\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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\Crefname{maincor}{Corollary}{Corollaries} \renewcommand*{\themainthm}{\Alph{mainthm}} \makeatletter \def\namedlabel#1#2{\begingroup \def\@currentlabel{#2}% \label{#1}\endgroup } \makeatother \fancypagestyle{firstpage}{% \fancyhf{} \renewcommand\headrulewidth{0pt} \fancyfoot[R]{Original text at \texttt{ \href{http://jeffhicks.net/snippets/index.php?tag=art_relationsOfLagrangianSubmanifolds}{snippets/art\_relationsOfLagrangianSubmanifolds}}} } \newcommand{\wt}{\widetilde} \newcommand{\wh}{\widehat} \newcommand{\wb}{\overline} \newcommand{\bb}{\mathbb} \newcommand{\scr}{\mathscr} \newcommand{\RR}{\mathbb R} \newcommand{\ZZ}{\mathbb Z} \newcommand{\CC}{\mathbb C} \newcommand{\TT}{\mathbb T} \newcommand{\NN}{\mathbb N} \newcommand{\PP}{\mathbb P} \newcommand{\LL}{\mathbb L} \newcommand{\II}{\mathbb I} \newcommand{\CP}{\mathbb{CP}} \newcommand{\del}{\nabla} \newcommand{\pp}{\mathbf{m}} \newcommand{\into}{\hookrightarrow} \newcommand{\emprod}{m} \newcommand{\tensor}{\otimes} 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\DeclareMathOperator{\codim}{codim} \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{\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}, year={2012}, publisher={OUP} } @article{craw2007explicit, title={Explicit methods for derived categories of sheaves}, author={Craw, 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Even in the simplest examples, this classification can become quite difficult. To give a meaningful answer to this question, we'll need a notion of equivalence between Lagrangian submanifolds. For submanifolds, a natural equivalence to consider is the isotopy class or the homotopy class. Let $\li_t: L\times I\to X$ be a homotopy of Lagrangian submanifolds. We now describe the \emph{flux class} of the homotopy, which is an element $\Flux_{\li_t}\in H^1(L;\RR)$. To define this class we prescribe its values on chains of $L$. Let $c\in C_1(L)$ be a chain. Then the homotopy can be applied to $c$ to give a 2-chain $\li_t(c)\in C_2(X)$. \begin{definition} \label{def:flux} The \emph{flux} of a Lagrangian homotopy $\li_t:L\times I\to X$ is the cohomology class defined by \[\Flux_{\li_t}(c):=\int_{\li_t(c)}\omega.\] \end{definition} To show that this is a cohomology class, we need to show that the flux homomorphism $\Flux_{\li_t}$ vanishes on boundaries $\partial b\in C_1(L_0)$ Without loss of generality, suppose that $I=[0,1]$, and let $b\in C_2(L)$ be a 2-chain. Then $\li_t(b)$ similarly gives a 3-chain in $X$ whose boundary components satisfy the relation $\li_t(\partial b)=i_0(b)-i_1(b)+\partial(\li_t(b))$. Applying the definition of the flux homomorphism to this relation yields: \[ \Flux_{\li_t}(\partial b)=\int_{\li_t(\partial b)}\omega=\int_{i_0(b)}\omega-\int_{i_1(b)}\omega + \int_{\partial(\li_t(b))}\omega\] The first two terms vanish because $i_0(b)$ and $i_1(b)$ are subsets of a Lagrangian submanifold so $\omega|_{\li_0(L)}=\omega|_{\li_1(L)}=0$. Because $\omega$ is closed, an application of Stoke's theorem shows that the last term vanishes as well. \begin{example} \label{exm:fluxInCylinder} Let $\CC^*$ be equipped with the symplectic form $\omega=\frac{1}{2\pi} d(\log r) \wedge d\theta$. Let $L_r$ be the Lagrangian \[L_r:=\{z \text{ such that } \log|z|=r\}.