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%version 1.3,\ \   16.05.2018, many errors found by Ivan Frolov
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\lhead{\tiny Hogde theory, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{12}{Hodge theory 12: Complex and K\"ahler structures on symmetric 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
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.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Almost complex, Hermitian and K\"ahler structures}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $M$ be a manifold. An endomorphism $I\in \End(TM)$, $I^2=-\Id_{TM}$
is called {\bf an almost complex structure}, and its $\1$-eigenbundle
is denoted as $T^{1,0}M\subset TM\otimes \C$.
An almost complex structure $I$ is called {\bf integrable}
if $[T^{1,0}M,T^{1,0}M]\subset  T^{1,0}M$. In this case
$(M,I)$ is called {\bf a complex manifold}.
A Riemannian metric on an almost complex manifold
is called {\bf Hermitian} if it is $I$-invariant.
\ed

\exercise
Let $U= V\oplus W$ be vector spaces.
Prove that their Grassmann algebras are decomposed
as follows: $\Lambda^n(U)= \bigoplus_{p+q=n}\Lambda^pV \otimes \Lambda^q W$.\footnote{This decomposition is not multiplicative.}
\ez

\definition
Consider the eigenvalue decomposition
$\Lambda^1(M, \C)= \Lambda^{1,0}(M)\oplus \Lambda^{0,1}(M)$
associated with the action of $I$, with 
$I\restrict{\Lambda^{1,0}(M)}=\1$ and $I\restrict{\Lambda^{0,1}(M)}=-\1$. 
It induces the decomposition on the de Rham algebra
\[
 \Lambda^k(M,\C)= \bigoplus_{p+q=k}\Lambda^p(\Lambda^{1,0}(M)) \otimes 
 \Lambda^q (\Lambda^{0,1}(M))
\]
as shown above. The bundles $\Lambda^p(\Lambda^{1,0}(M))$ and
$\Lambda^q (\Lambda^{0,1}(M))$ are denoted $\Lambda^{p,0}(M)$
and $\Lambda^{0,q}(M)$, and the component $\Lambda^p(\Lambda^{1,0}(M)) \otimes 
 \Lambda^q (\Lambda^{0,1}(M))$ is denoted $\Lambda^{p,q}(M)$.
The decomposition $\Lambda^k(M,\C)=\bigoplus_{p+q=k}\Lambda^{p,q}(M)$
is called {\bf the Hodge decomposition}, the sections
of $\Lambda^{p,q}(M)$ are called $(p,q)$-forms.
\ed

\exercise
Let $(M,I)$ be an almost complex manifold, and
$h$ an $I$-invariant Riemannian form.
\enum 
\ite Prove that $\omega(x, y)=h(Ix, y)$
is a (1,1)-form. 
\ite[!] Prove that any Hermitian form $h$ is
obtained from a (1,1)-form $\omega$ such that $\omega(x, Ix)> 0$
for all non-zero tangent vectors $x\in T_mM$.
\ee
\ez

\exercise
Prove that any almost complex manifold admits a Hermitian
metric. 
\ez

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

\definition
Let $(M, I, h)$ be an almost complex Hermitian manifold
The form $\omega(x, y)=h(Ix, y)$ is called {\bf the fundamental
form} of $M$. The triple $(M, I, \omega)$ is called
{\bf a K\"ahler triple} if $I$ is integrable and $\omega$ is closed.
In this case $M$ is called {\bf the K\"ahler manifold},
$h$ {\bf the K\"ahler metric} and $\omega$
{\bf the K\"ahler form.}
\ed

\remark
Recall that {\bf symplectic manifold}
is a manifold equipped with a non-degenerate, closed
2-form. Clearly, the K\"ahler form is closed
and non-degenerate.
\er

\exercise
Find a complex, compact manifold not admitting a 
K\"ahler metric.
\ez

\exercise[**]
Find a complex, compact manifold not admitting a 
K\"ahler metric, but admitting a symplectic structure.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Symmetric spaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
{\bf Homogeneous space} is a manifold with transitive
action of a Lie group (often assumed connected).
\ed

\exercise
Let $M$ be a connected manifold with transitive action of a Lie group $G$,
and $H$ be a stabilizer of a point $x\in M$ (in this case,
$H$ is called {\bf the isotropy group} of $x$).
\enum
\ite
Prove that $M$ is identified with the space of orbits $G/H$.
\ite[!] Let $x, y\in M$, and $H_x, H_y$ be the corresponding
isotropy groups. Prove that $H_x$ and $H_y$ are conjugate by
some element of $G$.
\ee
\ez

\exercise
Let $M=G/H$ be a homogeneous space with compact $H$. Assume that
$M$ is connected and all non-unit $g\in G$ act non-trivially.
\enum
\ite[!] Prove that $M$ admits a $G$-invariant Riemannian structure.
\ite[*] Prove that the natural map from the isotropy group
of $x$ to $GL(T_x M)$ is injective.
\ite[**] Is it always injective if $H$ is not necessarily compact?
\ee
\ez

