\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{\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, Alastair}, publisher={Citeseer} } 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A source of inspiration for us will be from smooth topology, where tools such as surgery, Morse theory, and cobordisms provide methods for generating new manifolds. An example where we can take a method from topology and directly import it into symplectic geometry is connect sum for Lagrangian curves inside of surfaces. In this setting, two Lagrangian curves which intersect at a point are modified at the point of intersection to produce a connected Lagrangian submanifold. See \cref{fig:polterovichSurgery}. The Polterovich connect sum is a generalization of this surgery to higher dimensions, which smooths out a transverse intersection between two Lagrangian submanifolds. The idea of construction is to take a \underline{\href{https://jeffhicks.net/snippets/index.php?tag=lemma:straightening}{ standard model for the transverse intersection}}, then construct a model neck in that canonical neighborhood. \begin{figure} \label{fig:polterovichSurgery} \centering \begin{tikzpicture} \draw[dashed] (0,0) circle[radius=2]; \draw (-2,0) -- (2,0) (0,2) -- (0,-2); \node at (-1,1) {$U$}; \node[right] at (0,1) {$L_1=i\mathbb R$}; \node[below] at (-1,0) {$L_0=\mathbb R$};\begin{scope}[shift={(5.5,0)}] \draw[dashed] (0,0) circle[radius=2]; \node at (-1,1) {$U$}; \node[below] at (-1,0) {$L_0\#L_1$}; \draw (-2,0) .. controls (-1.5,0) and (-1.5,0) .. (-1,0) .. controls (-0.5,0) and (0,0.5) .. (0,1) .. controls (0,1.5) and (0,1.5) .. (0,2) ; \draw (2,0) .. controls (1.5,0) and (1.5,0) .. (1,0) .. controls (0.5,0) and (0,-0.5) .. (0,-1) .. controls (0,-1.5) and (0,-1.5) .. (0,-2); \end{scope} \end{tikzpicture} \caption{The connect sum of two Lagrangian curves in a surface} \end{figure}\begin{proposition}\cite{polterovich1991surgery} \label{prp:polterovichSurgery} Let $L_1, L_2\subset X$ be two Lagrangian submanifolds intersecting transversely at a single point $p$. Then there exists a Lagrangian submanifold $L_1\#_p L_2\subset X$ which \begin{itemize} \item topologically is the connect sum of $L_1$ and $L_2$ at $p$. \item Agrees with $L_1\cup L_2$ outside of a small neighborhood of $p$ in the sense that \[L_1\#_p L_2|_{X\setminus U}=L_2\cup L_2|_{X\setminus U}.\] \end{itemize} \end{proposition} \begin{proof} \label{prf:polterovichSurgery} There exists a \underline{\href{https://jeffhicks.net/snippets/index.php?tag=lem:straightening}{ standard model}} of two Lagrangian submanifolds intersecting transversely at a point. Therefore, it suffices to construct a Lagrangian surgery neck for the standard intersection neighborhood $X=\CC^n$, $L_1=\RR^n$ and $L_2=\jmath\RR^n$. We start by picking a \emph{surgery profile curve}, \begin{align*} \gamma: [-R, R] \to& \CC\\ t\mapsto& (a(t)+\jmath b(t)) \end{align*} with the property that $a(t), b(t)$ are non-decreasing, and there exists a value $t_0$ so that \begin{itemize} \item $\gamma(t)=t$ for $t< t_0$, and \item $\gamma(t)=\jmath t$ for $t>t_0$. \end{itemize} We denote the are bounded between the real axis, imaginary axis, and curve $\gamma$ by $\lambda$. An example is drawn in \cref{fig:polterovichSurgeryProfile}. \begin{figure} \label{fig:polterovichSurgeryProfile} \centering \begin{tikzpicture} \fill[fill=gray!20] (-1.5,0) .. controls (-0.5,0) and (-0.5,0) .. (0,0) .. controls (0,0.5) and (0,0.5) .. (0,1.5) .. controls (0,0.5) and (-0.5,0) .. (-1.5,0); \draw (0,3) -- (0,-3); \draw (3,0) -- (-3, 0); \draw[red] (-3, 0)--(-1.5,0) (0,1.5)--(0, 3); \draw[red] (-1.5,0) .. controls (-0.5,0) and (0,0.5) .. (0,1.5); \node[above left] at (-0.5,0.5) {$a(t)+\jmath b(t)$}; \node[above right] at (1.5,0) {$c$}; \draw[dotted] (0,0) ellipse (1.5 and 1.5); \draw[dotted] (0,0) ellipse (2.5 and 2.5); \draw[->] (-2,-2.5) .. controls (-1.5,-2) and (-0.25,-0.4) .. (-0.25,0.1); \node at (-2.55,-2.5) {width}; \end{tikzpicture}\caption{Surgery Profile for Polterovich surgery} \end{figure}This data provides a construction for the Lagrangian surgery neck: \[ L_1\#_\gamma L_2:=\left\{(\gamma(t)\cdot x_1,\ldots, \gamma(t)\cdot x_n) \text{ such that } x_i \in \RR^n,t\in \RR, \sum_{i} x_i^2=1\right\}. \] Note