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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{\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 4: Another proof of Frobenius theorem; real analytic manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] October 3, 2020 } \end{center} \newpage {\bf \blue Distributions (reminder)} \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 Frobenius form vanishes. \newpage {\bf \blue Smooth submersions (reminder)} \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 Proof of Frobenius Theorem: Lie group action} \definition A {\bf \blue Lie group} is a smooth manifold equipped with a group structure in such a way that the group operations are smooth. {\bf \blue An action} of a Lie group $G$ on a manifold $M$ is a smooth map $G\times M \arrow M$ inducing the group action. \definition {\bf\blue The Lie algebra} of a Lie group is an algebra of left-invariant vector fields. Since the left action of $G$ on itself is free and transitive, {\bf \purple the Lie algebra of $G$ is in bijective correspondence with $T_e G$.} \remark Let $\rho:\; G\times M \arrow M$ be a Lie group action on a manifold $M$. Then $d\rho$ induces a map from the Lie algebra of $G$ to the Lie algebra of the manifold $M$. {\bf \purple The corresponding vector fields are tangent to the orbits of $G$.} {\bf \green Claim 1:} Suppose that $G$ is a Lie group acting on a manifold $M$. Let $B\subset TM$ be sub-bundle generated by the vector fields from Lie algebra of $G$. Then {\bf \red $B$ is integrable,} that is, Frobenius theorem holds of $B\subset TM$. \proof The orbits of the $G$-action are tangent to $B\subset TM$. \endproof \newpage {\bf \blue Proof of Frobenius Theorem: preliminaries} {\bf \green Exercise 1:} Let $u, v$ be commuting vector fields on a manifold $M$, and $e^{tu}$, $e^{tv}$ be corresponding diffeomorphism flows. {\bf \red Prove that $e^{tu}$, $e^{tv}$ commute.} {\bf \green Remark 1:} Let $\pi:\; M \arrow M_1$ be a smooth submersion, and $v\in TM$ a vector fields which satisfies \[ d\pi(v)\restrict x= d\pi(v)\restrict y \ \ \ \ (*) \] for any $x, y\in \pi^{-1}(z)$ and any $z\in M_1$. {\bf \purple In this case, the vector field $d\pi(v)$ is well defined on $M$. } {\bf \green Exercise 2:} Let $\pi:\; M \arrow M_1$ be a smooth submersion, and $u, v\in TM$ vector fields which satisfy (*). Consider the vector fields $u_1:= d\pi(u)$ and $v_1:= d\pi(v)\in TM_1$ defined as in Remark 1. {\bf \purple Prove that the commutator $[u, v]$ satisfies (*) and, moreover, $[u_1, v_1]= d\pi([u, v])$.} \newpage {\bf \blue Proof of Frobenius Theorem: commuting vector fields} {\bf \green Exercise 1:} Let $u, v$ be commuting vector fields on a manifold $M$, and $e^{tu}$, $e^{tv}$ be corresponding diffeomorphism flows. {\bf \red Prove that $e^{tu}$, $e^{tv}$ commute.} {\bf \green Solution. Step 1:} The statement is local, and trivial in any open set where $u=0$, hence it suffices to prove it in a coordinate chart where $u$ is non-degenerate. Since all non-degenerate vector fields can be linearized, {\bf \purple we can always assume that the vector field $u$ is a coordinate vector field, $u = d/dx_1$.} Then $e^{tu}(x_1, ..., x_n)=(x_1+t, ..., x_n)$. {\bf \green Step 2:} For any vector field $v=\sum_i a_i d/dx_i$, one has $\left[u, \sum_i a_i \frac d{dx_i}\right]= \sum \frac{da_i}{dx_1} \frac d{dx_i}$. Therefore, {\bf \purple $[u, v]=0$ is equivalent to the coefficients $a_i$ being constant in $x_1$}. This implies that the parallel transport along $x_1$ preserves $v$. Therefore, it also preserves $e^{tv}$, and the corresponding diffeomorphisms commute. \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 $m\in M$ has a neighbourhood $U\ni m$ and {\bf \red a smooth submersion $U\stackrel \pi \arrow V$ such that $B$ is its vertical tangent space: $B= T_\pi M$.} \pstep The ``if'' part is clear. The statement of Frobenius Theorem is local, hence we may replace $M$ be a small neighbourhood of a given point. We are going to show that $B$ locally has a basis of commuting vector fields. By Exercise 1, {\bf \purple these vector fields can be locally integrated to a commutative group action,} and Frobenius Theorem follows from Claim 1. {\bf \green Step 2:} Consider a smooth submersion $M \arrow M_1$ inducing an isomorphism from $B$ to $TM_1$. Let $\zeta_1, ..., \zeta_k$ be the coordinate vector fields on $M_1$. Since \\ $d\sigma:\; B\restrict x \arrow T_{\sigma(x)}M_1$ is an isomorphism, there exist unique vector fields $\xi_1, ..., \xi_k\in B$ such that $d\sigma(\xi_i)=\zeta_i$. By Exercise 2, $d\sigma([\xi_1, \xi_j]) = [\zeta_i, \zeta_j]=0$. However, $[\xi_1, \xi_j]$ is a section of $B$, and $d\sigma:\; B\restrict m \arrow T_{\sigma(m)}M_1$ is an isomorphism, hence $d\sigma([\xi_1, \xi_j])=0$ implies $[\xi_1, \xi_j]=0$. We have shown that {\bf \purple $B$ admits a basis of commuting vector fiels.