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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf 
Unobstructed symplectic packing (2)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

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


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\newcommand{\Hyp}{\operatorname{Hyp}}


\newpage

{\bf \blue  Gromov Capacity (reminder)}

\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 Complex manifolds (reminder)}

{\bf\green DEFINITION:} Let $M$ be a smooth manifold. 
An {\bf \blue almost complex structure} is an operator
$I:\; TM \arrow TM$ which satisfies $I^2 = - \Id_{TM}$.

The eigenvalues of this operator are $\pm \1$.
The corresponding eigenvalue 
decomposition is denoted $TM=T^{0,1}M\oplus T^{1,0}(M)$.

{\bf\green DEFINITION:}
An almost complex structure is {\bf \blue integrable}
if $\forall X,Y \in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$.
In this case $I$ is called {\bf \blue a complex structure operator}.
A manifold with an integrable almost complex structure
is called {\bf \blue a complex manifold}. 

{\bf\green THEOREM:} (Newlander-Nirenberg)\\
{\bf \purple This definition is equivalent to the usual one.}

\newpage

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


{\bf\green DEFINITION:} A Riemannian metric $g$ on
an almost complex manifold $M$ is called 
{\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$.
In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$,
hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric.

{\bf\green DEFINITION:} The differential 
form $\omega\in \Lambda^{1,1}(M)$ is called
{\bf \blue the Hermitian form} of $(M,I,g)$.

{\bf\green THEOREM:} Let $(M,I,g)$ be an almost complex Hermitian
manifold. {\bf \purple Then the following conditions are equivalent.}

(i) The complex structure $I$ is integrable, and 
the Hermitian form $\omega$ is closed.

(ii) One has $\nabla(I)=0$, where $\nabla$ is the Levi-Civita connection 
\[ \nabla:\; \End(TM) \arrow \End(TM)\otimes \Lambda^1(M).\]

{\bf\green DEFINITION:} A complex Hermitian manifold $M$
is called {\bf \blue K\"ahler} if either of these conditions hold.
The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ 
is called {\bf \blue the K\"ahler class} of $M$. The set
of all K\"ahler classes is called {\bf \blue the K\"ahler cone}.

\newpage

{\bf \blue K\"ahler structure on a blow-up}

\definition
Let $S$ be a total space of a line bundle
$\calo(-1)$ on $\C P^n$, identified with a space
of pairs $(z\in \C P^n, t\in z)$, where $t$ is a point
on a line $z\subset \C^{n+1}$ representing $z$.
The forgetful map $\pi:\; S \arrow \C^{n+1}$ is called {\bf \blue
a blow-up of $\C^{n+1}$ in 0}. Given an open ball
$B\subset \C^{n+1}$, the map $\pi:\; \pi^{-1}(B)\arrow B$
is called {\bf \blue a blow-up of $B$ in 0}.
To blow up a point in a complex manifold $M$, we remove
a ball $B$ around this point, and replace it with a
blown-up ball $\tilde B$, gluing $B\backslash x\subset \tilde B$ 
with $B\backslash x\subset M$.

\problem Suppose that $M$ is K\"ahler, and $\tilde M$ is
its blow-up. {\bf \purple Find a K\"ahler metric on $\tilde M$ and write
it explicitly.}

{\bf \green Answer:} {\bf \red Symplectic blow-up!}


\remark In this talk, I would often drop all $\pi$
and other constants from the equations. 

\newpage

{\bf \blue Symplectic quotient}

\definition
Let $\rho$ be an $S^1$-action on a symplectic
manifold $(M,\omega)$ preserving the symplectic structure,
and $\vec v$ its unit tangent vector. Cartan's formula
gives $0=\Lie_{\vec v}\omega= d(\omega\cntrct {\vec v})$,
hence $\omega\cntrct {\vec v}$ is a closed 1-form.
{\bf \blue Hamiltonian}, or {\bf \blue moment map}
of $\rho$ is an $S^1$-invariant function $\mu$ such that 
$d\mu =\omega\cntrct {\vec v}$, and {\bf \blue symplectic
quotient} $M\2\!_c S^1$ is $\mu^{-1}(c)/S^1$.

\remark
In these assumptions, 
restriction of the symplectic form 
$\omega$ to $\mu^{-1}(c)$ vanishes on $\vec v$, hence it
is {\bf \purple obtained as a pullback of a closed 2-form $\omega_{\!\2\!}$
on $M\2\!_c S^1$.}

\theorem
{\bf \red The form $\omega_{\!\2\!}$ is a symplectic form on $M\2\!_c S^1$.}
In other words, {\bf \purple 
the symplectic quotient is a symplectic manifold.}

\remark
If, in addition, $M$ is equipped with a K\"ahler structure
$(I,\omega)$, and $S^1$-action preserves the complex structure,
the symplectic quotient $M\2\!_c S^1$ inherits the K\"ahler structure.
In this case it is called {\bf\blue a K\"ahler quotient}. Whenever
the $S^1$-action can be integrated to holomorphic $\C^*$-action,
{\bf \purple the K\"ahler quotient is identified with an open subset
of its orbit space.}

\remark
The moment map is defined by $d\mu =\omega\cntrct {\vec v}$
uniquely up to a constant. However, the symplectic quotient
$M\2\!_c S^1=\mu^{-1}(c)/S^1$ {\bf \purple depends heavily on the
choice of $c\in \R$}.


