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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 9: Complex manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize NRU HSE, Moscow} \\[15mm] {\small Misha Verbitsky, February 21, 2018 } \end{center} \newpage {\bf \blue Complex structure on vector spaces} \definition Let $V$ be a vector space over $\R$, and $I:\; V \arrow V$ an automorphism which satisfies $I^2 = - \Id_V$. Such an automorphism is called {\bf\blue a complex structure operator} on $V$. {\bf \blue We extend the action of $I$ on the tensor spaces $V\otimes V \otimes ... \otimes V \otimes V^*\otimes V^* \otimes ... \otimes V^*$ by multiplicativity:} $I(v_1 \otimes ... \otimes w_1 \otimes ... \otimes w_n)= I(v_1) \otimes ... \otimes I(w_1)\otimes ... \otimes I(w_n)$. {\bf \red Trivial observations:}\\ 1. {\bf \purple The eigenvalues $\alpha_i$ of $I$ are $\pm \1$.} Indeed, $\alpha_i^2=-1$. 2. {\bf \purple $V$ admits an $I$-invariant, positive definite scalar product (``metric'') $g$.} Take any metric $g_0$, and let $g:= g_0 + I(g_0)$. 3. {\bf \purple $I$ is orthogonal for such $g$.}\\ Indeed, $g(Ix,Iy)=g_0(x,y)+ g_0(Ix,Iy)=g(x,y)$. 4. {\bf \purple $I$ diagonalizable over $\C$.} Indeed, any orthogonal matrix is diagonalizable. 5. {\bf \purple There are as many $\1$-eigenvalues as there are $-\1$-eigenvalues.} \newpage {\bf \blue Comples structure operator in coordinates} This implies that in an appropriate basis in $V \otimes_\R \C$, the almost complex structure operator is diagonal, as follows: \[\left[ \begin{array}{c|c} \begin{array}{cccc} \1 & & &\\ & \1 & & \\ & & \ddots & \\ & & & \1 \end{array} & 0 \\ \hline 0 & \begin{array}{cccc} -\1 & & &\\ & -\1 & & \\ & & \ddots & \\ & & & -\1 \end{array} \end{array}\right] \] We also obtain its normal form in a real basis: \[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \\ & & 0 & -1 \\ & & 1 & 0 \\ & & & & \ddots \\ & & & & & \ddots \\ & & & & & & 0 & -1 \\ & & & & & & 1 & 0 \end{bmatrix} \] \newpage {\bf \blue Hermitian structures} % Denote by $\nu$ the {\bf \blue real structure operator,} %$\nu(\sum \lambda_i w_i)=\sum \bar \lambda_i w_i$, where $w_i\in V$ %is a basis. Then $\nu(I(z))=I(\nu(z))$, that is, {\bf \red $I$ is real}. %For any $\1$-eigenvector %$w$, one has $I(\nu(w))=\nu(I(w))=\nu(\1 w)=-\1 w$, %hence {\bf \purple %$\nu$ exchanges $\1$-eigenvectors and $-\1$-eigenvectors.} \definition An $I$-invariant positive definite scalar product on $(V,I)$ is called {\bf \blue an Hermitian metric}, and $(V,I,g)$ -- an Hermitian space. \remark Let $I$ be a complex structure operator on a real vector space $V$, and $g$ -- a Hermitian metric. Then {\bf \red the bilinear form $\omega(x,y) := g(x, Iy)$ is skew-symmetric.} Indeed, $\omega(x,y) = g(x, Iy) = g(Ix, I^2y) = -g(Ix, y) = -\omega(y, x)$. \definition A skew-symmetric form $\omega(x,y)$ is called {\bf \blue an Hermitian form on $(V,I)$}. \remark In the triple $I, g, \omega$, {\bf \purple each element can recovered from the other two.} \newpage {\bf \blue The Grassmann algebra} \definition Let $V$ be a vector space. Denote by $\Lambda^i V$ the space of antisymmetric polylinear $i$-forms on $V^*$, and let $\Lambda^*V:=\bigoplus \Lambda^i V$. Denote by $T^{\otimes i}V$ the algebra of {\bf \green all} polylinear $i$-forms on $V^*$ (``tensor algebra''), and let $\Alt:\; T^{\otimes i}V\arrow \Lambda^i V$ be {\bf \blue the antisymmetrization}, \[\Alt(\eta)(x_1, ..., x_i):= \frac 1{i!