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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Symplectic geometry\\[15mm]
\small lecture 1: Symplectic manifolds and symplectic capacities}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
September 04, 2021
}
\end{center}

\newpage

{\bf \blue Symplectic manifolds}

\definition
A {\bf \blue symplectic form} is a closed,
non-degenerate 2-form $\omega\in \Lambda^2 (M)$.

\remark {\bf \purple A $2n$-manifold equipped with a non-degenerate
antisymmetric 2-form $\omega$ is oriented.} Indeed, 
$\omega^n$ is a volume form.

\definition
{\bf\blue An almost complex structure}
is an operator $I:\; TM \arrow TM$
such that $I^2= -\Id$.

\theorem
Let $\omega$ be a non-degenerate 
2-form on a manifold.
{\bf \red Then there exists an almost complex structure
$I$ such that the form $x, y \arrow \omega(x, I(y))$
is symmetric and positive definite.}
Moreover, {\bf \red the space of such almost complex
structures is contractible.}

\proof See the exercises. \endproof

\corollary
{\bf \purple A manifold admits a non-degenerate 
antisymmetric 2-form if and only if it
it admits an almost complex structure.}


\newpage

{\bf \blue Existence of symplectic structures}

\theorem {\bf \blue (Gromov)}\\
Let $M$ be a non-compact manifold
admitting an almost complex structure. {\bf \red Then $M$
admits a symplectic structure.}

\proof
{\em \green Eliashberg-Mishachev, Introduction to the h-principle.} 


By contrast, existence of a symplectic
structure {\bf \purple puts great restrictions
on the topology of a compact almost complex manifold.}

For instance, the top power of a symplectic form
is the volume form, hence 
{\bf \red its cohomology class is non-zero.}


\newpage

{\bf \blue Moser isotopy }

\definition
Let $(M, \omega)$ and $(M', \omega')$
be symplectic manifolds. A diffeomorphism
$\phi:\; M \arrow M'$ is called
{\bf\blue a symplectomorphism}
if $\phi^*\omega'=\omega$.

\remark {\bf \blue Symplectic geometry}
is a field of mathematics which {\bf \purple studies
symplectic manifolds up to an isomorphism
and the group of symplectomorphisms.}


\theorem {\bf \blue (Moser's isotopy lemma)}\\
Let $M$ be a compact symplectic manifold,
and $\omega_t$, $t\in [0,1]$ 
a smooth deformation of a symplectic form.
Assume that the cohomology class $[\omega_t]\in H^2(M)$
is constant in $t$. {\bf \red Then there exists a diffeomorphism flow
$V_t\in \Diff(M)$ mapping $\omega_0$ to $\omega_t$.}

\proof Next lecture.

\newpage

{\bf \blue Darboux' theorem}

\theorem {\bf \blue (Darboux)}
Let $(M, \omega)$ be a symplectic manifold.
Then in a neighbourhood of each
point {\bf \red there exist coordinates
$p_1, ..., p_n, q_1, ..., q_n$
such that $\omega= \sum_i dp_i \wedge dq_i$.}

\proof Next lecture (deduced from Moser isotopy).

\remark
From this theorem one can deduce
that {\bf \red the symplectomorphism group $\Symp(M)$ 
is simple and infinitely transitive}, that is, acts
transitively on discrete subsets $Z\subset M$
of the same cardinality. The same is true for
the diffeomorphism group, and the proof is 
more or less the same.

\definition
{\bf \blue Darboux coordinates}
on a symplectic manifold is a coordinate
system $p_1, ..., p_n, q_1, ..., q_n$
such that $\omega= \sum_i dp_i \wedge dq_i$


\newpage

{\bf \blue Volume and capacity}

\definition
{\bf\blue Standard symplectic structure} on $\R^{2n}$
with coordinates $p_1, ..., p_n,  q_1, ..., q_n$
is $\omega:= \sum_i dp_i \wedge dq_i$ (``Darboux coordinates'').

\definition
{\bf \blue A symplectic ball} of radius $r$
is the standard ball 
\[ 
B^{2n}:=\left\{(p_1, ..., p_n,  q_1, ..., q_n)\in 
\R^{2n}\ \ |\ \ \sum_i p_i^2 + q_i ^2 < r^2\right\}
\]
equipped with the standard symplectic form
$\omega:= \sum_i dp_i \wedge dq_i$

\definition
{\bf \blue Symplectic volume}
of a symplectic manifold $(M, \omega)$, $\dim_\R M =2n$,
is $\Vol(M, \omega):= \int_M \omega^n$.

