\documentclass[11 pt]{article} \usepackage{amsmath,amsthm,amsfonts,amssymb,titlesec} \usepackage{hyperref} \usepackage{tikz} \usepackage{verbatim} \usepackage{accents} \usepackage[citestyle=alphabetic,bibstyle=alphabetic,backend=bibtex]{biblatex} \usepackage{todonotes} \usepackage[american]{babel} \usepackage{fancyhdr} \hypersetup{colorlinks=false} \usetikzlibrary{calc, decorations.pathreplacing,shapes.misc} \usetikzlibrary{decorations.pathmorphing} \usepackage[left=1in,top=1in,right=1in]{geometry} \usepackage[capitalize]{cleveref} \newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} \newcommand{\xxx}{T base with combinatorial potential data } \newcommand{\Xxx}{T base with combinatorial potential data } \newcommand{\xxxc}{combinatorial potential stratified space } \newcommand{\Xxxc}{combinatorial potential stratified space } \newcommand{\argument}{symplectic character } \newcommand{\arguments}{symplectic characters } \newcommand{\snip}[2]{#1} 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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\CritVal}{CritV} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\FS}{FS} \DeclareMathOperator{\Vect}{Vect} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib} @article{ballard2012hochschild, title={Hochschild dimensions of tilting objects}, author={Ballard, Matthew and Favero, David}, journal={International Mathematics Research Notices}, volume={2012}, number={11}, pages={2607--2645}, 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This means that one cannot distinguish two symplectic manifolds simply based on the symplectic geometry of a small neighborhood of a point. In this section, we sketch a proof of this fact. \begin{theorem} \label{thm:moser1} Let $V_0$ and $V_1$ be closed connected $n$ manifolds with volume forms $\Omega_0$ and $\Omega_1$. Suppose that they have the same total volume, that is \[\int_{V_0} \Omega_0 = \int_{V_1} \Omega_1.\] Let $\phi_0: V_0\to V_1$ be a diffeomorphism. Then $\phi_0$ is isotopic to a volume preserving diffeomorphism $\phi_1$; that is \[\phi_1^*\Omega_1=\Omega_0.\] \end{theorem} \begin{proof} Without loss of generality, let $V_0=V_1$ and let $\phi_0$ be the identity. Let $\Omega_t=(1-t)\Omega_0+t\Omega_1$. We want to find $\{\phi_t\}$ with $\phi_t^*\Omega_t=\Omega_0$. We could equivalently describe such an isotopy $\phi_t$ by the vector field \[V_t:=\left(\frac{d}{dt}\phi_t\right)\circ\phi_t.\] Taking Lie derivatives, we have the pullback condition is equivalent to \[\frac{d}{dt}(\phi_t^*\Omega_t)=\phi_t^*\left(\mathcal L_{V_t} \Omega_t+\frac{d}{dt}\Omega_t\right)\] So we are looking for $V_t$ with $\mathcal L_{V_t}\Omega_t+\frac{d}{dt}\Omega_t=0$. Simplifying further gives us \[ d\iota_{V_t}\Omega_t=-\frac{d}{dt}\Omega_t=\Omega_0-\Omega_1\] By Stoke's theorem, the right term is exact as $\int_X\Omega_0=\int_X \Omega_1$. Therefore there exists some $n-1$ form $\eta$ so that $d\iota_{V_t}\Omega_t=d\eta$. So now we have to solve $\iota_{V_t}\Omega_t=\eta$. Since $\Omega_t$ is a volume form, then there exists a $V_t$ satisfying this equation. \end{proof} This tells us that symplectomorphisms of surfaces are boring because any diffeomorphism can be made into a symplectomorphism. In fact, every