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


\begin{document}

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\listok{6}{Symplectic handout 6: K\"ahler reduction}

We freely use the definitions given in assignment 3
(``Symplectic reduction''). A {\bf complex manifold}
is a manifold equipped with an almost complex
structure $I$ which satisfies 
$[T^{1,0}M, T^{1,0}M]\subset T^{1,0}M$, where $T^{1,0}M \subset TM \otimes_\R \C$
is the eigenspace of $I$  with the eigenvalue $\1$.
An almost complex structure that satisfies 
$[T^{1,0}M, T^{1,0}M]\subset
T^{1,0}M$ is called {\bf integrable}.



\definition
Recall that a Riemannian metric $g$ 
on an almost complex manifold $(M,I)$ is called {\bf Hermitian}
if $g(Ix, Iy) = g(x,y)$. Denote by $\omega$ the {\bf Hermitian 
form} $\omega(x, y):= g(Ix. y)$; it is easy to see
that $\omega$ is anti-symmetric. A Hermitian metric on a 
complex manifold $(M,I)$ is called {\bf K\"ahler}
if $d\omega=0$. A complex manifold equipped with a
K\"ahler metric is called {\bf a K\"ahler manifold}.
\ed

\exercise
Prove that the Fubini-Study form $\omega_{FS}$
constructed in Assignment 3 is closed, and defines
a K\"ahler structure on $\C P^n$.
\ez

\definition
{\bf A foliation} on a manifold $M$
is a sub-bundle $B\subset TM$ such that $[B, B] \subset
B$. By Frobenius theorem, this is equivalent to
a local decomposition $U= S\times R$,
of any sufficiently small open set $U\subset M$, 
with $B\subset TU$ equal to the tangent bundle to the fibers
of the projection $U \arrow R$.
{\bf A leaf} of a foliation is a maximal connected immersed submanifold
$Z\arrow M$ which satisfies $T_z Z= B\restrict z$ at
each $z\in Z$. {\bf Projection to the leaf space}
is a smooth submersion $U\arrow R$, defined 
locally in $U$, and mapping $U$ to the set of
leaves of $B$ on $U$.
\ed

\definition\label{_transve_stru_Definition_}
 {\bf Transversal Riemannian
structure/symplectic structure/almost complex structure} 
on a foliated manifold
$(M, B\subset TM)$ is a scalar product/skew-symmetric
form/almost complex structure
on $TM/B$ which is locally obtained as a 
pullback of a Riemannian/symplectic/almost complex structure on
the leaf space. 
\ed

\exercise
Let $G$ be a compact Lie group acting on
a manifold $M$. Prove that
all orbits have dimension $\dim G$
if and only if for some basis
$g_1, ... g_n \in \Lie G$
the corresponding vector fields 
on $M$ are linearly independent everywhere.
\ez

\definition
In this case, the action of $G$ on $M$
is called {\bf locally free.}
\ed

\exercise
Let $M$ be a manifold equipped with 
a locally free action of a compact Lie
group $G$ and $B\subset TM$
the bundle of vectors tangent to the $G$-action.
Suppose that $TM/B$ is equipped with a $G$-invariant
metric and almost complex structure. Prove that these
structures are transversal, in
the sense of the Definition
\ref{_transve_stru_Definition_},
and the orbit space is 
almost complex Hermitian.
\ez

\exercise\label{_symple_red_alm_co_Exercise_}
Let $M$ be an almost K\"ahler manifold,
$G$ a compact Lie group acting on $M$ locally freely by 
Hamiltonian isometries,
$t\in \g^*$ a central element, and
$\mu:\; M \arrow \g^*$ an equivariant
moment map.  Denote by $K \subset T\mu^{-1}(t)$
the bundle of vectors tangent to the orbits of
$G$ on $\mu^{-1}(t)$.
\enum
\ite
Prove that $K\subset T\mu^{-1}(t)$ is a foliation,
and $(\mu^{-1}(t), K\subset T\mu^{-1}(t))$ is equipped
with a transversal Riemannian and a
transversal symplectic structure,
obtained by restricting the
Riemannian and symplectic forms to $K^\bot$.

