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%version 1.0,\ \   14.10.2020
%version 1.1,\ \   15.10.2020: A couple of corrections from Nikita Klemyatin



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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{4}{Complex geometry handout 4: K\"ahler metrics
on homogeneous spaces}

%{\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
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%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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{\small 
In this and all subsequential handouts, you are allowed to invoke the 
Newlander-Nirenberg theorem. 
}

\definition
A ``homogeneous
almost complex manifold'' is a manifold equipped 
with a transitive Lie group action preserving the
almost complex structure. In the same way one defines
objects such as ``homogeneous Riemannian manifold''
or ``homogeneous K\"ahler manifold'': in all these
cases the Lie group acts transitively, preserving
the relevant geometric structure.
\ed

\exercise
Let $(M,I)=G/H$ be a homogeneous
almost complex manifold, $x\in M$, and $h\in \St_{x}(M)$
an element of the isotropy group, acting on $T_xM$
as $2\Id$, that is, $h(x)=2x$. Prove that $I$ is integrable.
\ez 

\exercise
Let $M$ be a homogeneous Riemannian manifold, $x\in M$, 
and $h\in \St_{x}(M)$ an element of the isotropy group,
acting on on $T_xM$ as identity. Prove that $h$ acts
trivially on $M$.
\ez 


\exercise
Construct a homogeneous, compact complex manifold $M$, $\dim_\C M=2$,
not admitting a K\"ahler metric.
\ez


\exercise
Find a complex structure on $SU(2)\times SU(2)$. Can it be K\"ahler?
\ez

\exercise
Find a complex structure on $SU(3)$. Can it be K\"ahler?
\ez

\exercise[*]
Let $G$ be a compact, semisimple Lie group, and $T$ its
maximal torus. Assume that $\dim_\R T$ is even. Prove that
$G$ admits a left-invariant complex structure.
\ez

\exercise
Construct a $U(1,n)$-invariant metric and complex structure
on $M:=\frac{U(1,n)}{U(1)\times U(n)}$. 
Prove that it is K\"ahler.
Prove that $M$ is biholomorphic to an open ball in $\C^n$.
\ez

\remark This metric on an open ball is called {\bf Bergman metric},
or {\bf complex hyperbolic metric}.
\er

\exercise
Construct an $SO(n+2)$-invariant K\"ahler structure on 
the Grassmannian $\Gr_\R(2,n):=\frac{SO(n+2)}{SO(n)\times SO(2)}$.
\ez





\end{document}