Symplectic capacity {\bf \purple is a way to distinguish
diffeomorphic symplectic manifolds of the same volume.}

\newpage
 
{\bf \blue Volume and capacity (2)}

\exercise
Let $\nu$ be a volume form on a compact manifold $M$,
$\nu'$ a volume form on $M'$, diffeomorphic to 
$M$. Assume that $\int_M\nu= \int_{M'}\nu'$.
{\bf \purple Prove that there exists a diffeomorphism
$\phi:\; M \arrow M'$
such that $\phi^* \omega' = \omega$.}

\exercise
Let $\nu$ be a volume form on a manifold $M$, not necessarily compact, 
and $\nu'$ a volume form on $M'$, diffeomorphic to 
$M$. Assume that $\int_M\nu= \int_{M'}\nu'$.
{\bf \purple Prove that for any open subset
$U\subset M$ with compact closure there exists a diffeomorphism
$\phi:\; M \arrow M'$
such that $\phi^* \omega'\restrict U = \omega\restrict U$.}

\question
Let $A, B\subset \R^{2n}$
be two manifolds homeomorphic to a ball,
of the same symplectic volume.
{\bf \red Are they symplectomorphic?}

{\bf \green ANSWER:} Not always,
because {\bf \purple the symplectic capacity might be different.}


\newpage

{\bf \blue Gromov capacity}

\definition
{\bf \green An (open) symplectic embedding} 
is an open embedding of symplectic manifolds,
symplectimorphic to its image. 

\definition
Let $(M,\omega)$ be a symplectic manifold,
and $r$ a supremum of radii of all symplectic
balls of the same dimension, admitting
a symplectic embedding to $M$.
The number $\capa(M,\omega):=\pi r^2$
is called {\bf \blue Gromov symplectic capacity}
of $M$.

\theorem {\bf \blue (Ekeland-Hofer)}\\
Let $\phi$ be an oriented diffeomorphism of symplectic manifold.
Then {\bf \red $\phi$ is a symplectomorphism if and only if $\phi$
preserves the symplectic capacity of all open subsets.}

\proof Later in these lectures. \endproof

\newpage

{\bf \blue Gromov Non-Squeezing Theorem}

\definition
{\bf \blue A symplectic cylinder} $C_r$ is $\R^{2n}\times B_r$,
where $\R^{2n}$ is equipped with the standard symplectic
form  $\sum_i dp_i\wedge dq_i$, and $B_r$ is the standard symplectic
ball of radius $r$ in $\R^2$.

\theorem {\bf \blue (Gromov)}
{\bf \red Symplectic capacity of a symplectic cylinder
$C_r$ is equal to $\pi r^2$.}

\proof  Later in these lectures. \endproof

\remark
The volume is not the only obstruction to symplectic embeddings.
Indeed, {\bf \purple the volume of the symplectic cylinder is infinite.}


{\bf \green Related:} This theorem is also
called ``Symplectic camel theorem'', or ``Gromov Non-Squeezing Theorem''.

\centerline{\epsfig{file=camel-and-needle.jpg,width=0.45\linewidth}}


\newpage

{\bf \blue Symplectic packing}

\exercise
Prove that {\bf \red there
exists a  non-degenerate 
$U(n+1)$-invariant symplectic form $\omega$ on $\C P^n$.}
Prove that such $\omega$ is unique, up to a constant multiplier.
Prove that {\bf \red this form is closed.}

\definition
The form $\omega$ is called {\bf\blue the Fubini-Study form}.
We usually fix the constant multiplier in such a way
that $\Vol (\C P^n, \omega)=1$.

\theorem
Let $v_N$ be a supremum of the total
volume of $N$ equal symplectic balls 
which admit a disjoint symplectic embedding to $\C P^2$ of volume 1.
Then 
 \setlength{\tabcolsep}{10pt}
\renewcommand{\arraystretch}{1.5}
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
N&1&2&3&4&5&6&7&8&9& N>9\\ \hline
\nu_N&1&\frac 1 2&\frac 3 4 & 1 & \frac {20}{25} &\frac {24}{25}
& \frac{63}{64} & \frac{288}{289} & 1 & 1 \\ \hline
\end{array}
\]

{\bf \blue The first 4 numbers are due to Gromov, last to Biran,
the rest are McDuff-Polterovich.}


\newpage

{\bf \blue Nagata conjecture}

\remark
{\bf \purple These numbers are related to Nagata conjecture},
which is still unsolved (Biran used Taubes' work on 
Seiberg-Witten invariants to avoid proving it).

\conjecture
Suppose $p_1, ..., p_r$ are very general points in 
$\C P^2$ and that $m_1, ..., m_r$ are positive integers. 
{\bf \red Then for 
any $r > 9$, any complex curve C in $\C P^2$ that passes through each of the 
points $p_i$ with multiplicity $m_i$ must satisfy
$\deg C > \frac{1}{\sqrt{r}}\sum_{i=1}^r m_i.$}

\remark Nagata conjecture was known already to Nagata 
when $r$ is a full square,
{\bf \purple and
unknown for all other $r$.} Biran's theorem is considered
as the {\bf \purple symplectic version of Nagata's conjecture.}

\newpage

{\bf \blue An exercise for Wednesday}


\exercise Let $\alpha_t\in\Lambda^k(M)$, $t\in [0,1]$ be
a smooth family of exact forms. Prove that
{\bf \red 
there exists a smooth family of forms $\eta_t\in\Lambda^{k-1}(M)$, $t\in [0,1]$,
such that $d\eta_t = \alpha_t$.}




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