\] Let $\li_r:L\times [r_0, r_1]\to X$ be the isotopy between $L_{r_0}$ and $L_{r_1}$. Let $e\in H_1(L_r)$ be the fundamental class of $L$. The amount of flux swept out between these two Lagrangian submanifolds is given by the difference of their $r$-values. \[\Flux_{i_r}(e)=\int_{S^1} \int_{r_0}^{r^1} \frac{1}{2\pi} d(\log |z|)\wedge d\theta = r_1-r_0.\] For this reason, the value $r$ is sometimes called the ``flux coordinate'' of the fibration $\CC^*\to \RR$. The flux class therefore provides a nice parameterization of the space of Lagrangian submanifolds in this example. \end{example} \begin{figure} \label{fig:fluxInCylinder} \centering \begin{tikzpicture} \fill[fill=gray!20, path fading = west] (-4.5,1.5) rectangle (-2.5,-0.5); \fill[fill=gray!20] (-2.5,-0.5) rectangle (2.5,1.5); \fill[fill=gray!20, path fading = east] (4.5,1.5) rectangle (2.5,-0.5); \fill[fill=blue!20] (-2.5,1.5) rectangle (2.5,-0.5); \begin{scope}[shift={(2.5,0)}, draw=gray] \fill[fill=blue!20] (0,0.5) ellipse (0.5 and 1); \begin{scope}[] \clip (0,2) rectangle (1,-1); \draw[red, fill=blue!20] (0,0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[] \clip (0,2) rectangle (-1,-1); \draw[dashed, red] (0,0.5) ellipse (0.5 and 1); \end{scope} \end{scope} \begin{scope}[shift={(-2.5,0)},draw=gray] \begin{scope}[] \clip (0,2) rectangle (1,-1); \draw[red] (0,0.5) ellipse (0.5 and 1); \end{scope} \begin{scope}[] \fill[ fill=blue!10] (0,0.5) ellipse (0.5 and 1); \clip (0,2) rectangle (-1,-1); \draw[dashed, red] (0,0.5) ellipse (0.5 and 1); \end{scope} \end{scope} \begin{scope}[] \draw (-3.5,1.5) -- (3.5,1.5); \begin{scope}[] \draw[dashed] (-3.5,1.5) -- (-5,1.5); \end{scope} \begin{scope}[shift={(8.5,0)}] \draw[dashed] (-3.5,1.5) -- (-5,1.5); \end{scope} \end{scope} \begin{scope}[shift={(0,-2)}] \draw (-3.5,1.5) -- (3.5,1.5); \begin{scope}[] \draw[dashed] (-3.5,1.5) -- (-5,1.5); \end{scope} \begin{scope}[shift={(8.5,0)}] \draw[dashed] (-3.5,1.5) -- (-5,1.5); \end{scope} \end{scope} \node at (-2.5,2) {$\log|z|=r_1$}; \node at (2.5,2) {$\log|z|=r_2$}; \node at (0,0.5) {Flux Swept}; \node at (-2.5,2.5) {$L_0$}; \node at (2.5,2.5) {$L_1$}; \node at (4,0.5) {$\mathbb C^*$}; \end{tikzpicture}\caption{A basic computation of flux swept out between two submanifolds of \(\CC^*\)} \end{figure}\begin{example} \label{exm:fluxInCotangent} Consider the symplectic manifold $X=T^*L$. Consider a closed one form $\eta \in \Omega^1(L, \RR)$. Use this to create a Lagrangian isotopy \begin{align*} \li_t: L\times [0, 1]\to& T^*L && (q,t) \mapsto& t\cdot \eta_p \end{align*} between the zero section and the graph of $\eta$. To each loop $\gamma\in L$, look at the cylinder $\li_t\circ \gamma:S^1\times I\to X$. This can be explicitly parameterized by \begin{align*} c: S^1\times I \to& T^*L&& (\theta, t) \mapsto& (\gamma(\theta), t\cdot \eta_{\gamma(\theta)}). \end{align*} The cotangent bundle has an exact symplectic form, $\omega=d\lambda$, so computing the flux can be simplified by using Stoke's theorem: \begin{align*} \Flux_{\li_t}(\gamma)=&\int_{c}\omega=\int_{c}d\lambda \\ =&\int_{S^1\times \{1\}}c^*\lambda - \int_{S^1\times\{0\}}c^*\lambda \\ =&\int_{S^1}\gamma^*\eta=\eta(\gamma) \end{align*} So $[\Flux_{\li_t}]=[\eta]\in \Omega(L)$. \end{example} This example highlights two important properties of Lagrangian submanifolds, and Lagrangian homotopy. First, there homotopies a stronger notion of equivalence for Lagrangian submanifolds beyond homotopy by looking at those homotopies which sweep out zero flux. \begin{definition} \label{def:exactHomotopy} We call $i_t: L\times I \to X$ an \emph{exact homotopy} if for every subinterval $J\subset I$, the flux of the restriction is exact, \[[\Flux_{i_{t}}|_J]=0\in H^1(L).