\newpage

{\bf \blue Symplectic blow-up}

\claim
Consider the standard $S^1$-action on $\C^n$, and
let $W\subset \C^n$ be an $S^1$-invariant open subset.
Consider the product $V:= W \times \C$ with
the standard symplectic structure and take the $S^1$-action
on $\C$ opposite to the standard one. 
{\bf \purple Then its moment map is $w-t$, where
$w(x)=|x|^2$ is the length function on $W$ and
$r(t)=|t|^2$ the length function on $\C$.}

\definition
{\bf \blue Symplectic cut} of $W$ is $(W \times \C)\2\!_c S^1$.

\remark
Geometrically, the symplectic cut is obtained as follows.
Take $c\in \R$, and let $W_c:=\; \{w\in W\ \ |\ \ |w|^2 \leq c\}$.
Then $W_c$ is a manifold with boundary $\6 W_c$, which is a sphere $|w|^2=c$.
Then $(W \times \C)\2\!_c S^1= (W_c \times \C)\2\!_c S^1$ is obtained
from $W_c$ by gluing each $S^1$-orbit which lies on $\6 W_c$
to a point. Combinatorially, $(W \times \C)\2\!_c S^1$ is 
$\C^n$ with 0 replaced with $\C P^{n-1}$.

\definition
In these assumptions,
{\bf \blue symplectic blow-up} of radius $\lambda=\sqrt c$ of $W$ in 0 is 
$(W \times \C)\2\!_c S^1$. {\bf \blue Symplectic blow-up} of a
symplectic manifold $M$ is obtained by removing a symplectic
ball $W$ and gluing back a blown-up symplectic ball
$(W \times \C)\2\!_c S^1$.

%\remark
%The {\bf \purple symplectic 
%form $\omega_c$ on the blow-up $(W \times \C)\2\!_c S^1$
%depends on $c$ as follows:} $\int_l \omega_c=c$, where $l\subset E$
%is a rational line on an exceptional divisor $E:=\pi^{-1}(c)$.


\newpage

{\bf \blue McDuff and Polterovich: symplectic packing from symplectic blow-ups}

\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 $x_1, ..., x_n$.

\remark
The choice of $x_i$ is irrelevant, because {\bf \purple the group of symplectic
authomorphisms acts on $M$ infinitely transitively.}

{\bf \green Theorem 1:} 
(McDuff-Polterovich)\\
Let $(M,\omega)$ be a symplectic manifold, $x_1, ..., x_n\in M$ distinct points,
and $c_1, ..., c_n$ a set of positive numbers.  Let $\pi:\; \tilde M \arrow M$
be a symplectic blow-up with centers in $x_i$, and $E_i\in H^2(\tilde M, \Z)$
the fundamental classes of its exceptional divisors. Then the
following conditions are equivalent.

(i) {\bf \red $M$ admits a symplectic
packing with radii $r_i=\pi^{-1}\sqrt{c_i}$} 

(ii) For any $\alpha \in [0,1]$,
{\bf \red there exists a 
form $\omega_\alpha(c_1, ..., c_n)$ cohomologically equivalent
to $\pi^* \omega- \sum \alpha\pi c_i E_i$}, symplectic
for $\alpha >0$, smoothly depending
on $\alpha$, and satisfying $\omega_0(c_1, ..., c_n)=\pi^*\omega$.
\endproof

\newpage

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

\remark
{\bf \purple In K\"ahler situation, the smooth dependence condition is 
trivial,} because for any two K\"ahler forms $\omega, \omega'$, 
straight interval connecting $\omega$ to $\omega'$ 
consists of K\"ahler forms
(indeed, {\bf \purple the set of K\"ahler forms is convex}). 
This brings the following corollary.

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


\newpage

{\bf \blue McDuff and Polterovich for tamed manifold}


\definition
An almost complex structure $I$ on $M$ {\bf\blue is tamed by a symplectic
form $\omega\in \Lambda^2 M$} if $\omega(x,Ix)>0$ for any non-zero tangent
vector $x\in TM$.

\theorem (McDuff-Polterovich, 1995)  Let $(M,\omega)$
be a compact symplectic manifold, 
$\tilde M\stackrel \nu \arrow M$ its symplectic blow-up in $x_1, ..., x_n$,
$E_i$ the corresponding exceptional divisors, $[E_i]$ their
fundamental classes, and $r_1,\ldots, r_k$ a collection
of positive numbers. Assume there exists an almost complex structure $I$ of
on $M$ tamed by $\omega$ and a symplectic form
$\tilde \omega$ on $\tilde M$ taming the pullback
almost complex structure $\tilde I$ so that 
$[\tilde \omega] = \nu^* [\omega]
- \pi \sum_{i=1}^k r_i^2 [E_i]$. {\bf \red Then $(M,\omega)$ admits a
symplectic embedding of $\bigsqcup\limits_{i=1}^k B^{2n} (r_i)$.}
\endproof

\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 $I, J,K$.