} \endproof \newpage {\bf \blue Complex manifolds (reminder)} \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 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.} \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) next lecture, probably. \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 Real analytic manifolds} \definition {\bf \blue A real analytic function} on an open set $U\subset \R^n$ is a function which admits a Taylor expansion near each point $x\in U$: \[ f(z_1+t_1, z_2+ t_2, ..., z_n +t_n)= \sum_{i_1, ..., i_n}a_{i_1, ..., i_n} t_1^{i_1}t_2^{i_2}...t_n^{i_n}. \] (here we assume that the real numbers $t_i$ satisfy $|t_i|<\epsilon$, where $\epsilon$ depends on $f$ and $M$). \remark Clearly, {\bf \purple real analytic functions constitute a sheaf.} \definition A {\bf\blue real analytic manifold} is a ringed space which is locally isomorphic to an open ball $B\subset \R^n$ with the sheaf of of real analytic functions. \newpage {\bf \blue Involutions} \definition {\bf \blue An involution} is a map $\iota:\; M \arrow M$ such that $\iota^2=\Id_M$. \exercise Prove that {\bf \purple any linear involution on a real vector space $V$ is diagonalizable,} with eigenvalues $\pm 1$. {\bf \green Theorem 1:} Let $M$ be a smooth manifold, and $\iota:\; M \arrow M$ an involutiin. {\bf \red Then the fixed point set $N$ of $\iota$ is a smooth submanifold.} \pstep {\bf \blue Inverse function theorem.} Let $m\in M$ be a point on a smooth $k$-dimensional manifold and $f_1, ..., f_k$ functions on $M$ such that their differentials $df_1, ..., df_k$ are linearly independent in $m$. Then $f_1, ..., f_k$ {\bf \purple define a coordinate system in a neighbourhood of $a$, giving a diffeomorphism of this neighbourhood to an open ball.} {\bf \green Step 2:} Assume that $d\iota$ has $k$ eigenvalues 1 on $T_m M$, and $n-k$ eigenvalues -1. Choose a coordinate system $x_1, ..., x_n$ on $M$ around a point $m\in N$ such that $dx_1\restrict m, ..., dx_k\restrict m$ are $\iota$-invariant and $dx_{k+1}\restrict m, ..., dx_n\restrict m$ are $\iota$-anti-invariant. Let $y_1=x_1+ \iota^*x_1$, $y_2=x_2+ \iota^*x_2$, ... $y_k=x_k+ \iota^*x_k$, and $y_{k+1}=x_{k+1}- \iota^*x_{k+1}$, $y_{k+2}=x_{k+2}- \iota^*x_{k+2}$, ... $y_n=x_{n}-\iota^*x_{n}$. Since $dx_i\restrict m= dy_i\restrict m$, these differentials are linearly independent in $m$. By Step 1, {\bf \purple functions $y_i$ define an $\iota$-invariant coordinate system on an open neighbourhood of $m$, with $N$ given by equations $y_{k+1}=y_{k+2}=...= y_n=0$.} \endproof \newpage {\bf \blue Real structures on complex vector spaces} \definition {\bf \blue An involution} is a map $\iota:\; M \arrow M$ such that $\iota^2=\Id_M$. {\bf \blue A real structure} on a complex vector space $V=\C^n$ is an $\R$-linear involution $\iota:\; V \arrow V$ such that $\iota(\lambda x) = \bar\lambda \iota(x)$ for any $\lambda\in \C$. \claim Let $\iota$ be a real structure on a complex vector space $V$, and $V^\iota\subset V$ the space of $V$-invariant vectors. {\bf \red Then $\dim_\R V^\iota=\dim_\C V$, and $V= V^\iota\otimes_\R \C$.} \pstep Let $x_1, ..., x_n$ be a basis in $V^\iota$, and $\sum_i \alpha_i x_i=0$ a linear relation in $V$, with $\alpha_i \in \C$. Then $0=\iota\left(\sum_i\alpha_i x_i\right)= \sum_i\bar\alpha_i x_i$. Averaging these two relations, we obtain $\sum \Re\alpha_i x_i=0$. Since $x_i$ are linearly independent over $\R$, this implies $\Re\alpha_i=0$ for all $i$. Applying the same argument to $\sum_i \1\alpha_i x_i=0$, we obtain that $\Im\alpha_i=0$ for all $i$. Then {\bf \purple the natural map $V^\iota\otimes_\R \C\arrow V$ is injective.} {\bf \green Step 2:} This map is also surjective. Indeed, for any $v\in V$, one has $\frac 1 2 (v+\iota(v))\in V^\iota$ and $\frac {\1} 2 (v-\iota(v))\in V^\iota$, hence $v$ can b expressed as a linear combination of vectors from $V^\iota$ with complex coefficients. \endproof \newpage {\bf \blue Real structures on complex manifolds} \definition A map $\Psi:\; M \arrow M$ 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 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 $\C^n$.} \end{document}