\ite Prove that these 2-forms
define an almost K\"ahler
structure on the orbit 
space.\footnote{{\bf An almost K\"ahler structure}
is a pair of compatible almost complex and symplectic
structures.}
\ee
\ez

\hint Use the arguments from Assignment 3. \eh


\exercise
Let $M$ be an almost K\"ahler manifold,
$G$ a compact Lie group acting on $M$ 
locally free by Hamiltonian isometries,
$t\in \g^*$ a central element, 
$\mu:\; M \arrow \g^*$ an equivariant
moment map.  Denote by $K \subset T\mu^{-1}(t)$
the bundle of vectors tangent to the orbits of
$G$. 
\enum
\ite Prove that $K \cap I(K) = 0$.
\ite
Prove that $T\mu^{-1}(t)/K = TM\restrict{\mu^{-1}(t)}/ K_\C$,
where $K_\C= K \oplus I(K)$.
Prove that the  complex structure operator on
$T\mu^{-1}(t)/K$ defined in Exercise \ref{_symple_red_alm_co_Exercise_}
coincides with the  complex structure on $TM\restrict{\mu^{-1}(t)}/ K_\C$
induced by the almost complex structure $I\in \End(TM)$.
\ee
\ez

\exercise
Let $G$ be a compact Lie group freely acting
on a manifold $M$. Consider the orbit space
$M/G$ with the quotient topology. Prove that
$M/G$ is homeomorphic to a smooth manifold.
\ez

\exercise
Let $M$ be a manifold equipped with
a locally free action on compact Lie group $G$.
Prove that $M/G$ is locally homeomorphic
to $\R^n/G$, where $G$ is a finite group.
\ez

\exercise
Construct a smooth manifold $M$ equipped with a free
action of $S^1$ such that $M/S^1$ is a 2-torus,
and a $\Z/2\Z$-quotient $M_1$ of $M$ with a locally
free action of $S^1$ such that $M_1/S^1$ is homeomorphic
to $S^2$.
\ez


\definition
Let $B\subset TM$ be a sub-bundle equipped with 
a complex structure operator $I\in \End B$, $I^2=-\Id$,
and $B\otimes_\R \C= B^{1,0} \oplus B^{0,1}$ the
corresponding eigenspace decomposition, 
$I\restrict{B^{1,0}}=\1$, $I\restrict{B^{0,1}}=-\1$.
The pair $(B, I)$ is called {\bf a CR-structure on $M$}
if $[B^{1,0}, B^{1,0}] \subset B^{1,0}$.
\ed

\exercise\label{_CR_on_submanifold_Exercise_}
Let $Z\subset M$ be a submanifold of a complex
manifold. Assume that $B:=TZ \cap I(TZ)$ has constant
rank. Prove that $(B, I \restrict B)$ is a CR-structure on $M$.
\ez

\exercise\label{_quot_complex_CR_Exercise_}
Let $(Z, B, I)$ be a CR-manifold, and 
$G$ a compact Lie group freely acting on $Z$.
Assume that $TZ= B \oplus K$, where $K \subset TZ$
is the subspace generated by the vector fields tangent to 
the $G$-action. Suppose that the operator $I$ on $B$
defines a transversal complex structure with respect
to the foliation tangent to $K$. 
Prove that the natural almost complex
structure on $Z/G$ induced by the action of $I$ on
$B= TZ/K= T(Z/G)$ is integrable.
\ez

\exercise
Let $M$ be a K\"ahler manifold,
$G$ a compact Lie group freely acting on $M$ 
by Hamiltonian isometries,
$t\in \g^*$ a central element,  and
$\mu:\; M \arrow \g^*$ an equivariant
moment map. Prove that 
the quotient $M\2 G:= \frac{\mu^{-1}(t)}{G}$ is equipped with
a natural K\"ahler structure.
\ez

\hint Use Exercise
\ref{_symple_red_alm_co_Exercise_} to show that
$M \2 G$ is almost K\"ahler,
Exercise \ref{_CR_on_submanifold_Exercise_}
to show that $\mu^{-1}(t)$ is a CR-manifold, and
Exercise \ref{_quot_complex_CR_Exercise_}
to prove that the complex structure on 
$M \2 G$ is integrable.
\eh
 
\exercise
Prove that the K\"ahler metric on the 
standard symplectic quotient $\C^n\2 S^1 = \C P^{n-1}$
is proportional to the Fubini-Study metric.
\ez



\end{document}