\corollary The group $SU(2)$ of orthogonal quaternions
acts on triples $(I,J,K)$ producing new hyperk\"ahler
structures.

{\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 Calabi-Yau and Bogomolov decomposition theorem (reminder)}


\remark {\bf \purple A hyperk\"ahler manifold is holomorphically
symplectic:} $\omega_J+\1 \omega_K$ is a holomorphic
symplectic form on $(M,I)$. 


{\bf\green THEOREM:} (Calabi-Yau) 
A compact, K\"ahler, holomorphically symplectic manifold
{\bf \red admits a unique hyperk\"ahler metric in any K\"ahler class.}


\claim
A compact hyperk\"ahler manifold $M$ {\bf \purple has maximal holonomy of 
Levi-Civita connection $Sp(n)$} if and only if {\bf \purple
$\pi_1(M)=0$, $h^{2,0}(M)=1$.}

\theorem (Bogomolov decomposition)\\
{\bf \red Any compact hyperk\"ahler manifold has a finite covering isometric
to a product of a torus and several maximal holonomy hyperk\"ahler
manifolds.}





%\newpage
%
%{\bf \blue Holomorphically symplectic manifolds}
%
%{\bf\green DEFINITION:} A holomorphically symplectic manifold 
%is a complex manifold equipped with non-degenerate, holomorphic
%$(2,0)$-form.
%
%
%\remark A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\
%$\omega_I:=  g(I\cdot, \cdot)$, $\omega_J:=  g(J\cdot, \cdot)$,
%$\omega_K:=  g(K\cdot, \cdot)$.}
%
%\claim 
%In these assumptions, $\omega_J + \1\omega_K$ is
%holomorphic symplectic on $(M,I)$.
%
%
%
%{\bf\green THEOREM:} (Calabi-Yau) 
%A compact, K\"ahler, holomorphically symplectic manifold
%admits a unique hyperk\"ahler metric in any K\"ahler class.
%
%{\bf\green DEFINITION:}  For the rest of this talk, 
%{\bf \red a hyperk\"ahler manifold
%is a compact, K\"ahler, holomorphically symplectic manifold.}
%
%

%\newpage
%
%{\bf \blue EXAMPLES.}
%
%%\example {\bf \purple A non-compact example:} $T^* \C P^n$ (Calabi).
%%
%%\remark $T^*\C P^1$ {\bf \blue
%%is a resolution of a singularity $\C^2/{\pm1}$.}
%%
%%\remark Let $M$ be a 2-dimensional complex manifold with 
%%holomorphic symplectic form outside of singularities, which are
%%all of form $\C^2/{\pm1}$. Then {\bf \purple its resolution is also
%%holomorphically symplectic.}
%
%\example Even-dimensional complex torus 
%$\C^{2n}/\Z^{4n}={\Bbb H}^{n}/\Z^{4n}$
%
%\example Take a 2-dimensional complex torus $T$,
%then $T/{\pm1}$ is an orbifold with 16 double points.
%Its resolution $\widetilde {T/{\pm1}}$ is called 
%{\bf \blue a Kummer surface}. {\bf \red
%It is holomorphically symplectic}.
%
%%\remark Take a symmetric square $\Sym^2 T$, with a natural
%%action of $T$, and let $T^{[2]}$ be a blow-up of a singular
%%divisor. {\bf \purple Then $T^{[2]}$ is naturally isomorphic to the
%%Kummer surface $\widetilde {T/{\pm1}}$.}
%
%\definition
%{\bf \blue A K3-surface} is a deformation of a Kummer surface.
%
%\theorem Any complex compact surface with $c_1(M)=1$
%and $H^1(M)=0$ {\bf \purple is isomorphic to K3.} Moreover, 
%{\bf \purple it is hyperk\"ahler.}
%
%\newpage 
%
%{\bf \blue Hilbert schemes} 
%
%%\remark {\bf\blue A complex surface} is a 2-dimensional complex manifold.
%
%\definition
%A {\bf\blue Hilbert scheme} $M^{[n]}$ of a complex surface $M$ is
%a classifying space of all ideal sheaves $I\subset \calo_M$ 
%for which the quotient $\calo_M/I$ has dimension $n$
%over $\C$.
%
%\remark 
%A Hilbert scheme {\bf \purple is obtained as a resolution of singularities}
%of the symmetric power $\Sym^n M$.
%
%\theorem (Fujiki, Beauville) {\bf \red A Hilbert scheme of
%a holomorphically symplectic manifold is hyperk\"ahler.}
%
%\example
%{\bf\blue A Hilbert scheme of K3}.
%
%\example
%Let $T$ is a torus. Then it acts on its Hilbert scheme
%freely and properly by translations. For $n=2$, the quotient $T^{[n]}/T$
%is a Kummer K3-surface. For $n>2$, it is called
%{\bf \blue a generalized Kummer variety}. 
%
%\remark There are 2 more ``sporadic'' examples
%of compact hyperk\"ahler manifolds, constructed by K. O'Grady.
%{\bf \purple All known compact hyperkaehler manifolds are
%finite quotients of the products
%of these 2 and the three series:} tori, Hilbert schemes of K3, and
%generalized Kummer.