}\sum_{\sigma\in \Sigma_i}(-1)^{\tilde \sigma}\eta(x_{\sigma_1}, ..., x_{\sigma_i}) \] where $\Sigma_i$ is the group of permutations, and $\tilde \sigma=1$ for odd permutations, and 0 for even. Consider the multiplicative operation (``wedge-product'') on $\Lambda^*V$, denoted by $\eta\wedge \nu:= \Alt (\eta \otimes \nu)$. The space $\Lambda^*V$ with this operation is called {\bf \blue the Grassmann algebra}. \remark {\bf \red It is an algebra of anti-commutative polynomials.} {\bf \green Properties of Grassmann algebra:} 1. $\dim \Lambda^i V:= \binom{\dim V}{i}$, $\dim \Lambda^* V=2^{\dim V}$. 2. $\Lambda^*(V \oplus W)= \Lambda^*(V) \otimes \Lambda^*(W)$. \newpage {\bf \blue The Hodge decomposition in linear algebra} \definition Let $(V,I)$ be a space equipped with a complex structure. {\bf \blue The Hodge decomposition} $V\otimes_\R \C:= V^{1,0}\oplus V^{0,1}$ is defined in such a way that $V^{1,0}$ is a $\1$-eigenspace of $I$, and $V^{0,1}$ a $-\1$-eigenspace. \remark Let $V_\C:= V \otimes_\R \C$. The Grassmann algebra of skew-symmetric forms $\Lambda^n V_\C:=\Lambda^n_\R V \otimes _\R C$ admits a decomposition \[ \Lambda^n V_\C= \bigoplus_{p+q=n} \Lambda^p V^{1,0} \otimes \Lambda^q V^{0,1} \] We denote $\Lambda^p V^{1,0} \otimes \Lambda^q V^{0,1}$ by $\Lambda^{p,q}V$. The resulting decomposition $\Lambda^n V_\C= \bigoplus_{p+q=n}\Lambda^{p,q}V$ is called {\bf \blue the Hodge decomposition of the Grassmann algebra}. \newpage {\bf \blue $U(1)$-representations and the weight decomposition } \remark The operator $I$ induces $U(1)$-action on $V$ by the formula $\rho(t)(v) = \cos t\cdot v + \sin t \cdot I(v)$. We extend this action on the tensor spaces by muptiplicativity. \remark {\bf \red Any complex representation $W$ of $U(1)$ is written as a sum of 1-dimensional representations $W_i(p)$}, with $U(1)$ acting on each $W_i(p)$ as $\rho(t)(v) = e^{\1 pt}(v)$. The 1-dimensional representations are called {\bf \blue weight $p$ representations of $U(1)$.} \definition A {\bf \blue weight decomposition} of a $U(1)$-representation $W$ is a decomposition $W= \oplus W^p$, where each $W^p=\oplus_i W_i(p)$ is a sum of 1-dimensional representations of weight $p$. \remark {\bf \red The Hodge decomposition $\Lambda^n V_\C= \bigoplus_{p+q=n}\Lambda^{p,q}V$ is a weight decomposition}, with $\Lambda^{p,q}V$ being a weight $p-q$-component of $\Lambda^n V_\C$. \remark $V^{p,p}$ is the space of $U(1)$-invariant vectors in $\Lambda^{2p}V$. Further on, {\bf \blue $TM$ is the tangent bundle on a manifold, and $\Lambda^iM$ the space of differential $i$-forms.} It is a Grassmann algebra on $TM$. \newpage {\bf \blue Holomorphic functions} \definition Let $I:\; TM \arrow TM$ be an endomorphism of a tangent bundle satisfying $I^2=-\Id$. Then $I$ is called {\bf \blue almost complex structure operator}, and the pair $(M,I)$ {\bf \blue an almost complex manifold}. \example $M=\C^n$, with complex coordinates $z_i=x_i + \1 y_i$, and $I(d/dx_i)=d/dy_i$, $I(d/dy_i)=-d/dx_i$. \example In complex dimension 1, {\bf \purple almost complex structure is the same as conformal structure with orientation} {\bf \red (prove it).