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%version 1.0,\ \   09.10.2020
%version 2.0,\ \   13.10.2020, last two exercises were wrong
%version 2.1,\ \   20.10.2020, formally integrable in 6b


\newcommand{\version}{version 2.1,\ \   20.10.2020}
\newcommand{\firstdate}{10.10.2020}

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\lhead{\tiny Complex geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{3}{Complex geometry handout 3: 
real analytic manifolds and integrability}

%{\scriptsize
%  {\bf Rules:} You may choose to solve only 
%``hard'' exercises (marked with !, * and **) 
%or ``ordinary'' ones (marked with ! or unmarked),
%or both, if you want to have extra stuff to work.
%To have a perfect score, a student must obtain
%(in average) a score of 10 points per week.
%
%If you have got credit for 2/3 of ordinary problems
%or 2/3 of ``hard'' problems, you receive  
%$6t$ points, where $t$ is a number depending on the
%date when it is done. Passing all ``hard'' 
%or all ``ordinary'' problems brings you $10t$ points.
%Solving of ``**'' (extra hard) problems is not
%obligatory, but each such problem gives you a credit
%for 2 ``*'' or ``!'' problems in the ``hard'' set.
%
%The first 3 weeks after giving a handout, $t=1.5$,
%between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
%The scores are not cumulative, only the
%best score for each handout counts.
%}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this handout, you are not allowed to invoke the 
Newlander-Nirenberg theorem.

\exercise
Let $\iota:\; M \arrow M$ be a real structure on a complex manifold.
Prove that $\iota$ maps complex submanifolds of $M$ to complex submanifolds.
\ez

\exercise
Let $M$ be a smooth manifold with a smooth action of a finite group $G$.
Prove that the fixed point set of $G$ is a smooth submanifold 
or find a counterexample.
\ez


\exercise
Let $M$ be a smooth manifold with a smooth action of the group $\Z$.
Prove that the fixed point set of $\Z$ is a smooth submanifold 
or find a  counterexample.
\ez

\definition
Let $(M,I)$ be an almost complex manifold.
Recall that $I$ is {\bf integrable} if $M$
admits a complex structure such that
in local holomorphic coordinates $z_1, ..., z_n$,
with $I$ mapping $dx_i$ to $dy_i$ and 
$dy_i$ to $-dx_i$, where $x_i$, $y_i$ are
real and imaginary parts of $z_i = x_i +\1 y_i$.
An almost complex structure $I$ is {\bf formally
integrable} if $[T^{1,0}M, T^{1,0}(M)]\subset T^{1,0}M$.
\ed

\exercise
Let $(M,I)$ be an almost complex manifold.
Assume that the bundle $\Lambda^{1,0}(M)$
is generated by differentials of holomorphic
functions. Prove that the almost complex structure
on $M$ is integrable.
\ez

\exercise
Let $(M,I)$ be an almost complex manifold,
and $d=d^{2,-1}+d^{1,0}+d^{0,1}+d^{-1,2}$
the Hodge decomposition of the de Rham differential.
\enum
\ite Prove that $d^{2,-1}$ and $d^{-1,2}$ are $C^\infty(M)$-linear.
\ite Assume that $d^{2,-1}=d^{-1,2}=0$. Prove that 
$I$ is formally integrable.
\ee
\ez

\definition
Let $G$ be a real Lie group, and $\g=T_e G$ its Lie algebra.
{\bf The left action} of $G$ on itself
is a map $L_g(x)=gx$, defined for any $g\in G$.
An almost complex structure is called
{\bf left invariant} if it is preserved by $L_g$
for all $g\in G$. 
\ed

\exercise
Let $G$ be a real Lie group, and $\g=T_e G$ its Lie algebra,
and $\g_\C:= \g\otimes_\R \C$ its complexification.
\enum
\ite
Prove that any complex structure operator on $\g=T_e G$
is uniquely extended to a left-invariant 
almost complex structure on $G$.
\ite
Let $I\in \End T_eG$ be a complex structure,
and $\g_\C= \g^{1,0}\oplus \g^{0,1}$ the
Hodge decomposition of its complexification.
Prove that $I$ defines a formally integrable
left-invariant complex structure on $G$
if and only if $\g^{1,0}\subset \g_\C$
is a Lie subalgebra.
\ee
\ez

\exercise
Let $(M,I)$ be an almost complex manifold.
\enum
\ite Earlier,
we denoted  the weight (-1,2) Hodge component of de Rham differential
by $d^{-1,2}$. Prove that 
$d^{-1,2}:\; \Lambda^{1,0}(M)\arrow \Lambda^{0,2}(M)$ is dual to the
complex conjugate of the
Nijenhuis tensor $\bar N:\; \Lambda^2(T^{0,1}M) \arrow T^{1,0}(M)$.
\ite
Prove that the operator
$d^{-1,2}$ satisfies $d^{-1,2}(d\phi)=0$
for any holomorphic function $\phi$.
\ee
\ez

\exercise
Let $(M,I)$ be an almost complex manifold.
Assume that the Nijenhuis tensor 
$\Lambda^2(T^{1,0}M) \arrow T^{0,1}(M)$
is surjective.  Prove that $(M,I)$
admits no local holomorphic functions.
\ez

\hint Use the previous exercise.
\eh

\exercise[*]
Construct a left-invariant 
almost complex structure $I$ on a Lie group $G$
such that $(G,I)$ admits no local holomorphic
functions.
\ez

\hint Use the previous exercise.
\eh




%\exercise
%Let $\iota:\; \C P^1 \arrow \C P^1$ map a point $x:y$ to 
%$\bar y^{-1}: \bar x^{-1}$. Prove that $\iota$ is a real structure
%which has no fixed points. Prove that the set $U\subset \C P^1\times \overline{\C P^1}$
%obtained as a complement to the ``antidiagonal'' $\{(x, \iota(x))\ \ |\ \ x\in \C P^1\}$
%is biholomorphic to the total space of the tangent bundle to $\C P^1$.
%\ez

 
\end{document}