\newpage

{\bf \blue Campana simple manifolds}

\definition
A complex 
manifold $M$, $\dim_\C M>1$,
 is called {\bf\blue Campana simple} if the union ${\goth U}$
of  all complex subvarieties $Z\subset M$ satisfying $0< \dim Z< \dim M$
has measure 0. A point which belongs to $M\backslash {\goth U}$ is called
{\bf \blue generic}.

\remark {\bf \red Campana simple manifolds are non-algebraic}.
Indeed, a manifold which admits a globally defined meromorphic function
$f$ is a union of zero divisors for the functions $f-a$, for all
$a\in \C$, and the zero divizor for
$f^{-1}$. Hence {\bf \purple Campana simple manifolds 
admit no globally defined meromorphic functions}.

\example
{\bf \purple 
A general complex torus has no non-trivial complex subvarieties,} hence
it is Campana simple.

\example
Let $(M,I,J,K)$ be a hyperk\"ahler manifold, and 
$L=aI+bJ+cK$, $a^2+b^2+c^2=1$ be a complex structure
induced by quaternions. Then for all such $(a,b,c)$
outside of a countable set, {\bf \purple all complex subvarieties
$Z\subset (M,L)$ are hyperk\"ahler, and (unless $M$ a finite
quotient of a product) $\bigcup_Z Z\neq M$} (V., 1994, 1996). {\bf \red
Therefore, $(M,L)$ is Campana simple.}

\conjecture (Campana)\\
Let $M$ be a Campana simple K\"ahler manifold.
{\bf \green Then $M$ is bimeromorphic to a finite quotient of a 
hyperk\"ahler orbifold or a torus.}

\newpage


{\bf \blue Demailly-Paun theorem}

\remark
Let $M$ be a compact K\"ahler manifold. Recall that
the cohomology space $H^2(M, \C)$ is decomposed as
$H^2(M, \C)= H^{2,0}(M)\oplus H^{1,1}(M)\oplus H^{0,2}(M)$
with $H^{1,1}(M)$ identified with the space of $I$-invariant
harmonic 2-forms, and $H^{2,0}(M)\oplus H^{0,2}(M)$
the space of $I$-antiinvariant
harmonic 2-forms. This decomposition is called {\bf \blue Hodge
decomposition}. The space $H^{1,1}(M)$ is a complexification
of a real space $H^{1,1}(M,\R)= \{\nu \in H^2(M,\R)\ \ |\ \ I(\nu)=\nu\}$.

\theorem
(Demailly-P\v aun, 2002)\\
Let $M$ be a compact K\"ahler manifold, and $\hat K(M)\subset H^{1,1}(M,\R)$
a subset consisting of all (1,1)-forms $\eta$ which satisfy
$\int_Z \eta^k>0$ for any $k$-dimensional complex subvariety 
$Z\subset M$. {\bf \red Then the K\"ahler cone of $M$ is one of the
connected components of $\hat K(M)$.}
\endproof

\newpage


{\bf \blue K\"ahler cone for blow-ups of Campana simple manifolds}

{\bf \green Theorem 2:}
Let $M$ be a Campana simple  compact K\"ahler manifold,
and $x_1, ..., x_n$ distinct generic points of $M$.
Consider the blow-up $\tilde M$ of $M$ in $x_1, ..., x_n$, 
let $E_i$ be the corresponding blow-up divisors, and
$[E_i]\in H^2(M,\Z)$ its fundamental classes. Decompose
$H^{1,1}(\tilde M,\R)$ as 
$H^{1,1}(\tilde M,\R)= H^{1,1}(M,\R)\oplus \bigoplus \R[E_i]$.
Assume that $\eta_0$ is a K\"ahler class on $M$.
{\bf \red Then for any $\eta=\eta_0 + c_i [E_i]$, the
following conditions are equivalent.}\\
\phantom{AAA}\ (i) $\eta$ is K\"ahler on $\tilde M$. \\
\phantom{AAA}\ (ii) all $c_i$ are negative, and
$\int_M \eta^{\dim_\C M}>0$.

{\bf \green Proof of (ii) $\Rightarrow$ (i). Step 1:}\\
All proper complex subvarieties of $\tilde M$ are either contained
in $E_i$, or do not intersect $E_i$. The condition
``$\eta_0$ is K\"ahler on $M$'' implies
$\int_Z \eta^k>0$ for all subvarieties not intersecting $E_i$.
Since $[E_i]$ restricted to $E_i$ is $-[\omega_{E_i}]$,
where $\omega_{E_i}$ is the Fubini-Study form, 
$c_i <0$ implies that $\int_Z \eta^k>0$ for all 
subvarieties which lie in $E_i$. 
Finally, the integral of $\eta$ over $\tilde M$ is positive
by the assumtion $\int_M \eta^{\dim_\C M}>0$.
{\bf \purple Therefore, the condition
(ii) implies that $\eta\in \hat K(\tilde M)$. }


