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Step 1:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Hodge theory \\[15mm] \small lecture 10: Newlander-Nirenberg theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize NRU HSE, Moscow} \\[15mm] {\small Misha Verbitsky, February 24, 2018 } \end{center} \newpage {\bf \blue Almost complex manifolds (reminder)} \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 $I(d/dx_i)=d/dy_i$, $I(d/dy_i)=-d/dx_i$. \definition Let $(V,I)$ be a space equipped with a complex structure $I:\; V \arrow V$, $I^2 =-\Id$. {\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. \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 Complex manifolds and almost complex manifolds (reminder)} \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 (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 \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 $\calo_M$, because $d\calo_M$ generates $\Lambda^{1,0}(M)$, and $\calo_M$ is determined by $I$, because $\calo_M=\{f\ \ |\ \ df\in \Lambda^{1,0}(M)\}$. \endproof \newpage {\bf \blue Frobenius form (reminder)} \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 $\Psi=0$. \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 $\Psi\in \Lambda^{2,0}M\otimes TM$ 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) later today. \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 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 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 $dx_i\restrict m= xy_i\restrict m$, 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} \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 $\lambda\in \C$. \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 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 $\C^n$.} \newpage {\bf \blue Real analytic manifolds and real structures} \proposition Let $M_\R\subset M_\C$ be a fixed point set of an antiholomorphic involution $\iota$, $U_i$ a complex analytic atlas, and $\Psi_{ij}:\; U_{ij}\arrow U_{ij}$ the gluing functions. {\bf \red Then, for an appropriate choice of coordinate systems all $\Psi_{ij}$ are real analytic on $M_\R$, and define a real analytic atlas on the manifold $M_\R$.} \pstep Let $z_1, ..., z_n$ be a holomorphic coordinate system on $M_\C$ in a neighbourhood of $m\in M_\R$ such that $\iota(dz_i)= d\bar z_i$ in $T_m^*M$. Such a coordinate system can be chosen by taking linear functions with prescribed differentials in $m$. {\bf \purple Replacing $z_i$ by $x_i:=z_i+\iota^*(\bar z_i)$, we obtain another coordinate system $x_i$ on $M$} (compare with Theorem 1). {\bf \green Step 2:} This new coordinate system satisfies $\iota^*x_i= \bar x_i$, hence $M_\R$ in these coordinates is giving by equation $\im x_1=\im x_2 = ... = \im x_n=0$. {\bf \purple All gluing functions from such coordinate system to another one of this type satisfy $\Psi_{ij}(\bar z_i)= \overline {\Psi_{ij}(\bar z_i)}$, hence they are real on $M_\R$.} \endproof \newpage {\bf \blue Real analytic manifolds and real structures (2)} \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 map. 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 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, $M_\C$ 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 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 is equal zero.} Therefore its Taylor series 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 anaytic 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} is the same manifold, with the same gluing functions, considered as real analytic maps. \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 real part of the sheaf $\calo_{M_\C}$ of holomorphic functions on $M_\C$. \endproof \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. {\bf \green Frobenius Theorem:} {\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 $M_\C=M\times \bar M$.} \claim 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.} \proof 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 $TM\otimes \C= T^{1,0}M\oplus T^{0,1}M$.} By Lemma 1 the same is true everywhere on $M_\C$. \endproof \corollary Let $(M,I)$ be a integrable almost complex manifold. {\bf \red Then $I$ is a real analytic almost complex structure.} \proof It was extended to $M_\C$ in the previous claim. \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 antiholomorphic 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 structures} \theorem Let $(M,I)$ be a real analytic almost complex manifold. {\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 $(M,I)$. Choosing a function with linearly independent differentials in $x\in M$, it would give a {\bf \purple holomorphic coordinate system in a neigbourhood of $(M,I)$,} and the transition functions between such coordinate systems are by construction holomorphic. \endproof \end{document}