\] \end{definition} \underline{\href{https://jeffhicks.net/snippets/index.php?tag=prp:exactSymplecticHamiltonian}{ Similar to the case of Hamiltonian isotopies}}, exact isotopies of Lagrangian submanifolds can be characterized in terms of the Lagrangian isotopy itself. Let $\li_t: L\times I\to X$ be a Lagrangian homotopy. This is a exact homotopy if there exists a Hamiltonian function $H_t:L\to \RR$ such that for all $v\in TL$, \[\omega\left(\frac{d}{dt}\li_t,v\right)=dH_t(v).\] \begin{proof} \label{prf:hamiltonianFlux} We show one direction here, which is that $\omega\left(\frac{d}{dt}\li_t,v\right)=dH_t(v)$ implies $\Flux_{\li_t}$ vanishes in cohomology. Let $c:S^1\to L$ be any 1-cycle in $L$. Parameterize a 2-chain $\li_t\circ c: S^1\times I \to X$ with coordinates $(\theta, t)$. The flux class applied to $c$ can be explicitly computed: \begin{align*} \Flux_{\li_t}(c)=&\int_{\li_t\circ c}\omega =\int_{I \times S^1} (\li_t\circ c )^* \omega\\ =&\int_I \int_{S^1} c^*\circ (\li_t)^* \omega =\int_I \int_{S^1} (c^* \iota_{\frac{d}{dt}\li_t}\omega ) dt\\ =&\int_I \left(\int_{S^1} (c^* dH_t) \right)dt =\int_I \left(\int_{S^1} d(c^*H_t)\right) dt \end{align*} By Stoke's theorem, the integral of an exact form over the circle is zero. For the reverse direction, fix a base point $x_0\in L$. For every point $x\in L$, pick a path $\gamma_x: [0,1]\to L$ with $\gamma_x(0)=x_0$ and $\gamma_x(1)=x$. Define the function $H_t: L\to \RR$ by \[dH_t(x_1):=\int_{\li_t\circ \gamma} \omega.\] Because the flux of the isotopy is zero, this integral does not depend on the choices of paths $\gamma_x$ and gives a well defined function on $L$. We now show that this function generates the Lagrangian isotopy. The vector field $\frac{d}{dt}\li_t$ is determined by the form $\iota_{\frac{d}{dt}\li_t}\omega$. Since $\iota_{\frac{d}{dt}\li_t}$ is closed, \begin{align*} \int_{\gamma_x} \iota_{\frac{d}{dt}\li_t}\omega =& \end{align*} \todo{We now check that this these two things match up by computing the vector field.} \end{proof} This shows that when $L$ is embedded, a time dependent Hamiltonian $H_t: L\times (-\epsilon, \epsilon)\to \RR$ defines an isotopy of embeddings $\li_t: L\times(-\epsilon, \epsilon)\to X$ for small values of $t$. We say this is the isotopy generated by the Hamiltonian $H$. Provided that $L$ is compact, we immediately get the following statement on the non-displacibility of Lagrangian submanifolds by small exact Lagrangian homotopies. \begin{proposition} \label{prp:lagrangianIntersection} Let $H: L\to \RR$ be a non-time dependent Hamiltonian . Let $\li_t: L\times[0, c]\to X$ be the isotopy generated by $H$. There exists $\epsilon>0$ such that for all $t_0, t_1\in [0, \epsilon]$, \begin{align*} |\li_{t_0}(L)\cap \li_{t_1}(L)|\geq \sum_{i=1}^n b_i(L). \end{align*} \end{proposition} \begin{proof} \label{prf:lagrangianIntersection} Let $\Crit(H)$ be the set of critical points of $H$. From \cref{clm:hamiltonianFlux}, whenever $x\in \Crit(H)$, $\frac{d\li}{dt}(x)=0$ and so $\li_{t_0}(x)=\li_{t_1}(x)$. Therefore, $|\li_{t_0}(L)\cap \li_{t_1}(L)|\subset\Crit(H)$. The proposition follows from the Morse inequalities. \end{proof} In fact, this can extends to the case of time-dependent Hamiltonians for sufficiently small $t$. \begin{corollary} Let $H_t: L\to \RR$ generate a Hamiltonian isotopy. Then there exists $\epsilon>0$ such that for all $t_0, t_1\in [0,\epsilon)$, \[|\li_{t_0}(L)\cap \li_{t_1}(L)|\geq \sum_{i=1}^n b_i(L).