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\DeclareMathOperator{\ind}{ind} \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 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For coprime integers $a, b\in \ZZ$, and $\theta\in S^1$, we write \[L_{(a, b),\theta}=\{(a\cdot t+\theta,b\cdot t), t\in S^1\}.\] for the Lagrangian $S^1$ of slope $(a, b)$ passing through the point $(\theta, 0)$. \begin{enumerate} \item Compute $\HF(L_{(a, b), \theta_1}, L_{(c, d), \theta_2})$. \item Write $L_0:=L_{(1,0), 0}, L_1:= L_{(0,1), 0}$. Let $L_2 = L_0\# L_1$. Find values $(a, b), \theta$ so that $L_2$ is Hamiltonian isotopic to $L_{(a, b), \theta}$. \item Let $\{x_{01}\}=L_0\cap L_1$, $\{x_{12}\}=L_1\cap L_2$, and $\{x_{20}\}=L_2\cap L_0$. Prove that \begin{align*} m^2(x_{12}, x_{01})=0 && m^2(x_{20}, x_{12})=0 && m^2x_{01}, (x_{20})=0 \end{align*} so that we have what appears to be an exact sequence \[L_0\xrightarrow{x_{01}} L_1 \xrightarrow{x_{12}} L_2 \xrightarrow{x_{20}} L_0[1].\] \item What happens in the previous computation if we replace $L_2$ with $L_2'$ which is Lagrangian (but not Hamiltonian) isotopic to $L_2$? \end{enumerate} \end{exercise} \begin{exercise} \label{exr:dehnTwistAsSurgery} Consider the space $T^*S^n$ which we describe as a symplectic submanifold of $\CC^{n+1}$ by the equation \[\{(x_0, \ldots, x_n,y_0, \ldots, y_n) \st \sum_{i} (x_i+\sqrt{-1} y_i)^2=1 \}\] \begin{enumerate} \item Consider the Hamiltonian given by $H=1/2|\vec y|^2$. Write down the Hamiltonian vector field on $T^*S^n$. \item Consider now the symplectic manifold with boundary \[B^*_1S^n=\{(x_0, \ldots, x_n,y_0, \ldots, y_n) \in T^*S^n, |\vec y|\leq 1\}.\] Show that there exists $t_0$ so that $\phi^{t}: B^*_1S^n\to B^*_1 S^n$, the time-$t_0$ Hamiltonian flow of $H$, acts by $-1$ on the boundary of $B^*_1S^n$. \item The symplectic Dehn twist is the map: \begin{align*} \tau^n: B^*_1 S^n\to& B^*_1 S^n\\ (\vec x, \vec p) \mapsto& -\phi^{t_0}(\vec x, \vec p). \end{align*} which fixes the boundary of $B^*_1 S^n$. Consider the Lagrangian submanifold \[F_q:=\{(1, \ldots, 0), (0, p_1, \ldots, p_n)\}\subset B^*_1S^n.\] Show that there is a Hamiltonian isotopy which identifies \[\tau_n(F_q)\sim S^n\# F_q.\] \item Consider in $T^2$ the Lagrangian submanifold $L:=L_{(1, 0),0}$ as before. Identify a small neighborhood $U$ of $L$ with $B^*_1(S^1)$, and define $\tau_L: T^2\to T^2$ by \[\tau_L(x)=\left\{\begin{array}{cc} \tau^1(x) &\text{if $x\in U$}\\ x &\text{otherwise}\end{array}\right.