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\begin{document}
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\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}
