, and $\calo_M$ is determined by $I$ as explained above. Therefore, an integrable almost complex structure defines a complex structure. Conversely, every complex structure gives a sub-bundle in $\Lambda^{1,0}(M)=d\calo_M\subset \Lambda^1(M,\C)$, and {\bf \purple such a sub-bundle defines an almost complex structure by Remark 1 in Lecture 1.} \endproof \newpage {\bf \blue Formal integrability (reminder)} \definition An almost complex structure $I$ on $(M,I)$ is called {\bf\blue formally integrable} if $[T^{1,0}M, T^{1,0}]\subset T^{1,0}$, that is, if $T^{1,0}M$ is involutive. \definition The Frobenius form $\Psi\in \Lambda^2(\Lambda^{1,0}M)\otimes T^{0,1}M$ is called {\bf \blue the Nijenhuis tensor}. \claim {\bf \purple If a complex structure $I$ on $M$ is integrable, it is formally integrable.} \proof Locally, the bundle $T^{1,0}(M)$ is generated by $d/dz_i$, where $z_i$ are complex coordinates. These vector fields commute, hence satisfy $[d/dz_i, d/dz_j]\in T^{1,0}(M)$. This means that the Frobenius form vanishes. \endproof \theorem {\bf \blue (Newlander-Nirenberg)}\\ {\bf\red A complex structure $I$ on $M$ is integrable if and only if it is formally integrable.} \proof (real analytic case) this lecture. \remark {\bf \purple In dimension 1, formal integrability is automatic.} Indeed, $T^{1,0}M$ is 1-dimensional, hence all skew-symmetric 2-forms on $T^{1,0}M$ vanish. \newpage {\bf \blue Holomorphic and antiholomorphic foliations} \definition Let $B\subset TM$ be a sub-bundle. The {\bf \blue foliation associated with $B$} is a family of submanifolds $X_t\subset U$, defined for each sufficiently small subset of $M$, called {\bf \blue the leaves of the foliation}, such that $B$ is the bundle of vectors tangent to $X_t$. In this case, $X_t$ are called {\bf \blue the leaves} of the foliation. \remark The famous ``Frobenius theorem'' says that {\bf \red $B$ is involutive if and only if it is tangent to a foliation.} \remark Let $(M,I)$ be a real analytic almost complex manifold, and $M_\C$ its complexification. Replacing $M_\C$ by a smaller neighbourhood of $M$, we may assume that the tensor $I$ is extended to an endomorphism $I:\; TM_\C \arrow TM_\C$, $I^2=-\Id$. {\bf \purple Since $TM_\C$ is a complex vector bundle, $I$ acts there with the eigenvalues $\1$ and $-\1$, giving a decomposition $TM_\C= T^{1,0}M_\C\oplus T^{0,1}M_\C$} \definition {\bf \blue Holomorphic foliation} is a foliation tangent to $T^{1,0}M_\C$, {\bf \blue antiholomorphic foliation} is a foliation tangent to $T^{0,1}M_\C$. \newpage {\bf \blue Antiholomorphic foliation on $M_\C=M\times \bar M$.} \remark Let $(M,I)$ be a integrable almost complex manifold, $M_\C =M\times \bar M$ its complexification, and $\pi, \bar \pi$ projections of $M_\C$ to $M$ and $\bar M$. {\bf \purple Then the fibers of $\bar\pi$ is a holomorphic foliation, and the fibers of $\pi$ is a holomorphic foliation.} \remark Let $TM_\C = T' \oplus T''$ be a decomposition of $TM_\C$ onto part tangent to fibers of $\bar \pi$ and tangent to fibers of $\pi$. {\bf \purple On $M_\R$ the decomposition $TM_\C = T' \oplus T''$ coincides with the decomposition $TM\otimes \C= T^{1,0}M\oplus T^{0,1}M$.} \corollary Let $(M,I)$ be a integrable almost complex manifold. {\bf \red Then $I$ is a real analytic almost complex structure.} \proof Extend $I$ to an operator on $M_\C$ acting as $\1$ on $T'$ and $-\1$ on $T''$. This operator is complex analytic because the decomposition $TM= T' \oplus T''$ is holomorphic. \endproof {\bf \green Corollary 1:} Let $(M,I)$ be a real analytic almost complex manifold. Then holomorphic functions on $M_\C$ which are constant on the leaves of antiholomoirphic foliation {\bf \red restrict to holomorphic functions on $(M,I)\subset M_\C$.} {\bf \green Proof:} Such functions are constant in the $(0,1)$-direction on $TM\otimes \C$. \endproof \newpage {\bf \blue Integrability of real analytic almost complex structure} %\theorem {\bf \blue (``linearization of a vector field'')} %Let $v\in TM$ be a nowhere vanishing vector field on $M$. %{\bf \purple Then there exists a family of 1-dimensional submanifolds %passing through each point of $M$ such that $v$ is tangent %to these submanifolds at each point of $M$.} \theorem {\bf \blue (Newlander-Nirenberg for real analytic manifolds)} Let $(M,I)$ be a real analytic almost complex manifold, $\dim_\R M=2$. {\bf \red Then $M$ is integrable.} \pstep Consider the complexification $M_\C$ of $M$, and let $TM_\C = T^{1,0}M_\C\oplus T^{0,1}M_\C$ be the decomposition defined above. By Frobenius theorem, there exists a foliation tangent to $T^{0,1}M_\C$ and one tangent to $T^{1,0}M_\C$. Since the leaves of these foliations are transversal, {\bf \purple locally $M_\C$ is a product of $M'$ and $M''$ which are identified with the space of leaves of $T^{0,1}M_\C$ and $T^{1,0}M_\C$.} {\bf \green Step 2:} Locally, functions on $M'$ can be lifted to $M'\times M''=M_\C$, giving functions which are constant on the leaves of the foliation tangent to $T^{0,1}M_\C$. By Corollary 1, such functions are holomorphic on $(M,I)$. Choose a collection of $n=\frac 1 2 \dim_\R M$ holomorphic functions $f_1, ... f_n$ on $M_\C$ which are constant on the leaves of $T^{0,1}M_\C$ and have linearly independent differentials in $x\in M\subset M_\C$. By inverse function theorem, {\bf \purple $f_1, ..., f_n$ is a holomorphic coordinate system in a neigbourhood of $x\in (M,I)$,} and the transition functions between such coordinate systems are by construction holomorphic. \endproof \end{document}