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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Symplectic geometry\\[15mm]
\small lecture 14: K\"ahler reduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
October 20, 2021
}
\end{center}

\newpage

{\bf \blue Characteristic foliation (reminder)}

\definition
Let $(M, \omega)$ be a symplectic manifold.
A submanifold $Z \subset M$ is called {\bf \blue
coisotropic} if $\dim_\R Z \geq \dim_\R M$ and
$\omega\restrict Z$ has rank $\dim_\R M- \dim_\R Z$
(minimal possible), or, equivalently, 
$(TZ)^{\bot_\omega} \subset TZ$, where
$(TZ)^{\bot_\omega}:= \{ x\in TZ\ \ |\ \ i_x\omega=0\}$

\definition
Let $Z\subset (M, \omega)$ be a coisotropic submanifold.
The bundle $(TZ)^{\bot_\omega}$ is called {\bf \blue
the characteristic bundle of $Z$}.

\theorem
Let  $Z\subset (M, \omega)$ be a coisotropic submanifold,
and $K\subset TM$ its characteristic bundle. {\bf \red Then
$[K, K]\subset K$, hence $K$ is tangent to a foliation} ${\cal F}$
which is called {\bf \blue the characteristic foliation of $Z$.} 
Moreover, {\bf \red the restriction $\omega\restrict Z$ is basic, and symplectic
on the leaf space of the characteristic foliation.}

\newpage

{\bf \blue Moment maps (reminder)}

\definition
$(M,\omega)$ be a symplectic manifold, and $G$ a 
Lie group acting on $M$ by symplectomorphisms.
{\bf \blue A moment map} $\mu$ of this action is a linear map
${\goth g}\arrow C^\infty M$
associating to each $g\in G$ its Hamiltonian.

\remark
It is more convenient to consider $\mu$
as an element of ${\goth g}^* \otimes_\R C^\infty M$,
or (and this is most standard) {\bf \red as a function
with values in ${\goth g}^*$}. 

\remark
Moment map {\bf \purple always exists} if $M$ 
is simply connected.

\definition 
A moment map $M \arrow  {\goth g}^*$
is called {\bf \blue equivariant}
if it is equivariant with respect to the 
coadjoint action of $G$ on ${\goth g}^*$.

\remark
$M\stackrel\mu \arrow  {\goth g}^*$ is a moment map
iff for all $g\in {\goth g}$, 
$\langle d\mu,g\rangle= i_{\rho_g}(\omega)$.
Therefore, {\bf \purple a moment map is defined up to 
a constant ${\goth g}^*$-valued function.}
An equivariant moment map is is defined up to 
{\bf \purple a constant ${\goth g}^*$-valued function
which is $G$-invariant}, that is, up to addition of a central 
vector $c\in \g^*$. 

\claim
{\bf \red
An equivariant moment map exists whenever $H^1(G, {\goth g}^*)=0$.}
In particular, when $G$ is reductive and $M$ is simply connected,
an equivariant moment map exists.
Further on, all moment maps will be tacitly considered equivariant.


\newpage

{\bf \blue Weinstein-Marsden theorem (reminder)}

%\definition A $G$-invariant $c\in \goth g^*$ is called
%{\bf\blue central}.

\definition
(Weinstein-Marsden)
$(M,\omega)$ be a symplectic manifold, $G$ a 
compact Lie group freely acting on $M$ by symplectomorphisms,
$M\stackrel\mu \arrow  {\goth g}^*$ an equivariant
moment map, and $c\in {\goth g}^*$ a central element.
The quotient $\mu^{-1}(c)/G$ is called {\bf \blue
symplectic reduction} of $M$, denoted by $M\2 G$.

\claim
{\bf \red The symplectic quotient $M\2 G$ is a symplectic
manifold of dimension $\dim M - 2 \dim G$.}\\
\pstep
$T_x (\mu^{-1}(c))= d\mu^{-1}(0)$.
However, the space 
$\langle d\mu, g\rangle \subset \Lambda^1 M$ is $\omega$-dual to the space 
$\tau(\goth g)$ of vector fields
tangent to the $G$-action, {\bf \purple hence 
$d\mu^{-1}(c)= \tau(\goth g)^\bot$.}

{\bf \green Step 2:}
Since $\mu$ is $G$-equivariant, $G$ preserves
$\mu^{-1}(c)$, hence $\tau(\goth g)\subset d\mu^{-1}(0)$.
This implies that {\bf \purple $\tau(\goth g)\subset TM$ is isotropic}
(that is, $\omega\restrict{\tau(\goth g)}=0$).
Its $\omega$-orthogonal complement in $T_x M$ 
is $T_x (\mu^{-1}(c))$ (Step 1).