isotopy of symplectic forms in the same cohomology can be realized by a family of symplectomorphisms. \begin{theorem} \label{thm:moser2} Let $X$ be a closed $(2n)$-manifold. Let $\{\omega_t\}$ be a smooth family of symplectic forms in the same cohomology class. Then there exists a smooth family of diffeomorphism $\{\phi_t\}$ with $\phi_0=\operatorname{id}_X$ and $\phi_t^*\omega_t=\omega_0$. \end{theorem} \begin{proof} Instead of finding $\phi_t$, we instead search for the vector $V_t$ generating the isotopy. Because the symplectic form should be invariant under this isotopy, we obtain the following condition on $V_t$: \begin{align*} 0=&\frac{d}{dt}(\phi_t^*\omega_t)=\phi_t^*\left(\mathcal L_{V_t}\omega_t+\frac{d}{dt}\omega_t\right)\\ =&\mathcal L_{V_t}\omega_t + \frac{d}{dt}\omega_t\\ =&d\iota_{V_t}\omega+\frac{d}{dt}\omega_t \end{align*} Since the cohomology class $[\omega_t]$ is constant, the time derivative $\frac{d}{dt}\omega_t$ is exact. Therefore there exists \footnote{Small technical remark: We need $\eta_t$ to depend smoothly on $t$, which can be achieved by using the Hodge decomposition.} $\eta_t$ with $d\eta_t=\frac{d}{dt}\omega_t$. This reduces our previous computation to \begin{align*} =&d\iota_{V_t}\omega_t+d\eta_t \end{align*} Since $\omega_t$ is nondegenerate, there exists unique $V_t$ with $\iota_{V_t}\omega_t+\eta_t=0$. \end{proof} We also have a relative version of the Moser theorem. \begin{theorem} \label{thm:relativemoser} Let $X$ be a manifold. Let $Y$ be a compact submanifold of $X$. Let $\omega_0, \omega_1$ be symplectic forms on $Y$ such that $\omega_0(p)=\omega_1(p)$ for all $p\in X$. Then there exists neighborhoods $U_0, U_1\supset X$ and a symplectomorphism $\phi:(U_0, \omega_0)\to (U_1, \omega_1)$ such that $\phi(p)=p$ for all $p\in X$.\end{theorem} \begin{proof} Let $\omega_t=(1-t)\omega_0+t\omega_1$ for $t\in [0, 1)$. Then $\omega_t$ is symplectic for all $t$ in a sufficiently small tubular neighborhood $N$ of $X$. We want to find $\phi_t$ with $\phi_t^*\omega_t=\omega_0$ for all $t$. Let $V_t=\frac{d}{dt}\phi_t\circ \phi_t$. Recall that \begin{align*} \frac{d}{dt}(\phi_t^*\omega_t)=&\phi^*_t(\mathcal L_{V_t}\omega_t+(\omega_1-\omega_0))\\ =&\phi^*_t(di_{V_t}\omega_t+(\omega_1-\omega_0)) \end{align*} Since $i^*\omega_0=i^*\omega_1$, we have that $[\omega_0]=[\omega_1]\in H^2(N)$ . Then there exists $\eta$ such that $d\eta=\omega_0-\omega_1$ and $\eta(p)=0$ for all $p\in X$. \begin{align*} =&\phi^*d(i_{V_t}\omega-\eta) \end{align*} Since $\omega_t$ is nondegenerate, there exists unique vector field $V_t$ with $i_{V_t}\omega_t=\eta$. Since $\eta(p)=0$ for all $p\in X$, it follows that $V_t(p)=0$ for all $p\in X$. For this $V_t$ we have that $\frac{d}{dt}(\phi^*_t\omega_t)=0$. Therefore $V_t$ generates a family of diffeomorphism $\{\phi_t\}$ which are the identity on $X$ and defined on a neighborhood of $X$. \end{proof} The application of this theorem to symplectic geometry shows that there are no local invariants of symplectic manifolds. If $(X, \omega_0)$ is symplectic then for all $p\in X$ there exists neighborhoods $U_0\supset \{p\}$ and $U_1\subset \RR^{2n}$ and a symplectomorphism $(U_0, \omega_0)\to (U_1, \omega_{std})$.\printbibliography \end{document}