\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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We call $\phi$ a \emph{symplectomorphism} if \[\phi^*\omega'=\omega.\] Given an embedding $\psi: X\to X'$, we say that $\psi$ is a \emph{symplectic embedding} if \[\psi^*\omega'=\omega\] \end{definition} One property of the canonical symplectic structure on the cotangent bundle $T^*Q$ is that the symplectomorphism type of $T^*Q$ is only dependent on the diffeomorphism type of $Q$ \cite{da2001lectures}. Let $f: Q_1\to Q_2$ be a diffeomorphism. Define the \emph{lift} of $f$ to be a map $f_\#: T^*Q_1\to T^* Q_2$ defined by \[f_\#(q,p)=(f(q), ((df_q)^*)^{-1}p\] The lift plays well with the tautological 1-form, in the sense that if $\lambda_i$ is the tautological 1-form for $T^*Q_i$, then $(f_\#)^*\lambda_2=\lambda_1$. A particularly interesting type of symplectomorphism is one which arises as an isotopy. A symplectic isotopy is a smooth 1-parameter family of maps $\phi_t: X\times I\to X$ which for fixed values of $t$ give a symplectomorphism, and start at the identity (in the sense that $\phi_0=\operatorname{id}_X$). These are examples of smooth isotopies and so they can equivalently be described as the flow of a time-dependent vector field. \begin{definition} \label{def:flow} Given a time-dependent vector field $V_t$, the \emph{flow associated to $V$} is the function $\phi_t: X\times \RR\to X$ which \begin{itemize} \item for all $t$, $\phi_t:X\to X$ is a diffeomorphisms and; \item is the identity at time 0 so that $\phi_0=\operatorname{id}_M$ and; \item generates the vector field $V_t$ in the sense that \[(V_t)_p:=\frac{d}{ds}\phi_s(\phi^{-1}_t(p))|_{s=t}.\] \end{itemize} We say that \emph{$ V_t$ is the infinitesimal generator} associated to $\phi_t$. \end{definition} There is an equivalence between vector fields and flows. \begin{theorem} \label{thm:existenceOfFlow} On compact manifolds, every time-dependent vector field has a well defined flow. \end{theorem} One can check that a smooth isotopy $\phi_t$ is a symplectic isotopy by verifying that the symplectic form is preserved by the infinitesimal generator $V_t$. This yields an easy to check criterion. \begin{proposition} \label{prp:exactsymplectichamiltonian} $\phi_t$ is a symplectic isotopy if and only if its infinitesimal generator $V_t$ satisfies \[d(\iota_{V_t}\omega)=0.\] \end{proposition} \begin{proof} We prove the forward direction, using that $\phi^*_t\omega=\omega$ for all $t$. By taking the derivative with respect to $t$ of both sides, we obtain: \begin{align*} 0=&\frac{d}{dt}\phi^*\omega =\phi^*_t\mathcal L_{V_t}\omega \end{align*} Since $\phi$ is a diffeomorphism, this is equivalent to the vanishing of $\mathcal L_{V_t}\omega$, \begin{align*} 0=& \mathcal L_{V_t}\omega \end{align*} Applying Cartan's formula, and using that $\omega$ is closed, \begin{align*} 0=&d(\iota_{V_t}\omega)+ \iota_{V_t}(d\omega)=d(\iota_{V_t}\omega). \end{align*} This means that $\iota_{V_t}\omega$ is closed. \end{proof} One could ask for the stronger condition of exactness for the 1-form $\iota_{V_t}\omega$. In this case, we can describe vector field $V_t$ by a function on $X$. \begin{definition} \label{def:hamiltonianVectorField} Let $(X, \omega)$ be a symplectic manifold, and $H: X\to \RR$ is a smooth function. The \emph{Hamiltonian vector field of $H$} is the unique vector field $V_H$ such that $dH=\iota_{V_H}\omega$. \end{definition} The uniqueness of this vector field arises from the non-degeneracy of the symplectic form $\omega$. This additionally means that to every exact symplectic isotopy we can associate a generating Hamiltonian