{\bf \green Step 3:}
Consider the {\bf \blue characteristic foliation}
${\cal F}$ on $\mu^{-1}(c)$. It is 
a bundle because $\mu^{-1}(c)\subset M$
is coisotropic. From Step 2 {\bf \purple we obtain that ${\cal F}=\tau(\goth g)$.}

{\bf \green Step 4:}
Since $\omega\restrict {\mu^{-1}(c)}$ is closed, it
satisfies  $\Lie_v (\omega)=0$ for all $v\in {\cal F}$.
This implies that it is {\em \green basic}, that is, 
lifted from the leaf space of characteristic foliation, 
identified with $M\2 G$.
\endproof

\newpage

{\bf \blue  K\"ahler manifolds}

\definition
{\bf \blue K\"ahler manifold}
is a complex manifold equipped with a compatible symplectic structure.

\example {\bf \purple A complex submanifold $Z \subset (M, I, \omega)$
of a K\"ahler manifold is
also K\"ahler.} Indeed, a restriction of a symplectic
form to $Z$ is closed; it is non-degenerate because
$\omega(x, Ix)$ is positive definite, hence
$\omega$ is non-degenerate on every $I$-invariant subspace.

\example
{\bf \purple 
The Fubini-Studi form $\omega$ on $\C P^n$ is K\"ahler.}

\definition
A complex submanifold of $\C P^n$ is called {\bf \blue projective}.

\example {\bf \purple All projective manifolds are K\"ahler}.

\example Compact complex tori {\bf \purple are always K\"ahler,
but not always projective.}


\newpage

{\bf \blue Symplectic reduction and a K\"ahler potential}

\definition
Let $d^c:= I dI^{-1}$.
{\bf \blue K\"ahler potential} on a K\"ahler manifold $(M, \omega)$
is a function $\psi$ such that $dd^c \psi=\omega$.

\proposition
Let $G$ be a real Lie group acting
on a K\"ahler manifold $M$ by holomorphic isometries,
and $\psi$ a $G$-invariant K\"ahler potential.
{\bf \red Then the moment map $\g \times M 
\stackrel {\mu_g} \arrow \R$
can be written as $g, m \arrow \Lie_{Iv}\psi$,}
where $v=\tau(g)\in TM$ is the tangent vector field
associated with $g\in \goth g$.

\proof
Since $\psi$ is $G$-invariant, and $I$ is $G$-invariant,
we  have $0 = \Lie_v d^c \psi = i_v (dd^c\psi) +
d(i_v d^c \psi)$. Using $\omega=dd^c \psi$,
we rewrite this equation as
$i_v\omega = -d(\langle d^c \psi, v\rangle)$,
giving an equation for the moment map 
$\mu_g= -\langle d^c \psi, v\rangle.$
Acting by $I$ on both sides, we obtain
$\mu_g= \langle d \psi, Iv\rangle = \Lie_{Iv}\psi$.
\endproof

\corollary Let $V$ be a Hermitian representation of
a compact Lie group $G$. {\bf \purple Then the corresponding moment
map can be written as $\mu_g(v)= \Lie_{Ig}|v|^2=
\frac 1 4 \langle v, Ig(v)\rangle.$} \endproof


\newpage

{\bf \blue Transversal complex and Riemannian structures}

\definition
{\bf\blue 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\blue 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\blue Projection to the leaf space}
is a smooth submersion $U\arrow R$,  mapping $U$ to the set of
leaves of $B$ on $U$.

\definition
 {\bf\blue 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 the bundle $(TM/B)$ which is locally obtained as a 
pullback of a Riemannian /symplectic / almost complex structure on
the leaf space. 

\newpage

{\bf \blue Transversal complex and Riemannian structures (2)}

\proposition
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 $h$ and a $G$-invariant almost complex structure $I$. 
{\bf \red  Then these
structures are transversal.}

\pstep
Let $h\in \Sym^2(T^*M)$ be a symmetric 2-form vanishing on
$B$ and positive definite and $G$-invariant on $TM/B$.
In Lecture 13 we proved that {\bf \purple a $B$-basic differential
form $\alpha$ is the one which satisfies $i_X\alpha=0$
and $\Lie_X \alpha=0$ for all vector fields $X\in B$.}
The same argument can be applied to show that
$h$ is basic whenever $\Lie_X h=0$ for all $X\in B$.
This follows because $h$ is $G$-invariant.

{\bf \green Step 2:} It remains to show that the
2-form $\omega\in \Lambda^2 M$ obtained 
as $\omega(x, y) = h(x, Iy)$ is basic.
This form vanishes on $B$ and is $G$-invariant,
hence it is basic by the same argument.
\endproof

\newpage

{\bf \blue Symplectic reduction for almost K\"ahler manifolds}

%\remark Recall that {\bf \blue an almost K\"ahler manifold}
%is a symplectic manifold  with compatible almost complex structure.

