\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig} %\usepackage[matrix,arrow]{xy} \newcommand{\green}{\color[rgb]{0,0.4,0}} \newcommand{\purple}{\color[rgb]{0.4,0,0.4}} \newcommand{\red}{\color[rgb]{0.7,0,0}} \newcommand{\blue}{\color{blue}} \def\eqref#1{(\ref{#1})} \newcommand{\goth}{\mathfrak} \newcommand{\g}{{\frak g}} \newcommand{\arrow}{{\:\longrightarrow\:}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\R}{{\Bbb R}} \newcommand{\Q}{{\Bbb Q}} \renewcommand{\H}{{\Bbb H}} \newcommand{\6}{\partial} \def\1{\sqrt{-1}\:} \newcommand{\restrict}[1]{{\left|_{{#1}}\right.}} \newcommand{\cntrct} % contraction with a vector field {\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}} \def\Bbb#1{\mathbb #1} \newcommand{\calo}{{\cal O}} \newcommand{\cac}{{\cal C}} \newcommand{\2}{{/\!\!/}} % Correcting TeX... %\let\oldtilde=\tilde %\renewcommand{\tilde}{\widetilde} \renewcommand{\bar}{\overline} \renewcommand{\phi}{\varphi} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} % Operatornames \newcommand{\even}{{\rm even}} \newcommand{\ev}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\it const}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\End}{\operatorname{End}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\St}{\operatorname{St}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Pic}{\operatorname{Pic}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Alt}{\operatorname{Alt}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sec}{\operatorname{Sec}} \newcommand{\Can}{\operatorname{Can}} \newcommand{\Sing}{\operatorname{Sing}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\coim}{\operatorname{coim}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\diam}{\operatorname{\sf diam}} \renewcommand{\max}{{\operatorname{max}}} \newcommand{\slope}{\operatorname{slope}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Kah}{\operatorname{Kah}} \newcommand{\Tw}{\operatorname{Tw}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Diff}{\operatorname{Diff}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\inbfpare}[1]{{% \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}% }} \newcommand{\comment}[1]{{}} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcommand{\question}{% {\bf \green QUESTION:\ }} \newcommand{\problem}{% {\bf \green PROBLEM:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Unobstructed symplectic packing 2 \hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} \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 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}