} \definition A function $f:\; M \arrow \C$ on an almost complex manifold is called {\bf \blue holomorphic} if $df \in \Lambda^{1,0}(M)$. \remark For some almost complex manifolds, {\bf \red there are no holomorphic functions at all}, even locally. Example: $S^6$ with a certain canonical ($G_2$-invariant) almost complex structure. \newpage {\bf \blue Holomorphic functions on $\C^n$} \theorem Let $f:\; M \arrow \C$ be a differentiable function on an open subset $M\subset \C^n$, with the natural almost complex structure. {\bf \red Then the following are equivalent.}\\ \ \ (1) {\bf \purple $f$ is holomorphic}.\\ \ \ (2) The differential $df:\; TM \arrow \C$, considered as a form on the vector space $T_x M=T_x\C^n=\C^n$ {\bf \purple is $\C$-linear.} \\ \ \ (3) For any complex affine line $L\in \C^n$, the restriction $f\restrict L=\C$ is {\bf \purple holomorphic (complex analytic) as a function of one complex variable.}\\ \ \ (4) $f$ is expressed as a sum of Taylor series around any point $(z_1, ..., z_n)\in M$: \[ f(z_1+t_1, z_2+ t_2, ..., z_n +t_n)= \sum_{i_1, ..., i_n}a_{i_1, ..., i_n} t_1^{i_1}t_2^{i_2}...t_n^{i_n}. \] (here we assume that the complex numbers $t_i$ satisfy $|t_i|<\epsilon$, where $\epsilon$ depends on $f$ and $M$). \proof (1) and (2) are tautologically equivalent. Equivalence of (1) and (3) is also clear, because a restriction of $\theta \in \Lambda^{1,0}(M)$ to a line is a $(1,0)$-form on a line, and, conversely, if $df$ is of type (1,0) on each complex line, it is of type (1,0) on $TM$, which is implied by the following linear-algebraic observation. \newpage {\bf \blue Holomorphic functions on $\C^n$ (2)} \theorem Let $f:\; M \arrow \C$ be a differentiable function on an open subset $M\subset \C^n$, with the natural almost complex structure. {\bf \red Then the following are equivalent.}\\ \ \ (1) {\bf \purple $f$ is holomorphic}.\\ \ \ (2) The differential $df:\; TM \arrow \C$, considered as a form on the vector space $T_x M=T_x\C^n=\C^n$ {\bf \purple is $\C$-linear.} \\ \ \ (3) For any complex affine line $L\in \C^n$, the restriction $f\restrict L=\C$ is {\bf \purple holomorphic (complex analytic) as a function of one complex variable.}\\ \ \ (4) $f$ is expressed as a sum of Taylor series around any point $(z_1, ..., z_n)\in M$. \lemma Let $\eta\in V^*\otimes \C$ be a complex-valued linear form on a vector space $(V,I)$ equipped with a complex structure. {\bf \purple Then $\eta\in \Lambda^{1,0}(V)$ if and only if its restriction to any $I$-invariant 2-dimensional subspace $L$ belongs to $\Lambda^{1,0}(L)$.} \exercise {\bf \red Prove it.} (4) clearly implies (2). (1) implies (4) by Cauchy formula. \newpage {\bf \blue Taylor decomposition from Cauchy formula} {\bf \purple Taylor series decomposition on a line is implied by the Cauchy formula:} \[ \int_{\6 \Delta} \frac{f(z)dz}{z-a}= 2\pi\1 f(a), \] where $\Delta\subset \C$ is a disk, $a\in \Delta$ any point, and $z$ coordinate on $\C$. Indeed, in this case, \[ 2\pi\1 f(a) = \sum_{i\geq 0} a^i\int_{\6 \Delta} f(z) (z^{-1})^{i+1}, \] because $\frac{1}{z-a}=z^{-1}\sum_{i\geq 0} (az^{-1})^i$. \newpage {\bf \blue Cauchy formula} Let's prove Cauchy formula, using Stokes' theorem. Since the space $\Lambda^{1,0}\C$ is 1-dimensional, $df\wedge dz=0$ for any holomorphic function on $\C$. This gives \claim A function on a disk $\Delta\subset \C$ {\bf \red is holomorphic if and only if the form $\eta:=f dz$ is closed} (that is, satisfies $d\eta=0$). \endproof Now, let $S_\epsilon$ be a radius $\epsilon$ circle around a point $a\in \Delta$, $\Delta_\epsilon$ its interior, and $\Delta_0:=\Delta\backslash \Delta_\epsilon$. Stokes' theorem gives \[ 0= \int_{\Delta_0} d\left(\frac{f(z)dz}{z-a}\right)= -\int_{S_\epsilon}\frac{f(z)dz}{z-a} + \int_{\6 \Delta} \frac{f(z)dz}{z-a}, \] hence Cauchy formula would follow if we show that $\lim\limits_{\epsilon\rightarrow 0}\int_{S_\epsilon}\frac{f(z)dz}{z-a}=2\pi\1 f(a).$ Assuming for simplicity $a=0$ and parametrizing the circle $S_\epsilon$ by $\epsilon e^{\1 t}$, we obtain \begin{multline*} \int_{S_\epsilon}\frac{f(z)dz}{z}= \int_0^{2\pi} \frac{f(\epsilon e^{\1t})}{\epsilon e^{\1t}} d(\epsilon e^{\1t})=\\ =\int_0^{2\pi} \frac{f(\epsilon e^{\1t})}{\epsilon e^{\1t}} \1 \epsilon e^{\1t}dt =\int_0^{2\pi}f(\epsilon e^{\1t})\1 dt \end{multline*} as $\epsilon$ tends to $0$, $f(\epsilon e^{\1t})$ tends to $f(0)$, and this integral goes to $2\pi\1f(0)$. \newpage {\bf \blue Sheaves} \definition A {\bf\blue presheaf of functions} on a topological space $M$ is a collection of subrings ${\cal F}(U)\subset C(U)$ in the ring $C(U)$ of all functions on $U$, for each open subset $U\subset M$, such that the restriction of every $\gamma\in{\cal F}(U)$ to an open subset $U_1\subset U$ belongs to ${\cal F}(U_1)$. \definition A presheaf of functions ${\cal F}$ is called {\bf\blue a sheaf of functions} if these subrings satisfy the following condition. Let $\{U_i\}$ be a cover of an open subset $U\subset M$ (possibly infinite) and $f_i\in{\cal F}(U_i)$ a family of functions defined on the open sets of the cover and compatible on the pairwise intersections: $$f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$$ for every pair of members of the cover. {\bf \purple Then there exists $f\in{\cal F}(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for all $i$.} \newpage {\bf \blue Sheaves and presheaves: examples} {\bf \green Examples of sheaves:} * Space of continuous functions * Space of smooth functions, any differentiability class * Space of real analytic functions {\bf \green Examples of presheaves which are not sheaves:} * Space of constant functions {\bf \purple (why?)} * Space of bounded functions {\bf \purple (why?)} \newpage {\bf \blue Ringed spaces} A {\bf\blue ringed space} $(M,{\cal F})$ is a topological space equipped with a sheaf of functions. A~{\bf \blue morphism} $(M,{\cal F})\stackrel\Psi\longrightarrow(N,{\cal F}')$ of ringed spaces is a continuous map $M\stackrel\Psi\longrightarrow N$ such that, for every open subset $U\subset N$ and every function $f\in{\cal F}'(U)$, the function $\psi^* f:=f\circ\Psi$ belongs to the ring ${\cal F}\big(\Psi^{-1}(U)\big)$. An {\bf\blue isomorphism} of ringed spaces is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ are morphisms of ringed spaces. \example Let $M$ be a manifold of class $C^i$ and let $C^i(U)$ be the space of functions of this class. {\bf \purple Then $C^i$ is a sheaf of functions, and $(M, C^i)$ is a ringed space.} \remark Let $f:\; X \arrow Y$ be a smooth map of smooth manifolds. Since a pullback $f^*\mu$ of a smooth function $\mu\in C^\infty(M)$ is smooth, {\bf \purple a smooth map of smooth manifolds defines a morphism of ringed spaces.