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  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf 
Unobstructed symplectic packing (1)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf TWENTY-THIRD G\"OKOVA
GEOMETRY / TOPOLOGY CONFERENCE\\
May 30 - June 4 (2016)\\ G\"okova, Turkey}
\end{center}


\newcommand{\Per}{\operatorname{\sf Per}}
\newcommand{\Gr}{\operatorname{\sf Gr}}
\newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}}
\newcommand{\Teich}{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}
\newcommand{\Symp}{\operatorname{Symp}}
\newcommand{\Hyp}{\operatorname{Hyp}}



\newpage

{\bf \blue Hyperk\"ahler manifolds (reminder)}


\definition
A {\bf \blue hyperk\"ahler structure} on a manifold $M$
is a Riemannian structure $g$ and a triple of complex
structures $I,J,K$, satisfying quaternionic relations
$I\circ J = - J \circ I =K$, such that $g$ is K\"ahler
for $I,J,K$.

{\bf\green REMARK:} This is equivalent to $\nabla I=\nabla J = \nabla K=0$:
the parallel translation along the connection preserves $I, J,K$.

{\bf\green DEFINITION:} 
Let $M$ be a Riemannian manifold, $x\in M$ a point.
The subgroup of $GL(T_xM)$ generated by parallel 
translations (along all paths) is called {\bf \blue 
the holonomy group} of $M$.

{\bf\green REMARK:}
{\bf \purple A hyperk\"ahler manifold can be defined as a manifold which has
holonomy in $Sp(n)$ } (the group of all endomorphisms preserving
$I,J,K$).


\newpage

{\bf \blue Main result of lecture 2:}

\definition
A symplectic structure $\omega$ on a hyperk\"ahler
manifold is called {\bf 
\blue standard} if $\omega$ is a K\"ahler form for
some hyperk\"ahler structure.

\remark
Any known symplectic structure on a hyperk\"ahler manifold
or a torus is of this type. {\bf \purple It was conjectured that non-standard
symplectic structures don't exist.}


\theorem {\bf \blue (E. Amerik, V.)}\\
Let $M$ be a maximal holonomy hyperk\"ahler manifold.
Then the period map $\Per:\; \Teich_s \arrow H^2(M,\R)$
{\bf \red is an open embedding on the set of all standard symplectic
structures}, and {\bf \red its image is the set of all cohomology classes
$v$ such that $q(\omega, \omega) >0$,} where $q$ is a quadratic form 
on cohomology defined below.

\newpage

{\bf \blue Bogomolov-Beauville-Fujiki form}

{\bf \green THEOREM:} (Fujiki)
Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is
hyperk\"ahler. Then $\int_M \eta^{2n}=c q(\eta,\eta)^n$,
for some primitive integer quadratic form $q$ on $H^2(M,\Z)$,
and $c>0$ an integer number.


{\bf \green Definition:} This form is called
{\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined
by the Fujiki's relation uniquely, up to a sign.} The sign is determined
from the following formula (Bogomolov, Beauville)
\begin{align*}  \lambda q(\eta,\eta) &=
   \int_X \eta\wedge\eta  \wedge \Omega^{n-1}
   \wedge \bar \Omega^{n-1} -\\
 &-\frac {n-1}{n}\left(\int_X \eta \wedge \Omega^{n-1}\wedge \bar
   \Omega^{n}\right) \left(\int_X \eta \wedge \Omega^{n}\wedge \bar \Omega^{n-1}\right)
\end{align*}
where $\Omega$ is the holomorphic symplectic form, and 
$\lambda>0$.

{\bf \green Remark:}  {\bf \red $q$ has signature $(b_2-3,3)$.}
It is positive
definite on $\langle \Omega, \bar \Omega, \omega\rangle$,
 where $\omega$ is a K\"ahler form. 


\newpage

{\bf \blue Ergodicity of mapping class group action}

\theorem (V., 2009)\\
Let $M$ be a maximal holonomy hyperk\"ahler manifold.
{\bf \purple Then the image of the mapping class group 
$\Gamma$ in $O(H^2(M,\Z))$ has finite index.}

\corollary {\bf \red $\Gamma$ acts on $\Teich_s$ with dense orbits.}

{\bf \green Proof:} We use a theorem of Calvin Moore:

\theorem (Calvin C. Moore, 1966)
Let $\Gamma$ be a lattice in a non-compact 
simple Lie group $G$ with finite center, and $H\subset G$ a 
non-compact semisimple Lie subgroup. {\bf \red Then the left 
action of $\Gamma$ on $G/H$ is ergodic.}

Applying this theorem to $\Gamma$ inside $G=SO(H^2(M,\R), q)$
and $H$ the stabilizer of $\omega\in H^2(M,\R)$,
we obtain that the action of $\Gamma$ on $\Teich_s\subset H^2(M,\R)$
{\bf \purple is ergodic on the set of symplectic form of a given volume}, 
hence has dense orbits. \endproof

\corollary 
Any continuous invariant of symplectic structures is constant!

\hfil

{\Large \bf \red Let's look for such invariants!}

\hfill




\newpage

{\bf \blue Full symplectic packing}

\definition
A {\bf \blue symplectic ball} is a ball of radius $r$ in
$\R^{2n}$, equipped with a standard symplectic structure
$\omega=\sum dp_i \wedge dq_i$.