\definition
{\bf A tensor} on a manifold $M$ is a section of the
{\bf tensor bundle} $TM^{\otimes p}\otimes T^*M^{\otimes q}$.
Whenever $G$ acts on $M$ by diffeomorphisms, it acts
on the space of tensors, because tensors are functorial.
\ed 

\exercise[!]
Let $M=G/H$ be a homogeneous space, and $H_x$ the isotropy group
of $x\in M$. Construct a bijective correspondence between $G$-invariant
tensors on $M$ and $H_x$-invariant vectors in 
$T_xM^{\otimes p}\otimes T^*_xM^{\otimes q}$.
\ez

\definition
A homogeneous space $M=G/H$ is called {\bf  symmetric space}
if $M$ admits a $G$-invariant Riemannian metric and
$H_x$ contains an involution $\iota$ which acts
as $-\Id$ on $T_x M$.
\ed

\definition
$SO(n)$ denotes the {\bf special orthogonal} group
(the group of all orthogonal matrices preserving the orientation).
$U(n)$ is {\bf unitary group} (the group of all complex-linear
matrices preserving a Hermitian form). $SU(n)$ is intersection
of $U(n)$ and $SL(n,\C)$.
\ed

\exercise
Consider the spaces $S^{2n}$, ${\Bbb C} P^n$, ${\Bbb H} P^n$
equipped with the natural action of $SO(2n+1)$, $U(n+1)$
and $Sp(n+1):= GL(n+1, {\Bbb H})\cap SO(4n+4)$.
Prove that they are symmetric spaces.
\ez

\exercise
Consider the Grassmannian 
$\Gr_\R(p,q):=\frac{SO(p+q)}{SO(p)\times SO(q)}$,
\enum
\ite[!] Prove that it is a symmetric space when  $p$ or $q$ is even.
\ite[*] Prove it for all $p, q$.
\ee
\ez



\definition
An {\bf odd tensor} on a symmetric space
is a tensor $\Psi\in TM^{\otimes p}\otimes T^*M^{\otimes q}$
for $p+q$ odd.
\ed

\exercise
Let $M=G/H$ be a symmetric space,
and $\Psi$ a $G$-invariant odd tensor. Prove that $\Psi=0$.
\ez

\exercise
Let $M=G/H$ be a symmetric space, and $\omega$ a 
$G$-invariant differential form. 
\enum
\ite
Prove that $d\omega=0$
and $\omega$ is even.
\ite[!] Suppose that $M$ is equipped
with a $G$-invariant Riemannian form. Prove that $\omega$
is harmonic.
\ite[!] Assume in addition that $M$ is compact and $G$ is connected.
Prove that any harmonic form is $G$-invariant.
\ee
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{K\"ahler structures on symmetric spaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise[!]
Let $M=G/H$ be a symmetric space, and $I$ a $G$-invariant
almost complex structure. Prove that $I$ is integrable.
\ez

\exercise[!]
$M=G/H$ be a symmetric space, $I$ a $G$-invariant
almost complex structure, and $h$ a $G$-invariant Hermitian form.
Prove that $(M, I, h)$ is K\"ahler.
\ez

\exercise
Construct a structure of symmetric space and a $G$-invariant
complex structure on the following spaces.
\enum
\ite $\C P^n$ (also prove that it is K\"ahler).
\ite[!] $\Gr_\R(2,n):=\frac{SO(n+2)}{SO(n)\times SO(2)}$
\ee
\ez

\exercise[!]
Let $M=G/H$ be a homogeneous space such that
the isotropy group $H_x$ acts on the (real) projectivization
${\Bbb P} T_x M$ transitively. Prove that the $G$-invariant
Riemannian metric on $M$ is unique up to a constant multiplier.
\ez


\exercise[*]
Consider the Grassmannian space
$\Gr_\R(p,q):=\frac{SO(p+q)}{SO(p)\times SO(q)}$.
Prove that $\Gr_\R(p,q)$  admits a $SO(p+q)$-invariant
metric. Prove that this metric is unique up to a constant
multiplier, when $p>2$ or $q>2$.
\ez


\exercise
Construct a $U(n+1)$-invariant Hermitian metric on $\C P^n$
(it is called {\bf Fubini-Study metric}).
\enum
\ite Prove that this metric is unique up to a constant.
\ite Prove that it is K\"ahler.
\ee
\ez

\definition
$U(p, q)$ is the group of all complex-linear matrices
preserving a pseudo-Hermitian metric $h$ of signature
$(p, q)$, with $h(x_1, ..., x_{p+q})= \sum_{i=1}^p |x_i|^2-
\sum_{i=q+1}^{p+q} |x_j|^2.$
\ed

\exercise
\enum
\ite[!]
Construct a $U(1,n)$-invariant metric and complex structure
on $M:=\frac{U(1,n)}{U(1)\times U(n)}$. 
\ite[!]
Prove that it is K\"ahler.
\ite[*] Prove that $M$ is biholomorphic to an open ball in $\C^n$.
\ite[**] Prove that all complex automorphisms of an open ball
are isometries with respect to this metric.
\ee
\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}