} \newpage {\bf \blue Complex manifolds} \definition {\bf \blue A holomorphic function} on $\C^n$ is a function $f:\; \C^n \arrow \C$ such that $df$ is complex linear, that is $df\in \Lambda^{1,0}(M)$. \remark Holomorphic functions form a sheaf. \definition {\bf \blue A complex manifold} $M$ is a ringed space which is locally isomorphic to an open ball in $\C^n$ with a sheaf of holomorphic functions. \remark In other words, {\bf \purple $M$ is covered with open balls embedded to $\C^n$} and transition functions (being coordinate functions for one ball considered in other coordinate system) {\bf \purple are holomorphic.} \newpage {\bf \blue Complex manifolds and almost complex manifolds} \definition {\bf \blue Standard almost complex structure} is $I(d/dx_i)=d/dy_i$, $I(d/dy_i)=-d/dx_i$ on $\C^n$ with complex coordinates $z_i=x_i+\1 y_i$. \definition A map $\Psi:\; (M,I)\arrow (N,J)$ from an almost complex manifold to an almost complex manifold is called {\bf \blue holomorphic} if $\Psi^*(\Lambda^{1,0}(N))\subset \Lambda^{1,0}(M)$. \remark This is the same as $d\Psi$ being complex linear; for standard almost complex structures, {\bf \purple this is the same as the coordinate components of $\Psi$ being holomorphic functions.} \definition {\bf \blue A complex manifold} is a manifold equipped with an atlas with charts identified with open subsets of $\C^n$ and transition functions holomorphic. \newpage {\bf \blue Integrability of almost complex structures} \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 \purple 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. \endproof \newpage {\bf \blue Frobenius form} \claim Let $B\subset TM$ be a sub-bundle of a tangent bundle of a smooth manifold. Given vector fiels $X,Y\in B$, consider their commutator $[X,Y]$, and lets $\Psi(X,Y)\in TM/B$ be the projection of $[X,Y]$ to $TM/B$. {\bf \red Then $\Psi(X,Y)$ is $C^\infty(M)$-linear in $X$, $Y$: } \[ \Psi(fX,Y)= \Psi(X,fY)=f\Psi(X,Y). \] \proof Leibnitz identity gives $[X,fY]=f[X,Y]+ X(f) Y$, and the second term belongs to $B$, hence does not influence the projection to $TM/B$. \endproof \definition This form is called {\bf \blue the Frobenius form} of the sub-bundle $B\subset TM$. This bundle is called {\bf\blue involutive}, or {\bf \blue integrable}, or {\bf \blue holonomic} if $\Psi=0$. \exercise {\bf \purple Give an example of a non-integrable sub-bundle.} \newpage {\bf \blue Formal integrability} \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 $\Psi\in \Lambda^{2,0}M\otimes TM$ 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) next 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 Distributions} \definition {\bf \blue Distribution} on a manifold is a sub-bundle $B\subset TM$ \remark Let $\Pi:\; TM \arrow TM/ B$ be the projection, and $x, y \in B$ some vector fields. Then $[fx, y]= f[x,y] - D_y (f) x$. This implies that {\bf \purple $\Pi([x,y])$ is $C^\infty(M)$-linear as a function of $x$ and $y$.