\newpage


{\bf \blue K\"ahler cone for blow-ups of Campana simple manifolds (cont.)}

{\bf \green Theorem 2:}
Let $M$ be a Campana simple  compact K\"ahler manifold,
and $x_1, ..., x_n$ distinct generic points of $M$.
Consider the blow-up $\tilde M$ of $M$ in $x_1, ..., x_n$, 
let $E_i$ be the corresponding blow-up divisors, and
$[E_i]\in H^2(M,\Z)$ its fundamental classes. Decompose
$H^{1,1}(\tilde M,\R)$ as 
$H^{1,1}(\tilde M,\R)= H^{1,1}(M,\R)\oplus \bigoplus \R [E_i]$.
Assume that $\eta_0$ is a K\"ahler class on $M$.
{\bf \red Then for any $\eta=\eta_0 + c_i [E_i]$, the
following conditions are equivalent.}\\
\phantom{AAA}\ (i) $\eta$ is K\"ahler on $\tilde M$. \\
\phantom{AAA}\ (ii) all $c_i$ are negative, and
$\int_M \eta^{\dim_\C M}>0$.

{\bf \green Proof of (ii) $\Rightarrow$ (i). Step 2:}\\
The form $\eta_0$ is K\"ahler on $M$, hence it lies
on the boundary of the K\"ahler cone of $\tilde M$,
and $\eta_0$ can be obtained as a limit 
\[ \eta_0=\lim_{\epsilon\rightarrow 0} \eta_0 + \epsilon c_i [E_i]
\]
of forms which lie in the same connected component of $\hat K(\tilde M)$.
Therefore, {\bf \purple $\eta$ belongs to the same connected component
of $\hat K(\tilde M)$ as a K\"ahler form.} By Demailly-P\v aun,
this implies that $\eta$ is K\"ahler.

{\bf \green Proof of (i) $\Rightarrow$ (ii).}\\
The numerical conditions of (ii) mean that 
$\eta\in \hat K(\tilde M)$, hence they are satisfied 
automatically, as follows from Step 1.
\endproof

\newpage


{\bf \blue Campana simple manifolds and symplectic packings}


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

{\bf \green Theorem 3:}
Let $(M,I,\omega_0)$ be a K\"ahler, compact, Campana simple manifold.
{\bf \red Then $M$ admits a full symplectic packing.}

{\bf \green Proof. Step 1:} 
Let $x_1, ..., x_n$ distinct generic points of $M$.
Consider the blow-up $\tilde M$ of $M$ in $x_1, ..., x_n$, 
let $E_i$ be the corresponding blow-up divisors, and
$[E_i]\in H^2(M,\Z)$ their fundamental classes. 
As follows from McDuff-Polterovich, 
existence of full symplectic
packing on $M$ is implied by existence of a K\"ahler form
$\omega(c_1, ..., c_n)$
on $\tilde M$  with cohomology class 
$[\omega(c_1, ..., c_n)]=[\omega_0]-\sum c_i[E_i]$
for all $(c_1, ..., c_n)$ satisfying 
$\int_{\tilde M}([\omega_0]-\sum c_i[E_i])^n>0$

{\bf \green Step 2:} Such a form exists by Theorem 2.
\endproof

\newpage

{\bf \blue Symplectic packing on hyperk\"ahler manifolds
  and compact tori with irrational symplectic form}

\definition
A symplectic form is called {\bf \blue irrational}
if its cohomology class is irrational, that is, lies
in $H^2(M,\R)\backslash \R\cdot H^2(M,\Q)$.

\theorem
Let $M$ be a hyperk\"ahler manifold or a compact
torus, $\omega$ an irrational, standard symplectic form,
and ${\cal T}$ the set of complex structures for which
$\omega$ is K\"ahler. {\bf \red Then the set ${\cal T}_0\subset
{\cal T}$ of Campana simple complex structures is dense
in ${\cal T}$ and has full measure in the corresponding
moduli space.}

{\bf \green Proof:} The Hodge loci of hyperk\"ahler manifolds admitting
a non-hyperk\"ahler subvariety have positive codimension, and
deformations of hyperk\"ahler subarieties never cover $M$.
\endproof

\corollary
Let $M$ be a hyperk\"ahler manifold or a compact
torus, equipped with a standard, irrational symplectic
form $\omega$. {\bf \red Then $M$ admits full symplectic packing.}

{\bf \green Proof:} By definition of a standard symplectic
form, there exists a complex structure $I$ such that
$\omega$ is K\"ahler. Deforming $I$ in ${\cal T}$, we obtain a 
  Campana simple complex structure for which
  $\omega$ is K\"ahler. Then $(M,\omega)$ admits
full symplectic packing by Theorem 3. \endproof

\newpage

{\bf \blue Symplectic cone and K\"ahler cone}

\definition
An almost complex structure $I$ {\bf \blue tames}
a symplectic structure $\omega$ if $\omega^{1,1}_I$ is
a Hermitian form on $(M,I)$.

