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


\begin{document}

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

\listok{0}{Symplectic geometry: exam}

{\scriptsize
Handouts score is given by the formula
$t=10a +5b+ 5c$, where $a$ is the number of 
completed handouts with at least 2/3 exercises credited
within 3 weeks since the exercise was distributed,
$b$ the number of handouts with 1/3 exercises credited
within 3 weeks, and $c$ the number of handouts with 2/3
exercises credited, if it is not already counted with 
$a$ and $b$.
 
Each student receives a random selection of 12 test
problems, 3 from each section (the output of the randomizer is printed 
on a separate sheet). The final score for the course is $s= 2 +p +[t/10]$, 
where $p$ is the total number of points
for the exam. The exam is oral. 
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symplectic forms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Construct two symplectic forms
$\omega_1, \omega_2$ on $\R^4$ 
such that $\omega_1+\omega_2$
has non-constant rank.
\ez

\exercise
Let $S$ be a 2-dimensional smooth manifold,
and $\omega_1, \omega_2$ two symplectic forms
in the same homotopy class. Prove that
$\omega_1+\omega_2$ is symplectic. Find
an example of symplectic forms  $\omega_1, \omega_2$ 
on $\R^2$ such that $\omega_1+\omega_2$ has non-constant rank.
\ez

\exercise
Let $U(n), O(2n), Sp(2n), GL(n, \C)$
be classical Lie groups embedded to $GL(2n, \R)$ in the
usual way. Prove that 
$O(2n)\cap  Sp(2n)= O(2n)\cap GL(n, \C) = Sp(2n) \cap GL(n, \C)= U(n)$.
\ez

\exercise[2 points]
Prove that $\pi_1(Sp(2n))= \Z$.
\ez

\exercise[2 points]
Let $V= \R^{2n}$ be a symplectic vector space,
and $X$ the Grassmanian of oriented Lagrangian subspaces in $V$.
Prove that $\pi_1(X)$ is infinite.
\ez



\exercise[2 points]
Let $U$ be the set of non-degenerate
bilinear antisymmetric 2-forms on $\R^{2n}$. Prove that $U$
has precisely 2 connected components.
\ez



\exercise
Prove that the group $Sp(2n)$ of symplectic matrices
acts transitively on the set of $n+k$-dimensional
coisotropic subspaces.
\ez

\exercise
Let $M$ be a compact symplectic manifold, and
$U\subset H^2(M, \R)$ the set
of all cohomology classes which are represented
by a symplectic form. Prove that $U$ is open
in $H^2(M, \R)$.
\ez

\exercise[2 points]
Let $M$ be a compact oriented manifold
which is smoothly fibered over 
a symplectic manifold $X$ with symplectic
fibers. Prove that $M$ admits a symplectic
structure or find a counterexample.
\ez

\exercise
Let $M=(S^1)^n \times S^3$.
Prove that $M$ does not admit
a symplectic structure for any $n$.
\ez



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symplectomorphisms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
{\bf An antisymplectic diffeomorphism} is a diffeomorphism
$\Psi$ of a symplectic manifold $(M, \omega)$
such that $\Psi^* \omega= -\omega$.
\ed

\exercise[2 points]
Find a compact symplectic manifold
$(M, \omega)$ admitting an antisymplectic diffeomorphism.
Find a compact symplectic manifold which does not admit
an antisymplectic diffeomorphism.
\ez


\exercise[3 points]
Let $\Psi$ be a diffeomorphism of a connected symplectic
manifold $(M, \omega)$ which satisfies $\Psi^*\omega=2 \omega$.
Prove that $\omega$ is exact, or find a counterexample.
\ez


\exercise
Let $\omega_i \in \Lambda^2 (M)$ be a sequence of
symplectic forms on a compact manifold $M$ converging
to a symplectic form $\omega$ in $C^0$-topology. Assume that
all $\omega_i$ are homologous. Prove that
almost all $(M,\omega_i)$ are symplectomorphic.
\ez


\exercise
Let $(M, I)$ be a compact almost complex manifold,
and $\omega_1, \omega_2\in \Lambda^2(M)$ be
almost K\"ahler forms. Assume that $\omega_1$ 
is homologous to $\omega_2$. Prove that
$(M, \omega_1)$ is symplectomorphic to
$(M, \omega_2)$.
\ez

\exercise
Let $M=\C^*$ be equipped with the standard
symplectic form $\omega = dx \wedge dy$.
\enum
\ite Find a symplectomorphism mapping a circle
of radius 1 to a circle of radius 2, or
prove it does not exist.
\ite Construct a symplectomorphism
mapping $M$ to $T^* S^1$ with the
standard symplectic structure, or prove it does not exist.
\ee
\ez

\exercise
Let $S\subset S^2$ be an equator, that is, a
geodesic circle, and $\phi:\; S^2 \arrow S^2$
a symplectomorphism. Prove that $\phi(S)\cap S \neq \emptyset$.
\ez

\exercise
Prove that the group of symplectomorphisms of $(M, \omega)$
acts transitively on $M$, for any connected symplectic manifold
$(M, \omega)$.
\ez




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lagrangian subvarieties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Construct a Lagrangian torus in $\C P^2$ with the standard
symplectic structure.
\ez