} \definition The map $[B,B]\arrow TM/B$ we have constructed is called {\bf \blue Frobenius bracket} (or {\bf \blue Frobenius form}); it is a skew-symmetric $C^\infty(M)$-linear form on $B$ with values in $TM/B$. \definition A distribution is called {\bf \blue integrable}, or {\bf \blue holonomic}, or {\bf \blue involutive}, if its Frobenuus form vanishes. \newpage {\bf \blue Smooth submersions} \definition Let $\pi:\; M \arrow M'$ be a smooth map of manifolds. This map is called {\bf \blue submersion} if at each point of $M$ the differential $D\pi$ is surjective, and {\bf \blue immersion} if it is injective. \claim Let $\pi:\; M \arrow M'$ be a submersion. Then each $m\in M$ has a neighbourhood $U\cong V\times W$, where $V, W$ are smooth and {\bf \red $\pi \restrict U$ is a projection of $V\times W=U\subset M$ to $W\subset M'$ along $V$.} \exercise {\bf \purple Deduce this result from the inverse function theorem.} \exercise {\bf \blue (``Ehresmann's fibration theorem'')}\\ Let $\pi:\; M \arrow M'$ be a smooth submersion of compact manifolds. Prove that $\pi$ is a locally trivial fibration. \definition {\bf \blue Vertical tangent space} $T_\pi M\subset TM$ of a submersion $\pi:\; M \arrow M'$ is the kernel of $D\pi$. \claim Let $\pi:\; M \arrow M'$ be a submersion and $T_\pi M\subset TM$ the vertical tangent space. {\bf \red Then $T_\pi M$ is an involutive subbundle.} \proof $D_\pi([X, Y])= [D_\pi(X), D_\pi(Y)]=0$ for any $X, Y \in \ker D_\pi$. \endproof \newpage {\bf \blue Frobenius theorem (statement)} {\bf \green Frobenius Theorem:} Let $B\subset TM$ be a sub-bundle. Then $B$ is involutive if and only if each point $x\in M$ has a neighbourhood $U\ni x$ and {\bf \red a smooth submersion $U\stackrel \pi \arrow V$ such that $B$ is its vertical tangent space: $B= T_\pi M$.} \remark The implication {\bf \blue ``$B=T_\pi M$'' $\Rightarrow$ ``Frobenius form vanishes''} was proven above. \definition The fibers of $\pi$ are called {\bf \blue leaves}, or {\bf \blue integral submanifolds} of the distribution $B$. Globally on $M$, {\bf \blue a leaf of $B$} is a maximal connected manifold $Z\hookrightarrow M$ which is immersed to $M$ and tangent to $B$ at each point. A distribution for which Frobenius theorem holds is called {\bf \blue integrable}. If $B$ is integrable, the set of its leaves is called {\bf \blue a foliation}. The leaves are manifolds which are immersed to $M$, but not necessarily closed. \newpage {\bf \blue Frobenius theorem: existence of integral submanifolds} \remark To prove the Frobenius theorem for $B\subset TM$, {\bf \purple it suffices to show that each point is contained in an interal submanifold.} In this case, the smooth submersion $U\stackrel \pi \arrow V$ is a projection to the leaf space of the distribution. \remark {\bf \red When $B$ is 1-dimensional} (in this case one says that {\bf \blue $B$ has rank 1}, denoted $\rk B=1$), {\bf \red Frobenius theorem follows from existence of the diffeomorphism flow associated with a vector field.} Indeed, locally we may assume that $B$ admits a non-degenerate section $v$. Let $V_t:\; M \times \R \arrow M$ be the corresponding flow of diffeomorphisms. Then $Z_m:=V_t(\{m\}\times \R$ is tangent to $v$ everywhere, hence it is a 1-dimensional manifold immersed in $M$. Clearly, $Z_m$ is a leaf this distribution. {\bf \purple Since $B$ is a tangent to a foliation, it is integrable.} Further on we shall need the following exercise. \exercise Let $V_t= e^{v_t}$ be a diffeomorphism flow on $M$, and $F\subset TM$ a vector bundle. Assume that $[v_t, F]\subset F$. {\bf \red Prove that then $V_t$ preserves $F\subset TM$.