{\bf \green THEOREM 1:}
Let $M$ be an almost K\"ahler manifold,
$G$ a compact Lie group acting on $M$ 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 $\mu^{-1}(t)$. 
Then $K\subset T\mu^{-1}(t)$ 
is tangent to a foliation, {\bf \red equipped
with a transversal Riemannian and a
transversal symplectic structure,}
obtained by restricting the
Riemannian and symplectic forms to $K^\bot$.
Moreover, these bilinear forms {\bf \red define
an almost K\"ahler structure on the orbit 
space $\frac{\mu^{-1}(t)}{G}$.}

\proof
These two bilinear forms are transversal because they are
$G$-invariant, and the symplectic form on $M/G$ is closed
by Marsden-Weinstein theorem.
\endproof

\corollary
Let $G$ be a compact group acting by Hamiltonian
symplectomorphisms on an almost K\"ahler manifold.
Assume that $G$ preserves the almost complex structure.
{\bf \red Then the symplectic reduction $M\2 G$ is also
almost K\"ahler.}
\endproof

\remark
Since $K\subset T\mu^{-1}(t)$ is isotropic,
one has $K \cap I(K)=0$. This implies
$T\mu^{-1}(t)/K = TM\restrict{\mu^{-1}(t)}/ K_\C$,
where $K_\C = K \oplus I(K)$. {\bf \purple This is another way
to equip the bundle $T\mu^{-1}(t)/K$ with an almost
complex structure.}


\newpage

{\bf \blue CR-manifolds}

\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 \blue a CR-structure on $M$}
if $[B^{1,0}, B^{1,0}] \subset B^{1,0}$.

\claim
Let $Z\subset M$ be a submanifold of an almost complex
manifold. Assume that $B:=TZ \cap I(TZ)$ has constant
rank. {\bf \red Then $(B, I \restrict B)$ is a CR-structure on $M$.}

\proof
Any $(1, 0)$-vector fields $X,Y\in B^{0,1}\subset TZ\otimes_\R \C$ can be smoothly
extended to a section $\tilde X, \tilde Y \in T^{1,0}M$.
A commutator $[\tilde X,\tilde Y]$ of $(1,0)$-vector fields on $M$ is of type
$(1,0)$. Commutator of vector fields tangent
to $Z$ remains tangent to $Z$. {\bf \purple Therefore,
$[\tilde X,\tilde Y]\bigg|_ Z \in B^{1,0}$,
giving $[X,Y]\in B^{1,0}$.}
\endproof

\newpage

{\bf \blue CR-manifolds and leaf spaces}

{\bf \green PROPOSITION 2:}
Let $(Z, B, I)$ be a CR-manifold, 
equipped with an action of a compact group $G$
compatible with the CR-structure.
Assume that $TZ= B \oplus K$, where $K \subset TZ$
is the subspace generated by the vector fields tangent to 
the $G$-action. Then the operator $I$ on $B$ {\bf \purple defines
a transversal complex structure} with respect to the
foliation tangent to $K$. Morever, {\bf \red the natural almost
complex structure on $Z/G$ induced by the action of $I$ on
$B= TZ/K= T(Z/G)$ is integrable.}

\proof
This operator is transversal because it is $G$-invariant,
and integrable because $[B^{1,0}, B^{1,0}] \subset B^{1,0}$.
\endproof

\newpage

{\bf \blue K\"ahler reduction}

\corollary
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. 
{\bf \red Then the quotient $M\2 G:= \frac{\mu^{-1}(t)}{G}$ is equipped with
a natural K\"ahler structure.}

\pstep
Let $Z:= \mu^{-1}(t)$ and let $K\subset TZ$ be the kernel of the
restriction of the K\"ahler form to $Z$. 

{\bf \purple The space $Z:= \mu^{-1}(t)$ is equipped with a CR-structure
because $Z$ is a subspace of a complex manifold
with $TZ \cap I(TZ)$ of constant rank. }
Indeed,
$I(TZ) \cap K=0$ because $K$ is isotropic, hence
\[ \rk(TZ \cap I(TZ))\leq \dim Z - \rk K= \dim M - 2 \codim Z.\]
However, the intersection $TZ \cap I(TZ)$ satisfies
$\rk(TZ \cap I(TZ))\geq \dim M - 2 \codim Z$ because
it is an intersection of two subspaces of codimension
$\codim Z$, giving $\rk(TZ \cap I(TZ))=\dim Z - \rk K$.

{\bf \green Step 2:}
The quotient $M\2 G:= Z/G$ is almost K\"ahler by Theorem 1.
The almost complex structure on $M\2 G$
is integrable by Proposition 2.
\endproof



\end{document}