\proposition
Let $(M,I,\omega)$ be an almost complex tamed 
symplectic manifold, and $\eta\in \Lambda^{2,0+0,2}(M, \R)$
a closed real $(2,0)+(0,2)$-form. {\bf \red Then $\omega+\eta$ 
is also a symplectic form.} Moreover, 
{\bf \red the complex structure $I$ is tamed by
$\omega+\eta$.}

{\bf \green Proof. Step 1:} 
Since $I(\eta)=-\eta$ for each $(2,0)+(0,2)$-form $\eta$,
one has $\omega(x,Ix)=\omega^{1,1}(x,Ix) >0$ for each
non-zero $x$. 

{\bf \green Step 2:} Since $\eta^{1,1}=0$, one has
$\omega+\eta(x,Ix)=\omega^{1,1}(x,Ix) >0$
for each non-zero $x$. Therefore,  $\omega+\eta$ 
is non-degenerate. 
\endproof

\definition
A {\bf\blue symplectic class} of a manifold $M$ is a 
cohomology class of a symplectic form on $M$.
{\bf \blue Symplectic cone} of a symplectic manifold
$M$ is a set $\Symp(M)\subset H^2(M,\R)$ of all
symplectic classes. {\bf \blue Taming cone} of $(M,I)$
is a cone of symplectic classes of all symplectic
form taming an almost complex structure $I$.

{\bf \green Corollary 1:}
Let $M$ be a K\"ahler manifold, and $\Kah(M)$ its K\"ahler
cone. {\bf \purple Then the taming cone of $M$ contains 
$\Kah(M)+H^{2,0+0,2}(M, \R)$.}
\endproof


\newpage

{\bf \blue Symplectic cone for blown-up tori and hyperk\"ahler
  manifolds}

%{\bf \green Claim 1:}
%Let $M$ be a hyperk\"ahler manifold or a torus, and $K(M)$
%the union of K\"ahler cones for all deformations
%of complex structures on $M$. {\bf \red Then $K(M)$ is open in
%$H^2(M,\R)$.}
%
%\proof V., arXiv:alg-geom/9501001 Lemma 5.6. \endproof

{\bf \green Theorem 4:} 
Let $(M,I, \omega_I)$ be a compact 
K\"ahler manifold obtained as a limit of Campana simple
manifolds. {\bf \red Then $(M,\omega_I)$ admits full symplectic packing.}

{\bf \green Proof. Step 1:} 
Let $B$ be an open neighbourhood of $I$
in the moduli space of complex structures on $M$, and
$B\stackrel \phi\arrow H^2(M, \R)$ a map putting $J$ to $(\omega_I)^{1,1}_J$.
By Kodaira stability theorem, $\phi(J)$ is a K\"ahler
class for $J$ sufficiently close to $I$.
Therefore, {\bf \purple there exists
a Campana simple complex structure $J$ such that
$\omega_J:= (\omega_I)^{1,1}_J$ is K\"ahler, arbitrarily
close to $I$ in $B$. }

{\bf \green Step 2:} By Theorem 3, 
{\bf \purple $\eta_J:= \omega_J + \sum c_i [E_i]$ is a K\"ahler class on a blow-up
of $(M, J)$, with blow-up points generic.}
Indeed, the condition $\int_M \eta_J^{\dim_\C M}>0$
remains true for $J$ sufficiently close to $I$.

{\bf \green Step 3:} Now, $\omega_I-\omega_J$ is 
by definition a $(2,0)+(0,2)$-cohomology class on $(M,J)$.
Therefore, $\eta=\omega_I + \sum c_i [E_i]$ is obtained from a K\"ahler form
$\eta_J$ by adding a $(2,0)+(0,2)$-form on $(M,J)$. This implies that
{\bf \purple $\eta$ belongs to taming cone,} and Theorem 4 follows from the
taming version of McDuff-Polterovich.
\endproof



\newpage

{\bf \blue Further directions}

1. We explored symplectic packing by symplectic balls.
What about a packing by other subsets $K\subset \R^{2n}$?

1A. Define a packing number $\nu(K,M)$ of $(K,\omega)$ to $M$ as a supremum
of all $\epsilon$ for which $(K, \epsilon \omega)$
admits a symplectic embedding to $M$. This function
is obviously semicontinuous on $K$ and $M$. When $K$
is a union of symplectic balls, and $M$ a hyperk\"ahler
manifold or a torus, $\nu(K,M)= \frac{\Vol(M)}{\Vol(K)}$.
Using ergodicity, it is possible to show that $\frac{\nu(K,M)}{\Vol(M)}$
is constant for irrational symplectic structures on such $M$.
Is it equal to $1$? If so, we have ``full packing by $K$''.

2. Replacing blow-ups by orbifold blow-ups
and balls by symplectic ellipsoids with rational axis length,
our argument would give full packing by ellipsoids 
(paper in preparation, jointly with M. Entov).


3. Let $\Symp$ be the infinite-dimensional Frechet manifold
of all symplectic forms on $M$, and $\Diff$ the diffeomorphism
group. The full packing phenomena seems to be related
to ergodicity of $\Diff$-action on $\Symp$: the packing
defines a semi-continuous, $\Diff$-invariant function
on $\Symp$, which should be a posteriori constant on
the set of all symplectic structures with dense $\Diff$-orbits.
One could study other semi-continuous quantities in relation
to $\Diff$-action and ergodicity. 



