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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
