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Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Symplectic geometry\\[15mm] \small lecture 1: Symplectic manifolds and symplectic capacities} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] September 04, 2021 } \end{center} \newpage {\bf \blue Symplectic manifolds} \definition A {\bf \blue symplectic form} is a closed, non-degenerate 2-form $\omega\in \Lambda^2 (M)$. \remark {\bf \purple A $2n$-manifold equipped with a non-degenerate antisymmetric 2-form $\omega$ is oriented.} Indeed, $\omega^n$ is a volume form. \definition {\bf\blue An almost complex structure} is an operator $I:\; TM \arrow TM$ such that $I^2= -\Id$. \theorem Let $\omega$ be a non-degenerate 2-form on a manifold. {\bf \red Then there exists an almost complex structure $I$ such that the form $x, y \arrow \omega(x, I(y))$ is symmetric and positive definite.} Moreover, {\bf \red the space of such almost complex structures is contractible.} \proof See the exercises. \endproof \corollary {\bf \purple A manifold admits a non-degenerate antisymmetric 2-form if and only if it it admits an almost complex structure.} \newpage {\bf \blue Existence of symplectic structures} \theorem {\bf \blue (Gromov)}\\ Let $M$ be a non-compact manifold admitting an almost complex structure. {\bf \red Then $M$ admits a symplectic structure.} \proof {\em \green Eliashberg-Mishachev, Introduction to the h-principle.} By contrast, existence of a symplectic structure {\bf \purple puts great restrictions on the topology of a compact almost complex manifold.} For instance, the top power of a symplectic form is the volume form, hence {\bf \red its cohomology class is non-zero.} \newpage {\bf \blue Moser isotopy } \definition Let $(M, \omega)$ and $(M', \omega')$ be symplectic manifolds. A diffeomorphism $\phi:\; M \arrow M'$ is called {\bf\blue a symplectomorphism} if $\phi^*\omega'=\omega$. \remark {\bf \blue Symplectic geometry} is a field of mathematics which {\bf \purple studies symplectic manifolds up to an isomorphism and the group of symplectomorphisms.} \theorem {\bf \blue (Moser's isotopy lemma)}\\ Let $M$ be a compact symplectic manifold, and $\omega_t$, $t\in [0,1]$ a smooth deformation of a symplectic form. Assume that the cohomology class $[\omega_t]\in H^2(M)$ is constant in $t$. {\bf \red Then there exists a diffeomorphism flow $V_t\in \Diff(M)$ mapping $\omega_0$ to $\omega_t$.} \proof Next lecture. \newpage {\bf \blue Darboux' theorem} \theorem {\bf \blue (Darboux)} Let $(M, \omega)$ be a symplectic manifold. Then in a neighbourhood of each point {\bf \red there exist coordinates $p_1, ..., p_n, q_1, ..., q_n$ such that $\omega= \sum_i dp_i \wedge dq_i$.} \proof Next lecture (deduced from Moser isotopy). \remark From this theorem one can deduce that {\bf \red the symplectomorphism group $\Symp(M)$ is simple and infinitely transitive}, that is, acts transitively on discrete subsets $Z\subset M$ of the same cardinality. The same is true for the diffeomorphism group, and the proof is more or less the same. \definition {\bf \blue Darboux coordinates} on a symplectic manifold is a coordinate system $p_1, ..., p_n, q_1, ..., q_n$ such that $\omega= \sum_i dp_i \wedge dq_i$ \newpage {\bf \blue Volume and capacity} \definition {\bf\blue Standard symplectic structure} on $\R^{2n}$ with coordinates $p_1, ..., p_n, q_1, ..., q_n$ is $\omega:= \sum_i dp_i \wedge dq_i$ (``Darboux coordinates''). \definition {\bf \blue A symplectic ball} of radius $r$ is the standard ball \[ B^{2n}:=\left\{(p_1, ..., p_n, q_1, ..., q_n)\in \R^{2n}\ \ |\ \ \sum_i p_i^2 + q_i ^2 < r^2\right\} \] equipped with the standard symplectic form $\omega:= \sum_i dp_i \wedge dq_i$ \definition {\bf \blue Symplectic volume} of a symplectic manifold $(M, \omega)$, $\dim_\R M =2n$, is $\Vol(M, \omega):= \int_M \omega^n$. Symplectic capacity {\bf \purple is a way to distinguish diffeomorphic symplectic manifolds of the same volume.} \newpage {\bf \blue Volume and capacity (2)} \exercise Let $\nu$ be a volume form on a compact manifold $M$, $\nu'$ a volume form on $M'$, diffeomorphic to $M$. Assume that $\int_M\nu= \int_{M'}\nu'$. {\bf \purple Prove that there exists a diffeomorphism $\phi:\; M \arrow M'$ such that $\phi^* \omega' = \omega$.