%
%\newpage
%
%{\bf \blue APPENDIX: Mumford-Tate group}
%
%{\bf \blue The Mumford-Tate group}
%
%\definition 
%Let $(M,I)$ be a K\"ahler manifold,
%and ${\cal I}\in \End(H^*(M,\R))$
%{\bf \blue the Hodge decomposition operator} acting on $(p,q)$-forms
%as a multiplication by $(p-q)\1$. 
%Consider the smallest rational
%Lie subalgebra containing ${\cal I}$,
%and let  $G_{MT}\subset GL(H^*(M,\R)$
%be the corresponding Lie group. It is called
%{\bf \blue the Mumford-Tate group of $(M,I)$},
%
%\claim
%The Mumford-Tate group
%$G_{MT}$ is a connected component of a 
%group $G\subset \Aut(H^*(M))$ {\bf \purple 
%stabilizing all rational $(p,p)$-vectors}
%in the tensor algebra $T^\otimes (H^*(M))$.
%
%{\bf \green Proof:} Follows from Chevalley's theorem
%which claims that an algebraic group is determined
%by its algebra of invariants.
%
%\definition
%Let $S$ be a holomorphic family of 
%complex structures of Kaehler type on a compact 
%manifold $M$. For any rational tensor
%$v\in T^\otimes (H^*(M,\Q))$, denote
%by $Z_v\subset S$ the set of all $I\in S$
%for which $v$ has type $(p,p)$. Let
%$Z$ be the union of all $Z_v$
%of positive codimension. We say
%that $I\in S$ is {\bf \blue Mumford-Tate generic}
%if $I\notin Z$. 
%
%\newpage
%
%
%{\bf \blue The Mumford-Tate generic complex structures}
%
%\remark
%(Lower semicontinuity of Mumford-Tate group)\\
%{\bf \purple $G_{MT}(M,I)$ is the same
%for all Mumford-Tate generic $I$} and, moreover,
%$G_{MT}(I')\subset G_{MT}(M, I)$
%for all $I'\in S$.
%
%\proposition
%Let $M$ be a complex torus, $\dim_\C M\geq 2$, and 
%$\omega\in H^{1,1}(M,R)$ a cohomology class. Consider 
%the set ${\cal T}$ of all complex structures $J$ on $M$
%such that $\omega\in H^{1,1}(M,J)$. 
%Then $\bigcap_{J\in {\cal T}}H^{p,p}(M,J)=\R \omega^p$.
%
%\proof The complex structures $J:\; \R^{2n}\arrow \R^{2n}$
%preserving $\omega$ generate the stabilizer $\St(\omega)$ of $\omega$ in
%$GL(\R^{2n})$. From Chevalley's theorem it follows that
%the algebra of tensor invariants of $\St(\omega)$ is generated
%by $\omega$. \endproof
%
%\corollary 
%In these assumptions, for a general $J\in {\cal T}$.
%the Mumford-Tate group $G_{MT}(M, J)$
%is $\St(\omega)$ for rational $\omega$, and equal to
%$GL(\R^{2n})$ for irrational $\omega$.
%
%\proof Since $J$ is generic,
%$G_{MT}(M, J)$ contains $W_I$ for all $I\in {\cal T}$.
%By the Proposition above, $G_{MT}(M, J)$ is the minimal
%subgroup of $GL(\R^{2n})$ containing $\St(\omega)$.
%By Chevalley's theorem, it's $\St(\omega)$ when
%$\omega$ is rational, and $GL(\R^{2n})$ otherwise.
%\endproof
%
%
%\newpage
%
%{\bf \blue Complex torus with irrational symplectic form}
%
%\theorem
%Let $M$ be a compact complex torus, $\dim_\C M \geq 2$, 
%and $\omega$ an irrational symplectic form which is standard
%in the sense of Definition 1. {\bf \red Then there exists a 
%Campana simple complex structure $I$ on $(M,\omega)$ 
%such that $(M,I,\omega)$ is K\"ahler.}
%
%\proof
%Since $\omega$ is standard, it is flat in appropriate coordinates
%on $M=\R^{2n}/\Z^{2n}$. Choosing MT-generic $J\in {\cal T}$,
%we obtain a complex torus without cohomology classes
%which are invariant under the Mumford-Tate group. However,
%a fundamental class of a complex subvariety is always
%Mumford-Tate invariant. Therefore, $(M,J)$
%has no proper complex subvarieties.
%\endproof
%
%
%\newpage
%
%{\bf \blue Bogomolov-Beauville-Fujiki form}
%
%{\bf\green DEFINITION:} A hyperk\"ahler manifold $M$ is called
%{\bf \blue simple} if $\pi_11(M)=0$, $H^{2,0}(M)=\C$.
%
%{\bf \purple Bogomolov's decomposition:} Any 
%hyperk\"ahler manifold admits a finite covering
%which is a product of a torus and several 
%simple hyperk\"ahler manifolds.
%
%
%{\bf \red Further on, all hyperk\"ahler manifolds
%are assumed to be simple}.
