\documentclass[10pt]{article}
\usepackage{url}

%version 1.0,\ \   22.09.2021
%version 1.1,\ \   29.09.2021, �������� � 3.2, 3.3
%version 1.2,\ \   02.10.2021, �������� � 3.8
%version 1.3,\ \   09.10.2021, �������� � 3.10
%version 2.0,\ \   13.10.2021, ������� ��������
%version 2.1,\ \   20.10.2021, some more
%version 2.2,\ \   23.10.2021, linear vector field in 3.7b

\newcommand{\version}{version 2.2,\ \   23.10.2021}
\newcommand{\firstdate}{22.09.2021}

\addtolength{\topmargin}{-15mm}
\addtolength{\textheight}{30mm}
\addtolength{\oddsidemargin}{-10mm}
\addtolength{\textwidth}{20mm}

\input{defs-listki-en.tex}


\setlength{\headheight}{15pt}
\pagestyle{fancy} 
\lhead{\tiny Symplectic geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\listok{3}{Symplectic handout 3: Symplectic reduction}

\definition
Let $(M, \omega)$ be a symplectic manifold.
A vector field $v\in TM$ is called
{\bf symplectomorphic} if $\Lie_v \omega=0$.
\ed

\exercise
Let $\rho$ be a closed $k$-form on $M$,
and $v$ a vector field. Prove that
$\Lie_v\rho=0$ if and only if $d(i_v\rho)=0$,
where $i_v(\rho) = \rho(v, \cdot, ..., \cdot)$
denotes the contraction operator.
\ez

\definition
A symplectomorphic vector field $v$ on $(M, \omega)$
is called {\bf Hamiltonian} if the 1-form $i_v(\omega)$ is exact.
{\bf Hamiltonian symplectomorphism} is $\Psi_1$,
where $\Psi_t$ is a flow of symplectomorphisms
associated with a smooth family $v_t$, $t\in [0,1]$ of
Hamiltonian vector fields, with $\Psi_0=\Id$,
and $\Psi_t^{-1}\frac{d\Psi_t}{dt}=v_t$.
\ed

\definition
Let $G$ be a Lie group acting on a symplectic manifold
$(M, \omega)$ by symplectomorphisms,
$\g=T_e G$ its Lie algebra, and $\rho:\; \g \arrow TM$
the corresponding Lie algebra action, that is,
the differential of the action of $G$ in $e\in G$.
Suppose that all vector fields in $\rho(\g)$ are Hamiltonian.
Consider a linear map
$\mu_1:\;  \g\arrow C^\infty M$ which takes each 
vector field $x\in \g$ to the Hamiltonian
of $\rho(x)$, and let $\mu:\; M \arrow \g^*$
take $X \in \g$ to $\mu_1(X)$.
The map $\mu:\; M \arrow \g^*$
is called {\bf the moment map} of 
the Lie group action.
It is called {\bf an equivariant moment map}
if $\mu(g(x)) = \coad(g)(\mu(x))$,
where $g\in G$, and $\coad$ denotes 
the coadjoint action of $G$ on $\g^*$.
\ed

\exercise
Let $G$ be a Lie group acting on a
simply connected symplectic manifold
by symplectomorphisms.
Prove that the moment map  $\mu:\; M \arrow \g^*$ 
always exists.\footnote{Here we don't assume that $\mu$ is
equivariant.} Prove that the set of moment maps
is a finite-dimensional affine space with 
free, transitive action of the space $\g^*$ considered as
an abelian Lie group.
\ez

\exercise
Let $G$ be a Lie group which acts on $M$ by Hamiltonian symplectomorphisms.
We say that this action is {\bf locally free} if each orbit is
diffeomorphic to a quotient of $G$ by a discrete subgroup. 
Prove that $G$ acts locally freely if and only if the moment map
$\mu$ is a submersion.
\ez

\exercise\label{_Poisson_via_moment_Exercise_}
Let $\mu:\; M \arrow \g^*$ be 
an equivariant moment map, 
$\rho:\; \g \arrow TM$ the Lie algebra action,
and $x, y\in \g$. Prove that
$\omega(\rho(x), \rho(y))= \mu([x, y])$.
\ez

\exercise\label{_central_Exercise_}
An covector $t\in \g^*$ is called {\bf central}
if it is $\coad$-invariant. Prove that
any central element vanishes on $[\g,\g]$.
\ez

\exercise
Prove that the space of equivariant moment maps
is a finite-dimensional affine space with 
free, transitive action of the space $(\g^*)^G$
of central covectors.
\ez


\exercise
Let  $\omega=\sum_i dx_i\wedge dy_i$ be the
standard symplectic form on $\C^n$, and $X\in \goth u(n)$ 
a complex linear vector field acting by isometries.
\enum
\ite Prove that the symplectic gradient
$\omega^{-1}(dH)$ of a function $H\in C^\infty(\C^n)$
can be written as $I(\grad H)$, where $\grad H= (dH)^\sharp$ is the usual
gradient. 
\ite Let $X$ be a vector field acting on $\C^n$ by
linear holomorphic isometries. Prove that $4 \omega= d I d (r)$,
where $r(z) =|z|^2$. Prove that 
$\Lie_X (I dr) = 0 = d(\langle I dr, X\rangle) + 2 i_X \omega$.
Deduce from this $\Lie_{IX}(dr)=d(\langle dr, IX\rangle) = 2 i_X \omega$.