\] For $a, b\in \ZZ$, and $\theta\in S^1$, find the Lagrangian submanifold $L_{(a', b'), \theta'}$ which is Hamiltonian isotopic to $\tau_L(L_{(a, b), \theta})$. \end{enumerate} \end{exercise} \begin{exercise} \label{exr:biranCornea3ended} Let $X$ be a compact symplectic manifold. Recall that a \emph{3-ended Lagrangian cobordism} $K: (L_0, L_1)\rightsquigarrow L_2$ is a closed Lagrangian submanifold $K\subset X\times \CC$ with the property that there exists a compact subset $U\subset \CC$ so that \[K|_{\pi_\CC^{-1}(\CC\setminus U)}=((L_0\times \RR_{>0})\cup( L_1\times (\sqrt{-1}+\RR_{>0}) )\cup( L_2\times (\RR_{<0})))|_{\pi_\CC^{-1}(\CC\setminus U)}.\] Suppose that $X$ is an exact symplectic manifold, which in turn makes $X\times \CC$ an exact symplectic manifold. Let $K$ be an exact 3-ended Lagrangian cobordism. \begin{figure} \label{fig:3EndedLagrangianCobordism} \centering \begin{tikzpicture} \fill[gray!20] (-3.5,2.5) rectangle (4.5,-3.5); \fill[fill=red!20] (0.5,-1) ellipse (3 and 2); \draw[fill=blue!20] (-3.5,-1.5) .. controls (-3,-1.5) and (-1.5,-1.5) .. (-1,-1.5) .. controls (-0.5,-1.5) and (0,-2) .. (1,-2) .. controls (2,-2) and (2,-1.5) .. (2.5,-1.5) .. controls (3,-1.5) and (4,-1.5) .. (4.5,-1.5) .. controls (4,-1.5) and (3,-1.5) .. (2.5,-1.5) .. controls (2,-1.5) and (2,0) .. (2.5,0) .. controls (3,0) and (4,0) .. (4.5,0) .. controls (4,0) and (3,0) .. (2.5,0) .. controls (2,0) and (2,0) ..(1,0.5) .. controls (0,1) and (-0.5,-1.5) .. (-1,-1.5); \node[left] at (-3.5,-1.5) {$L_0\times \mathbb R_{\ll 0}$}; \node[right] at (4.5,-1.5) {$L_1\times \mathbb R_{\gg 0}$}; \node[right] at (4.5,0) {$L_2\times(\sqrt{-1}+ \mathbb R_{\gg 0})$}; \node at (0.5,-1) {$\pi_{\mathbb C}(K)$}; \node at (3.5,1.5) {$\mathbb C$}; \node at (-1.5,0) {$U$}; \end{tikzpicture} \caption{The projection to the \(\CC\) coordinate of a 3-ended Lagrangian cobordism} \end{figure}\begin{enumerate} \item Show that $L_0, L_1$ and $L_2$ are exact Lagrangian submanifolds in $X$. \item Consider the curve $\gamma^-\subset \CC$. Show that for any exact Lagrangian submanifold $L\subset X$, $\CF(L\times \gamma^-, K)=\CF(L, L_0)$ as a vector space. \item Give $X\times \CC$ an almost complex structure of the form $J_X\times J_\CC$. Suppose that we have a finite energy pseudoholomorphic strip $u: \RR\times [0, 1]\to X\times \CC$ with $u(t, 0)\in L\times \gamma^-$ and $u(t, 1)\in K$, and ends limiting to intersections of $L\times \gamma^-\cap K$. Show that $\pi_\CC(u)\in \text{Im}(\gamma^-)\cap \RR_{<0}$ (the location of the red cross in the figure). From this, conclude that if $J_X$ is chosen so that all pseudoholomorphic strips with boundary on $L, L_2$ are regular, that $\CF(L\times \gamma^-, K) = \CF(L_2, K)$ \emph{as chain complexes}. \begin{figure} \label{fig:3EndedLagrangianCobordismGammaMinus} \centering \begin{tikzpicture} \fill[gray!20] (-3.5,2.5) rectangle (4.5,-3.5); \draw[fill=blue!20] (-3.5,-1.5) .. controls (-3,-1.5) and (-1.5,-1.5) .. (-1,-1.5) .. controls (-0.5,-1.5) and (0,-2) .. (1,-2) .. controls (2,-2) and (2,-1.5) .. (2.5,-1.5) .. controls (3,-1.5) and (4,-1.5) .. (4.5,-1.5) .. controls (4,-1.5) and (3,-1.5) .. (2.5,-1.5) .. controls (2,-1.5) and (2,0) .. (2.5,0) .. controls (3,0) and (4,0) .. (4.5,0) .. controls (4,0) and (3,0) .. (2.5,0) .. controls (2,0) and (2,0) ..