\definition
Let $M$ be a compact symplectic manifold
of volume $V$. We say that $M$ {\bf \blue admits a full, or unobstructed,
symplectic packing} if for any disconnected union $S$
of symplectic balls of total volume less than
$V$,  $S$ admits a symplectic embedding to $M$.


\definition
A symplectic structure $\omega$ on a torus is called {\bf 
\blue standard} if there exists a flat torsion-free connection
preserving $\omega$. 


\theorem {\bf \blue (Latschev, McDuff, Schlenk, 2011)}\\
{\bf \red All 4-dimensional tori with standard symplectic structures
admit full symplectic packing.}

\newpage

{\bf \blue Main result}


\definition
Let $M$ be a compact symplectic manifold
of volume $V$. We say that $M$ {\bf \blue admits a full, or unobstructed,
symplectic packing} if for any disconnected union $S$
of symplectic balls of total volume less than
$V$,  $S$ admits a symplectic embedding to $M$.



\definition
A symplectic structure $\omega$ on a torus is called {\bf 
\blue standard} if there exists a flat torsion-free connection
preserving $\omega$. 
A symplectic structure $\omega$ on a hyperk\"ahler manifold
is called {\bf 
\blue standard} if $\omega$ is a K\"ahler form for
some hyperk\"ahler structure.

\remark
Any known symplectic structure on a hyperk\"ahler manifold
or a torus is of this type. {\bf \red It was conjectured that non-standard
symplectic structures don't exist.}


\theorem (M. Entov, V.)\\
Let $M$ be a compact even-dimensional torus, 
or a hyperk\"ahler manifold (such as a K3 surface),
and $\omega$ a standard symplectic form. {\bf \red Then $(M, \omega)$
admits a full symplectic packing.}

\remark In this talk, {\bf \purple all tori are compact,
even-dimensional, and satisfy $\dim_\R M\geq 4$.}

\newpage

{\bf \blue  Gromov Capacity}

\definition
Let $M$ be a symplectic manifold. Define
{\bf\blue Gromov capacity} $\mu(M)$ as the supremum 
of radii $r$, for all symplectic embeddings from
a symplectic balls $B_r$ to $M$.

\definition
Define {\bf \blue symplectic volume}
of a symplectic manifold $(M,\omega)$
as $\int_M \omega^{\frac 1 2\dim_\R M}$.

\remark
Gromov capacity is obviously bounded by the symplectic volumes:
a manifold of Gromov capacity $r$ has volume $\ge \Vol(B_r)$.
However, {\bf \purple there are manifolds of infinite volume with finite
Gromov capacity.}

\theorem
{\bf \blue (Gromov)} \\
Consider {\bf \blue a symplectic cylinder}
$C_r:=\R^{2n-2}\times B_r$ with the product symplectic
structure. Then the Gromov capacity of $C_r$ is $r$.

\remark
This result was used by Gromov to study
symplectic packing in $\C P^2$. He proved that {\bf \purple there is
no full symplectic packing,} and found precise bounds.

\newpage

{\bf \blue Symplectic packing in $\C P^2$ (Gromov, McDuff, Polterovich, Biran)}

\theorem
Let $v_N$ be a supremum of number $V$ such that
a collection of $N$ equal symplectic balls of total
volume $V$ can be embedded to symplectic $\C P^2$ 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 few numbers are due to Gromov, last to Biran,
the rest are McDuff-Polterovich.}

\remark
{\bf \red 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 given positive integers. 
{\bf \red Then for 
any $r > 9$ any 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 $r$, even when $p_1, ..., p_r$ are generic.}


\newpage

{\bf \blue Ekeland-Hofer theorem}

\theorem {\bf \blue (Ekeland-Hofer)} \\
Let $M$, $N$ be symplectic manifolds, and 
$\phi:\; M \arrow N$ a diffeomorphism. Suppose that for all
sufficiently small, convex open sets $U\subset M$, Gromov capacity
satisfies $\mu(U)= \mu(\phi(U))$. 
{\bf \red Then $\phi$ is a symplectomorphism.}

\remark
This can be used to define {\bf \purple 
$C^0$- (continuous) symplectomorphisms.}

\remark
Ekeland-Hofer theorem implies a theorem of Gromov-Eliashberg:
{\bf \purple symplectomorphism group is $C^0$-closed in the group of
diffeomorphisms.}

\newpage

{\bf \blue McDuff and Polterovich for K\"ahler manifolds}


\definition
Let $M$ be a symplectic manifold, $x_1, ..., x_n\in M$ distinct points,
and $r_1, ..., r_n$ a set of positive numbers. We say that $M$ {\bf \blue admits 
symplectic packing} with centers $x_1, ..., x_n$ and radii $r_1, ..., r_n$
if there exists a symplectic embedding from a disconnected union 
of symplectic balls of radii $r_1, ..., r_n$ to $M$ mapping centers
of balls to $x_1, ..., x_n$.

\theorem {\bf \blue (McDuff, Polterovich, 1995)}\\
Let $(M, \omega)$ be a K\"ahler manifold,
 $\tilde M\stackrel \nu \arrow M$ its blow-up in $x_1, ..., x_n$,
$E_i$ the corresponding exceptional divisors, and $[E_i]$ their
fundamental classes. Assume that the class
$\nu^*\omega- \sum_i c_i [E_i]$ is K\"ahler,
for some $c_i >0$. {\bf \red Then  $M$ admits a symplectic
packing with radii $r_i=\pi^{-1}\sqrt{c_i}$.}

\remark {\bf \purple Converse is also true} 
(next lecture).



\end{document}