\ite   Prove that the moment map for the action
of the Lie group $\R= e^{tX}$
can be written as $\mu(v) = \frac 1 4 \Lie_{IX} (r)$.
\ee 
\ez



\exercise
Let $\mu:\; M \arrow \g^*$ be 
an equivariant moment map, and 
$t\in \g^*$ a central covector.
Prove that for any
and any $x, y \in \rho(g)\restrict {T(\mu^{-1}(t))}$, 
one has $\omega(x, y) =0$.
\ez

\hint
Use Exercise \ref{_Poisson_via_moment_Exercise_} 
and Exercise \ref{_central_Exercise_}.
\eh


\exercise\label{_fiber_of_mu_coisotro_Exercise_}
Let $G$ be a connected Lie group which acts on 
$(M,\omega)$ locally freely by Hamiltonian symplectomorphisms,
$\mu:\; M \arrow \g^*$ 
an equivariant moment map, $t\in \g^*$ a central element
and $Z:= \mu^{-1}(t)$.
\enum
\ite
Prove that $Z$ is coisotropic
(that is, for each  $z\in Z$
one has $(T_z Z)^\bot \subset T_zZ$).
\ite Prove that $(T_z Z)^\bot=\rho(\g)$,
where $\rho:\; \g \arrow TM$
is the corresponding action of the Lie algebra 
$\Lie(G)$.
\ee
\ez

\hint
Use the previous exercise.
\eh


\exercise
Let $Z\subset M$ be a coisotropic submanifold,
and $B\subset TZ$ the sub-bundle defined as
$B:= (TZ)^\bot$. Prove that $[B, B]\subset B$.
\ez

\remark 
By Frobenius theorem, the condition 
$[B, B]\subset B$ means that $B$ is 
is a tangent bundle to a foliation 
on $M$. This foliation is called
{\bf the characteristic foliation}
of $Z\subset M$. Further on, you are
allowed to apply Frobenius theorem,
if you can state it correctly.
\er

\exercise
Let $Z\subset M$ be a coisotropic
subvariety in $(M, \omega)$, and $\pi:\; Z \arrow Z_0$ 
the projection to the leaf space
of the characteristic foliation.
\enum
\ite Let $\pi:\; X \arrow Y$ 
be a smooth submersion, and $\alpha\in \Lambda^k (X)$
a closed form. Prove that $\alpha = \pi^* \alpha_0$
if and only if $i_v \alpha=0$ for any vector field
$v$ tangent to the fibers of $\pi$.

\ite
Prove that $Z_0$ is equipped with
a symplectic form $\omega_0$ such that
$\pi^* \omega_0 = \omega\restrict Z$.
\ee
\ez

\exercise
In assumptions of Exercise \ref{_fiber_of_mu_coisotro_Exercise_},
let $Z:= \mu^{-1}(t)$, where $t\in \g^*$ is
a central covector. 
\enum
\ite Prove that $Z$ is $G$-invariant.
\ite Prove that the leaves of the characteristic
foliation on $Z$ coincide with the orbits of $G$.
\ite Assume that $G$ is compact and its action is free. 
Prove that the quotient space
$\frac{\mu^{-1}(t)}{G}$ is a smooth manifold
which is equipped with a natural symplectic structure.
\ee
\ez

\hint Use the previous exercise. \eh

\definition
The manifold $M\2 G:=\frac{\mu^{-1}(t)}{G}$
is called {\bf the symplectic reduction} of $M$.
\ed


\exercise
Consider the natural action of the group
$U(n+1)$ on $\C P^n$. 
\enum
\ite Prove that there exists
an $U(n+1)$-invariant non-degenerate 2-form on $\C P^n$.
\ite Prove that such a form is always closed.
\ite Prove that it is unique up to a constant 
multiplier.\footnote{This form is called {\bf the Fubini-Study symplectic form}.}
\ee
\ez

\exercise
Consider the action of $U(1)$ on $\C^{n+1}$ by complex rotations.
Prove that the symplectic quotient $\C^{n+1}\2 U(1)$ is $\emptyset$, a point,
or $\C P^n$ with the Fubini-Study symplectic form,
depending on the choice of the moment map.
\ez


\exercise[*]
Let $\gamma\subset S^1 \times S^1$
be a circle $S^1\times \{x\}$,
and $g$ a Hamiltonian diffeomorphism
of $S^1\times S^1$.
Prove that $g(\gamma)\cap \gamma$ is non-empty.
\ez


 
\end{document}



\enum
\ite
Prove that the central covector on the Lie algebra
${\goth u}(n+1)$ is unique up to a constant multiplier.
\ite Prove that the symplectic reduction 