\exercise
Let $Z\subset M$ be a submanifold.
Define {\bf the conormal bundle} $CZ$
as the space of all pairs $z, v\in T^*M$
where $z\in Z$ and the form $v\in T^*_z M$ 
vanishes on $T_z Z \subset T_z M$.
Prove that $CZ$ is a Lagrangian submanifold
in $T^*M$ with the standard symplectic structure.
\ez

\exercise
Let $\pi:\; T^*M \arrow M$ be the standard
projection, and $L_1, L_2$ two Lagrangian sections of
$\pi$ (that is, Lagrangian submanifolds such that
$\pi:\; L_i \arrow M$ is a diffeomorphism). Assume that
$h^1(M)=0$. Prove that $L_1 \cap L_2 \neq \emptyset$.
\ez

\exercise
Let $M =T^2$ and let $\phi:\; t \arrow 2t$ be the 
standard 4:1 covering. Construct a symplectic
structure on $M \times M$ such that the
graph of $\phi$ is Lagrangian.
\ez

\exercise
Let $M= M_1 \times M_2$, where $M_1, M_2\cong T^2$.
Find a symplectic structure on $M$
such that the homology class $[M_1] + n [M_2]$ can be
represented by a Lagrangian submanifold, for any given $n\in \Z^{>0}$.
\ez

\exercise[2 points]
Prove that $M=T^*S^2$ admits a symplectic structure $\omega$ such that
all Lagrangian submanifolds $L\subset M$ diffeomorphic to
$S^2$ are contractible in $M$.
\ez

\exercise[2 points]
Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold,
and $M \arrow X$ a smooth submersion with Lagrangian
fibers diffeomorphic to $\R^n$. Prove that
$M$ is symplectomorphic to $T^* X$.
\ez

\remark Let $\eta$ be a 1-form on a manifold $M$.
{\bf The zero set} of $\eta$ is the set of all
points $x\in M$ such that $\eta \restrict {T_x M}=0$.
\er

\exercise
Let $X\subset (M, \omega)$ be a Lagrangian submanifold.
Prove that there is a 1-form $\eta$ defined
in a neighbourhood $U$ of $X$ such that
$X$ is the zero set of $\eta$ and $\omega\restrict
U=d\eta$.
\ez

\exercise
Let $\eta$ be a 1-form on $M$, $\dim_\R M=2n$, such that
the 2-form $\omega:=d\eta$ is symplectic.
Assume that the zero set of $\eta$
is an $n$-dimensional submanifold of $M$. Prove that
it is $\omega$-Lagrangian.
\ez


\exercise[3 points]
Construct an exact symplectic form 
on $\C P^3 \backslash \C P^1$.
\ez




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hamiltonians}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $X$ be a Hamiltonian vector field
on a symplectic manifold $(M,\omega)$,
and $e^{tX}$ the corresponding diffeomorphism flow.
Prove that $(e^{tX})^* H= H$, where $H$ is the
Hamiltonian of $X$.
\ez

\exercise
Let $X_1, X_2$ be Hamiltonian vector fields
on $(M_1, \omega_1)$ and  $(M_2, \omega_2)$.
Prove that $X_1 + X_2$ is a Hamiltonian vector field
on $M_1 \times M_2$.
\ez 


\exercise
Let $(M, \omega)$ be a  symplectic manifold,
and $F=(f_1, ..., f_n)$ a collection of functions
such that the differentials $df_i$ are linearly
independent, and the corresponding Poisson brackets
vanish: $\{f_i, f_j\}=0$, for all $i, j$.
\enum
\ite Prove that $2n \leq \dim_\R M$.
\ite Suppose that $2n = \dim_\R M$.
Prove that the level sets
$F^{-1}(c)$ are Lagrangian for any $c\in \R^n$.
\ee
\ez


\exercise
Let $X\in TM$ be a vector field on $M$,
and $e^{tX}$ the corresponding diffeomorphism flow.
Prove that its action on $T^*M$ is always Hamiltonian.
\ez



\exercise
Construct a Hamiltonian vector field
with non-closed orbits
on a 4-dimensional torus equipped
with the standard symplectic structure.
\ez



\exercise
Let $T^*X\stackrel \pi \arrow X$ be the
Lagrangian fibration on $T^*X$ with the standard
symplectic structure. Consider the group $G$ generated
by Hamiltonian symplectomorphisms associated
with functions $f\in \pi^* C^\infty X$.
Prove that $G$ is commutative and acts
transitively on the set of Lagrangian
sections of $\pi$, if $H^1(X)=0$.
\ez


\exercise
Let $G$ be a compact group freely acting
on a manifold $M$. Prove that the induced
action on $T^*M$ admits an equivariant moment map.
\ez

\exercise[2 points]
Let $G$ be a compact group freely acting
on a manifold $M$. Prove that
$T^*M\2 G= T^*(M/G)$.
\ez

\exercise[2 points]
Let $G=SO(3)$ act on $S^2$
by standard rotations. Denote by $\mu$ the
natural embedding $\mu:\; S^2 \arrow \R^3$.
Prove that after an appropriate 
identification of $\goth{so}(3)$ with $\R^3$,
the moment map can be identified with $\mu$.
\ez

\exercise
Let $S\subset M$ be a smooth hypersurface
in a symplectic manifold $M$, given 
as a level set of a function $H$,
\[ S=\{ x\in M \ \ | \ H(x)=0\}.
\]
Prove that the corresponding Hamiltonian
vector field $X_H$ preserves $S$, and
is tangent to the characteristic foliation on $S$.
\ez





\end{document}