function. The Hamiltonian isotopies give a large set of easy-to-describe symplectic isotopies, and the relation between Hamiltonian isotopies and all symplectic isotopies has a nice interpretation in terms of the topology of $X$. \begin{corollary} If $H^1(X, \RR)=0$, then every symplectic isotopy is Hamiltonian. \end{corollary} \begin{example} \label{exm:hamiltonsEquations} Suppose that we are working in $\RR^{2n}=(q_1, \ldots , q_n,p_1, \ldots p_n) $, equipped with the standard symplectic form $\omega= \sum_{i=1}^n dp_i \wedge dq_i$. We can compute the Hamiltonian vector field for $H: \RR^{2n}\to \RR$, in local coordinates as \[V_H=\sum_{k=1}^n \left(a_k \partial_{q_k}+b_k \partial_{p_k}\right)\] where the functions $a_k$ and $b_k$ are given by the formulas \begin{align*} \frac{\partial p_k}{\partial H}=-b_k & & \frac{\partial q_k}{\partial H}=a_k. \end{align*} \end{example} A notable feature of Hamiltonian flow $V_H$ is that it preserves the level sets of $H$, as \[V_H(H)=dH(V_H)=\omega(V_H, V_H)=0.\] \begin{example} \label{exm:sphereRotation} Consider $S^2=\{(x_0, x_1, x_2)\;|\;x_0^2+x_1^2+x_2^2=1\}$ equipped with the symplectic form agreeing with the standard metric induced from $\RR^3$. Take the Hamiltonian \begin{align*} H:S^2\to&\RR\\ (x_0,x_1,x_2)\mapsto& x_2 \end{align*} as drawn in \cref{fig:hamiltonianOnSphere}. Since Hamiltonian flow preserves the level sets of $H$, we know that the latitudinal slices are orbits under the action of the Hamiltonian flow. To show that the Hamiltonian flow uniformly rotates the sphere, consider the map $\phi:S^2\setminus\{(0,0,1), (0,0,-1)\}\into S^1\times \RR\subset \RR^3$, where $S^1\times \RR=\{(x_0, x_1, x_2)\;|\; x_0^2+x_1^2=1\}$, and the embedding is given by the latitudinal projection. This projection (the \emph{Gall-Peters} map projection) is area-preserving, and so $\phi$ is a symplectic embedding. In these new coordinates, $\omega=d\theta\wedge dx_2$ and $H=x_2$. In the Gall-Peters' coordinates, $V_H=\partial_\theta$. \begin{figure} \label{fig:hamiltonianOnSphere} \centering \begin{tikzpicture} \draw[fill=gray!20] (-2,2) ellipse (1.5 and 1.5); \begin{scope}[] \clip (-4,2) rectangle (0,3); \draw[dashed] (-2,2) ellipse (1.5 and 0.5); \end{scope} \begin{scope}[] \clip (-4,2) rectangle (0,1); \draw (-2,2) ellipse (1.5 and 0.5); \end{scope} \draw (2.5,3.5) -- (2.5,0.5); \draw[->] (0.5,2) -- (1.5,2); \node[fill=white] at (1,2) {$H$}; \draw[dotted] (2.5,4) -- (2.5,3.5) (2.5,0.5) -- (2.5,0); \node at (-2,4.5) {$S^2$}; \node at (2.5,4.5) {$\mathbb R$}; \node[right] at (3,3.5) {Fixed Point}; \node[right] at (3,2) {Turning Counterclockwise}; \node[right] at (3,0.5) {Fixed Point}; \node[circle, fill=black, scale=.3] at (-2,3.5) {}; \node[circle, fill=black, scale=.3] at (-2,0.5) {}; \node[circle, fill=black, scale=.3] at (2.5,0.5) {}; \node[circle, fill=black, scale=.3] at (2.5,3.5) {}; \end{tikzpicture}\caption{The Hamiltonian flow of the standard height function rotates the sphere counterclockwise relative to the north pole} \end{figure} \end{example} The Hamiltonian flow is sometimes called the symplectic gradient. In the setting where we have a compatible triple $(X, \omega,g, J)$, the Hamiltonian flow and gradient are related by the almost complex structure. \begin{lemma} \label{lem:gradients} Let $(X, \omega, g, J)$ be a compatible triple. Let $H:X\to \RR$ be a Hamiltonian. \[\grad H = J V_H\] \end{lemma} \begin{proof} This is a direct computation. On any test vector $v$, \begin{align*} g(J V_H, v)=\omega(V_H, v)=dH(v)= g(\grad(H), v) \end{align*} Because $g$ is nondegenerate, $\grad(H)=JV_H$. \end{proof}\printbibliography \end{document}