\] \end{corollary} Secondly, every cohomological class is realizable as a flux class in the cotangent bundle. This observation motivates the following theorem, which shows that this behavior occurs more generally. \begin{theorem}\cite{weinstein1971symplectic} \label{thm:weinsteinNeighborhood} Let $\li: L\to X$ be a compact Lagrangian submanifold of $(X, \omega)$. Then there exists a neighborhood of the zero section $T^*_\epsilon L\subset TL$ and a symplectic embedding $\phi: T^*_\epsilon L\to X$ which agrees on the zero section, in the sense that \[\phi|_L=\li.\]\end{theorem} In other words, a symplectic neighborhood of a Lagrangian submanifold is determined by the diffeomorphism class of the Lagrangian. This local model will be useful later, as many of our constructions involving Lagrangian submanifolds will involve restricting to a Weinstein neighborhood, performing the construction there, and then implanting our construction into the original symplectic manifold. An example of such a construction is the following. \begin{lemma} \label{lem:straightening} Let $L\subset T^*Q$ be a Lagrangian submanifold. Suppose that the intersections between $L$ and $Q$ are transverse. Then there exists a Hamiltonian isotopy $\phi$ of $L$ so that \[\phi(L)\cap B^*_\epsilon Q=\bigcup_{p\in L\cap Q} T^*_pQ.\] \end{lemma} \begin{proof} \label{prf:straightening} At each $q\in L\cap Q$, consider the Lagrangian submanifold $T^*qQ$. Take local coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ identifying $q$ with the origin so that $T^*qQ$ is the linear subspace in the $p_i$ directions. We can take a Weinstein neighborhood $T^*_\epsilon(T^*_qQ)$ of $T^*_qQ$, whose cotangent bundle structure is \begin{align*} T^*_\epsilon(T^*_qQ)\to T^*qQ && (q_1, \ldots, q_n, p_1, \ldots, p_n)\mapsto (p_1, \ldots, p_n). \end{align*} Since the intersection $L\cap Q$ is transverse, the projection $T_qL\to T_q(T^*qQ)$ is surjective. Therefore when restricted to a small enough neighborhood of $\in U\subset T^*_qQ$ the Lagrangian $L|_{T^*_\epsilon U}$ presents itself as a section of $T^*_\epsilon U\to U$. Therefore, there exists a one form $\eta\in \Omega^1(U)$ so that $L|_{T^*U}$ is parameterized by $(p, \eta_p)$. By taking an even smaller $U$, we may assume that $U$ is a contractible neighborhood, and $\eta=dH$ is an exact one-form. Pick $\rho$ a function which vanishes in a neighborhood of $q\in U$, and takes the value $1$ in a neighborhood of $\partial U$. Consider the Lagrangian section of $T^*U$ parameterized by $d(\rho \cdot H)$. This section is Hamiltonian isotopic to $L|_{T^*_\epsilon U}$ relative boundary. Additionally, $d(\rho\cdot H)$ agrees with $U=T^*_qQ$ in a small neighborhood of $q$. The Lagrangian submanifold $L\setminus (L|_{T^*U})\cup (d(\rho\cdot H))$ is Hamiltonian isotopic to $L$. \end{proof} \printbibliography \end{document}