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\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 1 \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 (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}