%
%
%{\bf \green THEOREM:} (Fujiki). 
%Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is
%hyperk\"ahler. Then $\int_M \eta^{2n}=q(\eta,\eta)^n$,
%for some integer quadratic form $q$ on $H^2(M)$.
%
%
%{\bf \green Definition:} This form is called
%{\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined
%by this relation uniquely, up to a sign.} The sign is determined
%from the following formula (Bogomolov, Beauville)
%\begin{align*}  c q(\eta,\eta) &=
%   (n/2)\int_X \eta\wedge\eta  \wedge \Omega^{n-1}
%   \wedge \bar \Omega^{n-1} -\\
% &-(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 
%$c=C^{n-1}_{2n-2}\int_M\wedge \Omega^{n}
%   \wedge \bar \Omega^{n}>0$.
%
%\newpage
%
%{\bf \blue The period map}
%
%{\bf \green Remark:} For any $J\in \Teich$,
%$(M,J)$ is also a simple hyperk\"ahler manifold, hence
%$H^{2,0}(M,J)$ is one-dimensional. 
%
%{\bf \green Definition:} Let 
%$P:\; \Teich \arrow {\Bbb P}H^2(M, \C)$
%map $J$ to a line $H^{2,0}(M,J)\in {\Bbb P}H^2(M, \C)$.
%The map $P:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ is 
%called {\bf\blue the period map}.
%
%\remark 
%{\purple $P$ maps $\Teich$ into an open subset of a 
%quadric,} defined by
%\[
%{\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \  q(l,l)=0, q(l, \bar l) >0.
%\]
%It is called {\bf \blue the period space} of $M$.
%
%\remark 
%${\Bbb Per}=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$
%
%{\bf \green THEOREM:} (Bogomolov)\\
%Let $M$ be a simple hyperk\"ahler manifold,
%and $\Teich$ its Teichm\"uller space. Then
% {\bf \red the period map $P:\; \Teich \arrow {\Bbb Per}$ is locally a
%diffeomorphism. }
%
%\remark Bogomolov's theorem implies that
%{\bf \purple $\Teich$ is smooth.} It is {\bf \red non-Hausdorff}
%even in the simplest examples.
%
%
%\newpage
%
%{\bf \blue Mumford-Tate group for hyperk\"ahler manifolds}
%
%\theorem
%Let $M$ be a hyperk\"ahler manifold, and
%$I$ a Mumford-Tate generic complex structure on $M$.
%{\bf \red Then $MT(M,I)=\Spin(H^2(M,\R),q)$,} where
%$\Spin(H^2(M,\R),q)$-action on $H^*(M,\R)$ is generated by
%all operators $W_{I'}$ for all complex structures in its
%deformation space.
%
%\proof By semicontinuity, $MT(M,I)$ contains all $W_{I'}$.
%In V., ``Mirror Symmetry for hyperkaehler manifolds'', 
%AMS/IP Stud. Adv. Math. 10 (1999) 115-156,
%it was shown that the group generated by all
%$W_{I'}$ is $\Spin(H^2(M,\R),q)$. \endproof
%
%\corollary
%Let $M$ be a hyperk\"ahler manifold, and
%$\omega\in H^{1,1}(M,R)$ a transcendental cohomology class. Consider 
%the set ${\cal T}$ of all K\"ahler deformations $J$ of the complex
%structure on $M$ such that $\omega\in H^{1,1}(M,J)$, and
%let $J$ be a Mumford-Tate generic point in ${\cal T}$.
%Then $MT(M,J)=\Spin(H^2(M,\R),q)$.
%
%\proof
%The group $MT(J)$ is a minimal rational subgroup of
%$\Spin(H^2(M,\R),q)$ containing the stabilizer $\St(\omega)$.
%For transcendental $\omega$,
%this subgroup coincides with $\Spin(H^2(M,\R),q)$ 
%\endproof
%
%\newpage
%
%{\bf \blue Campana simple hyperk\"ahler manifolds}
%
%\theorem
%Let $M$ be a hyperk\"ahler manifold. Consider the
%$SU(2)$-action on its cohomology induced by quaternions.
%Assume that all rational $(p,p)$-classes are
%$SU(2)$-invariant. Then $M$ is Campana simple.
%
%\proof From V., 
%{\em Tri-analytic subvarieties of hyper-K\"ahler
%  manifolds,} GAFA {\bf 5} no. 1 (1995), 92-104,
%it follows that all subvarieties of $M$ are hyperk\"ahler,
%and from Misha Verbitsky,
%{\em Deformations of trianalytic subvarieties of hyperk\"ahler manifolds},
% Selecta Math. (N.S.) 4 (1998), no. 3, 447--490, it
% follows that all such manifolds are Campana simple.
%\endproof
%
%\corollary
%Any hyperk\"ahler manifold with $MT(M,I)\supset \Spin
%(H^2(M,\R))$ is Campana simple. \endproof



\end{document}


