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%version 1.0,\ \   25.06.2020
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% of almost complex manifolds, 01.10.2020}


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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{1}{Complex geometry handout 1: Complex structure operators}

%{\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.
%}

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\exercise
Prove that the group $GL(n, \C)$ of complex
automorphisms of $\C^n$ is connected.
\ez

\definition
Let $V$ be a vector space. A {\bf complex structure}
on $V$ is an operator $I\in \End(V)$ which satisfies
$I^2=-\Id$.
\ed

\exercise
Let $V$ be a real 4-dimensional vector space,
equipped with a Euclidean metric, 
and $S$ the space of all orthogonal
complex structures $I\in \End(V)$.
Prove that $S$ is a disconnected union
of two 2-dimensional spheres.
\ez

\exercise
Let $V$ be a real 4-dimensional vector space,
equipped with a scalar product $g$ of signature
$(3,1)$. Prove that the space $\Lambda^2(V)$
of anti-symmetric 2-forms is equipped with
an $SO(3,1)$-invariant complex structure.
\ez

\exercise
Let $V$ be a 4-dimensional real vector space,
and $U$ the space of antisymmetric 2-forms
$\Omega$ on $V\otimes_\R \C$ such that 
$\Omega\wedge \Omega=0$ and $\Omega\wedge \bar \Omega\neq 0$.
Denote by ${\Bbb P}U$ the projectivization of $U$,
that is, the quotient $U/\C ^*$, under the standard
$\C^*$-action. Construct a $GL(V)$-invariant
bijective correspondence between ${\Bbb P}U$
and the space of complex structures on $V$.
\ez

\definition
Let $M$ be a manifold. An endomorphism $I\in \End(TM)$, $I^2=-\Id_{TM}$
is called {\bf an almost complex structure}. An $I$-invariant
Riemannian form is called {\bf Hermitian form}.
A smooth map $\phi$ of almost complex manifold is called
{\bf holomorphic} if ifs differential commutes with $I$.  
\ed

\definition
Let $g_1, g_2$ be Riemannian metrics on a smooth manifold $M$.
They are said to be {\bf conformal}, or {\bf conformally
equivalent} if there exists a smooth function $\lambda\in C^\infty M$
such that $g_1 = \lambda\cdot g_2$. {\bf Conformal structure}
is a metric up to conformal equivalence.
\ed


\exercise
Let $I$ be an almost complex structure on a 
manifold $M$ of real dimension 2. Prove that
all Hermitian metrics on $(M,I)$ are
conformally equivalent.
\ez 

\exercise
Let $(M,I)$ be an almost complex manifold, $\dim_\C M=n$
Prove that $(M,I)$ always admits a Hermitian 
metric $g$. Consider the orientation form $\omega^n$, obtained as
the top exterior power of the corresponding Hermitian form $\omega$.
Prove that the orientation defined by $\omega^n$
is independent from the choice of $g$.
\ez



\exercise
Let $f:\; M \arrow N$ be an oriented diffeomorphism
of almost complex Hermitian manifolds
of real dimension 2. Prove that
$f$ is holomorphic if and only
if it preserves the conformal structure.
\ez


\exercise
Prove that the space of almost complex structures
on a 2-dimensional manifold is a disconnected 
union of two contractible sets.
\ez

\exercise
Let $M$ be a manifold admitting a non-degenerate 2-form.
Prove that $M$ admits an almost complex structure.
\ez

\exercise
Prove that the space of almost complex structures
is homotopy equivalent to the space of 
non-degenerate 2-forms.
\ez

 
\end{document}