(1,0.5) .. controls (0,1) and (-0.5,-1.5) .. (-1,-1.5); \node[left] at (-3.5,-1.5) {$L_0\times \mathbb R_{\ll 0}$}; \node[right] at (4.5,-1.5) {$L_1\times \mathbb R_{\gg 0}$}; \node[right] at (4.5,0) {$L_2\times(\sqrt{-1}+ \mathbb R_{\gg 0})$}; \node at (0.5,-1) {$\pi_{\mathbb C}(K)$}; \node at (3.5,1.5) {$\mathbb C$}; \node at (-1.5,0) {$U$}; \draw (-3.5,-3) .. controls (-3,-3) and (-3,-3) .. (-2.5,-3) .. controls (-2,-3) and (-2,-3) .. (-2,-2.5) .. controls (-2,-2) and (-2,1) .. (-2,1.5) .. controls (-2,2) and (-2,2) .. (-1.5,2) .. controls (-1,2) and (4,2) .. (4.5,2); \node[fill=gray!20] at (1.5,2) {$\gamma^-$}; \end{tikzpicture}\caption{Profile of the curve \(\gamma^-\)} \end{figure} \item Consider now the curve $\gamma^+\subset \CC$. Using a similar argument, one can prove that there are no pseudoholomorphic strips $u:\RR\times [0, 1]\to X\times \CC$ with $\lim_{t\to\infty} u(s, t)=z_2$ and $\lim_{t\to-\infty} u(s, t)=z_0$. What can you conclude about the relationship between $\CF(L\times \gamma^+, K)$, $\CF(L, L_0)$ and $\CF(L, L_1)$? \begin{figure} \label{fig:3EndedLagrangianCobordismGammaPlus} \centering \begin{tikzpicture} \fill[gray!20] (-3.5,2.5) rectangle (4.5,-3.5); \draw[fill=blue!20] (-3.5,-1.5) .. controls (-3,-1.5) and (-1.5,-1.5) .. (-1,-1.5) .. controls (-0.5,-1.5) and (0,-2) .. (1,-2) .. controls (2,-2) and (2,-1.5) .. (2.5,-1.5) .. controls (3,-1.5) and (4,-1.5) .. (4.5,-1.5) .. controls (4,-1.5) and (3,-1.5) .. (2.5,-1.5) .. controls (2,-1.5) and (2,0) .. (2.5,0) .. controls (3,0) and (4,0) .. (4.5,0) .. controls (4,0) and (3,0) .. (2.5,0) .. controls (2,0) and (2,0) ..(1,0.5) .. controls (0,1) and (-0.5,-1.5) .. (-1,-1.5); \node[left] at (-3.5,-1.5) {$L_0\times \mathbb R_{\ll 0}$}; \node[right] at (4.5,-1.5) {$L_1\times \mathbb R_{\gg 0}$}; \node[right] at (4.5,0) {$L_2\times(\sqrt{-1}+ \mathbb R_{\gg 0})$}; \node at (0.5,-1) {$\pi_{\mathbb C}(K)$}; \node at (3.5,1.5) {$\mathbb C$}; \node at (-1.5,0) {$U$}; \draw (-3.5,-3) .. controls (2,-3) and (2,-3) .. (2.5,-3) .. controls (3,-3) and (3,-3) .. (3,-2.5) .. controls (3,-2) and (3,1) .. (3,1.5) .. controls (3,2) and (3,2) .. (3.5,2) .. controls (4,2) and (4,2) .. (4.5,2); \node[fill=gray!20] at (0.5,-3) {$\gamma^+$}; \end{tikzpicture}\caption{Profile of the curve \(\gamma^+\)} \end{figure} \item Observe that $L\times \gamma^-$ and $L\times \gamma^+$ are Hamiltonian isotopic. Exhibit a long exact sequence whose terms are $\HF(L, L_0), \HF(L, L_1)$ and $\HF(L, L_2)$. \end{enumerate} \end{exercise} \printbibliography \end{document}