} \exercise Let $\nu$ be a volume form on a manifold $M$, not necessarily compact, and $\nu'$ a volume form on $M'$, diffeomorphic to $M$. Assume that $\int_M\nu= \int_{M'}\nu'$. {\bf \purple Prove that for any open subset $U\subset M$ with compact closure there exists a diffeomorphism $\phi:\; M \arrow M'$ such that $\phi^* \omega'\restrict U = \omega\restrict U$.} \question Let $A, B\subset \R^{2n}$ be two manifolds homeomorphic to a ball, of the same symplectic volume. {\bf \red Are they symplectomorphic?} {\bf \green ANSWER:} Not always, because {\bf \purple the symplectic capacity might be different.} \newpage {\bf \blue Gromov capacity} \definition {\bf \green An (open) symplectic embedding} is an open embedding of symplectic manifolds, symplectimorphic to its image. \definition Let $(M,\omega)$ be a symplectic manifold, and $r$ a supremum of radii of all symplectic balls of the same dimension, admitting a symplectic embedding to $M$. The number $\capa(M,\omega):=\pi r^2$ is called {\bf \blue Gromov symplectic capacity} of $M$. \theorem {\bf \blue (Ekeland-Hofer)}\\ Let $\phi$ be an oriented diffeomorphism of symplectic manifold. Then {\bf \red $\phi$ is a symplectomorphism if and only if $\phi$ preserves the symplectic capacity of all open subsets.} \proof Later in these lectures. \endproof \newpage {\bf \blue Gromov Non-Squeezing Theorem} \definition {\bf \blue A symplectic cylinder} $C_r$ is $\R^{2n}\times B_r$, where $\R^{2n}$ is equipped with the standard symplectic form $\sum_i dp_i\wedge dq_i$, and $B_r$ is the standard symplectic ball of radius $r$ in $\R^2$. \theorem {\bf \blue (Gromov)} {\bf \red Symplectic capacity of a symplectic cylinder $C_r$ is equal to $\pi r^2$.} \proof Later in these lectures. \endproof \remark The volume is not the only obstruction to symplectic embeddings. Indeed, {\bf \purple the volume of the symplectic cylinder is infinite.} {\bf \green Related:} This theorem is also called ``Symplectic camel theorem'', or ``Gromov Non-Squeezing Theorem''. \centerline{\epsfig{file=camel-and-needle.jpg,width=0.45\linewidth}} \newpage {\bf \blue Symplectic packing} \exercise Prove that {\bf \red there exists a non-degenerate $U(n+1)$-invariant symplectic form $\omega$ on $\C P^n$.} Prove that such $\omega$ is unique, up to a constant multiplier. Prove that {\bf \red this form is closed.} \definition The form $\omega$ is called {\bf\blue the Fubini-Study form}. We usually fix the constant multiplier in such a way that $\Vol (\C P^n, \omega)=1$. \theorem Let $v_N$ be a supremum of the total volume of $N$ equal symplectic balls which admit a disjoint symplectic embedding to $\C P^2$ of volume 1. Then \setlength{\tabcolsep}{10pt} \renewcommand{\arraystretch}{1.5} \[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline N&1&2&3&4&5&6&7&8&9& N>9\\ \hline \nu_N&1&\frac 1 2&\frac 3 4 & 1 & \frac {20}{25} &\frac {24}{25} & \frac{63}{64} & \frac{288}{289} & 1 & 1 \\ \hline \end{array} \] {\bf \blue The first 4 numbers are due to Gromov, last to Biran, the rest are McDuff-Polterovich.} \newpage {\bf \blue Nagata conjecture} \remark {\bf \purple These numbers are related to Nagata conjecture}, which is still unsolved (Biran used Taubes' work on Seiberg-Witten invariants to avoid proving it). \conjecture Suppose $p_1, ..., p_r$ are very general points in $\C P^2$ and that $m_1, ..., m_r$ are positive integers. {\bf \red Then for any $r > 9$, any complex curve C in $\C P^2$ that passes through each of the points $p_i$ with multiplicity $m_i$ must satisfy $\deg C > \frac{1}{\sqrt{r}}\sum_{i=1}^r m_i.$} \remark Nagata conjecture was known already to Nagata when $r$ is a full square, {\bf \purple and unknown for all other $r$.} Biran's theorem is considered as the {\bf \purple symplectic version of Nagata's conjecture.} \newpage {\bf \blue An exercise for Wednesday} \exercise Let $\alpha_t\in\Lambda^k(M)$, $t\in [0,1]$ be a smooth family of exact forms. Prove that {\bf \red there exists a smooth family of forms $\eta_t\in\Lambda^{k-1}(M)$, $t\in [0,1]$, such that $d\eta_t = \alpha_t$.} \end{document}