} \newpage {\bf \blue Basic sub-bundles (1)} \definition Let $B\subset TM$ be an involutive sub-bundle. A sub-bundle $F\subset TM$ is called {\bf \blue basic} for $B$ if $F\supset B$ and for all $b\in B, b'\in F$, one has $[b, b']\in F$. \remark One should think of basic sub-bundles as of {\bf \purple sub-bundles preserved by all diffeomorphisms obtaned from exponentiation of a vector field $v\in B$.} \lemma Let $B\subset TM$ be an integrable distribution, $\pi:\; M \arrow M_1$ projection to the leaf space of $B$, and $F\supset B$ a sub-bundle of $TM$ containing $B$. Then the following conditions are equivalent: {\bf \red (a) $F$ is basic for $B$.\\ (b) There exists a sub-bundle $F_1\subset TM_1$ such that $\pi^{-1}(F_1) = F$.} \proof Next slide. \newpage {\bf \blue Basic sub-bundles (2)} \lemma Let $B\subset TM$ be an integrable distribution, $\pi:\; M \arrow M_1$ projection to the leaf space of $B$, and $F\supset B$ a sub-bundle of $TM$ containing $B$. Then the following conditions are equivalent: {\bf \red (a) $F$ is basic for $B$.\\ (b) There exists a sub-bundle $F_1\subset TM_1$ such that $\pi^{-1}F_1 = F$.} \pstep Consider coordinates $x_1, ..., x_n$ on $M$ such that $x_{k+1}=\pi^*(x_{k+1}', ..., x_n=\pi^*(x_n)$, where $x_i', i= k+1, k+2,..., n$ are coordinates on $M_1$, and $\frac {d}{dx_1}, ..., \frac {d}{dx_k}$ generate $B$. Locally such coordinates always exist, because $B$ is integrable. Denote by $G$ a subgroup of $\Diff(M)$ obtained by exponents of $\frac {d}{dx_1}, ..., \frac {d}{dx_k}$. Since $[B, F]\subset F$, the corresponding diffeomorphisms preserve $F$. {\bf \purple Therefore, $F$ is a $G$-invariant sub-bundle of $TM$.} {\bf \green Step 2:} {\bf \purple Any $G$-invariant sub-bundle $F\supset B$ is obtained as $\pi^{-1}(F_1)$ for some sub-bundle $F_1 \subset TM_1= M/G$.} Indeed, since the action of $G_1$ is free, the bundle $F$ is generated over $C^\infty M$ by $G$-invariant sections. However, any $G$-invariant bundle $F$ containing $B$ is generated by $G$-invariant sections, which can be lifted from $M/G$ {\bf \purple (check this)}. {\bf \green Step 3:} Conversely, if $F$ is lifted from $M_1=M/G$, it is $G$-invariant, hence $e^{tb}(b')\subset F$, and this gives $[b, b']\subset F$ {\bf \purple (check this)}. \endproof \newpage {\bf \blue Frobenius theorem (proof)} {\bf \green Frobenius Theorem:} Let $B\subset TM$ be a sub-bundle. Then $B$ is involutive if and only if each point $x\in M$ has a neighbourhood $U\ni x$ and {\bf \red a smooth submersion $U\stackrel \pi \arrow V$ such that $B$ is its vertical tangent space: $B= T_\pi M$.} \pstep Consider a rank 1 sub-bundle $B_1 \subset B$. Using the diffeomorphism flow as above, we prove that $B_1$ is integrable. Since $[B_1, B]\subset B$, the bundle $B$ is basic with respect to $B_1$. {\bf \purple Therefore, $B= \pi^{-1}(B')$ for some $B'\subset TM_1$, where $M_1$ is the leaf space of $B_1$.} {\bf \green Step 2:} Let $\pi:\; M \arrow M_1$ be the projection to the leaf space. Then $B= \pi^{-1}(B')$, where $\rk B'=\rk B-1$. Using induction in $\rk B$, we can assume that $B'$ is integrable. Let $\pi_0:\; M_1 \arrow M_0$ be the projection to the leaf space of $B'$, defined locally in $M$. {\bf \purple Then $\pi\circ\pi_0:\; M \arrow M_0$ is the projection to the leaf space of $B$.} \endproof \end{document}