that when $t < t_0$ this parameterizes $(\RR\setminus B_r(0))\subset \CC^n$, and when $t > t_0$ the chart parameterizes $(\jmath \RR \setminus B_r(0))\subset \CC^n$. Therefore, this construction satisfies the condition that the surgery Lagrangian agrees with the surgery components outside of a small neighborhood of the surgery point. This Lagrangian has the topology of $S^{n-1}\times \RR$, which is the local model for the connect sum $\RR^n\#_0\RR^n$. Then \cref{prp:polterovichSurgery} follows by taking $L_1\#_\gamma L_2$ for any suitable choice of $\gamma$. \end{proof} The order of the summands plays an important role in Lagrangian surgery, as rarely are the Lagrangian $L_1\#L_2$ and $L_2\#L_1$ isotopic. The surgery construction does not uniquely specify \emph{the} Lagrangian surgery of two Lagrangian submanifolds, different choices of curves $\gamma$ produce different Lagrangian submanifolds. Given a Lagrangian isotopy $\li_t: L\to C$ we associate \underline{\href{https://jeffhicks.net/snippets/index.php?tag=def:flux}{ flux cohomology class}} $\Flux_{\li_t}\in H^1(L)$. We can similarly associate a flux class to a Lagrangian surgery. A given surgery profile curve $\gamma$ can be extended to a family of Lagrangian surgery profiles by scaling the profile curve by a parameter $s$. This gives us a family of surgeries $L_1\#_{s\gamma} L_2$, which are Lagrangian isotopic. The flux of the surgery is defined to be the flux of this isotopy. \begin{proposition} \label{prp:fluxOfSurgery} Let $L_0, L_1$ be two Lagrangian submanifolds which intersect transversely at a point, and let $U$ be a neighborhood of the point in which we implant a surgery neck. Suppose two profile curves $\gamma_0, \gamma_1$ have the same flux $\lambda$. Then there exists a Hamiltonian supported on $U$ whose time one identifies $L_1\#_{\gamma_0} L_2$ and $L_1\#_{\gamma_1} L_2$.\end{proposition} The flux of $\gamma$ is the area bounded by $\gamma$ and the two axis. This is sometimes called the width or neck-width of the surgery. We will write $L_1\#_{\lambda}L_2$ a Lagrangian connect sum determined by a surgery profile curve with flux $\lambda$. \begin{example} \label{exm:polterovichSurgery} We now visualize the Polterovich surgery for Lagrangian sections of $T^*\RR^n$. The Lagrangians which we consider are two sections of the cotangent bundle. Let $L_1$ be the graph of $d(q_1^2+ \cdots+ q_n^2)$, and let $L_2$ be the graph of $d(-q_1^2-\cdots -q_n^2)$. In dimension 2, we can then draw $L_1\# L_2$ and $L_2\# L_1$ as multisections of the cotangent bundle. These multisections are sketched in \cref{fig:surgeryDim4}. Note that one of surgeries creates a Lagrangian which has a ``neck'' visible in the projection to the base of the cotangent bundle. The other surgery is generically a double-section of the cotangent bundle, except over the fiber of the intersection point where it is instead an $S^1$. \begin{figure} \label{fig:surgeryDim4} \centering \begin{tikzpicture}[scale=.7] \begin{scope}[shift={(-12.5,1)}] \draw (-5,5) rectangle (5,-5); \node at (3,4) {$L_1= d(x^2_1+x_2^2)$}; \begin{scope} \draw (-3,0)--(3,0) (0,3)--(0,-3); \foreach \r in { .5, 1,1.5, 2,2.5, 3} { \foreach \t in {0,...,36} { \draw[->, red] ({\r * cos(10*\t)},{ \r * sin(10*\t) })-- ({1.1*\r * cos(10*\t)},{ 1.1*\r * sin(10*\t) }); %\draw[->, blue] ({\r * cos(10*\t)},{ \r * sin(10*\t) })-- ({0.9*\r * cos(10*\t)},{ 0.9*\r * sin(10*\t) }); } } \end{scope} \end{scope} \begin{scope}[shift={(-12.5,-11.5)}] \draw (-5,5) rectangle (5,-5); \node at (3,4) {$L_2=d(-x^2_1-x_2^2)$}; \begin{scope} \draw (-3,0)--(3,0) (0,3)--(0,-3); \foreach \r in { .5, 1,1.5, 2,2.5, 3} { \foreach \t in {0,...,36} { %\draw[->, red] ({\r * cos(10*\t)},{ \r * sin(10*\t) })-- ({1.1*\r * cos(10*\t)},{ 1.1*\r * sin(10*\t) }); \draw[->, blue] ({\r * cos(10*\t)},{ \r * sin(10*\t) })-- ({0.9*\r * cos(10*\t)},{ 0.9*\r * sin(10*\t) }); } } \end{scope} \end{scope} \begin{scope}[] \begin{scope}[shift={(0,1)}] \draw (-5,5) rectangle (5,-5); \node at (3,4) {$L_1\#L_2$ in $\mathbb R^2$}; \begin{scope} \draw (-3,0)--(3,0) (0,3)--(0,-3); \foreach \r in { .5, 1,1.5, 2} { \foreach \t in {0,...,36} { \draw[->, red] ({(\r+1) * cos(10*\t)},{ (\r+1) * sin(10*\t) })-- ({(1.1*\r +1)* cos(10*\t)},{ (1.1*\r+1) * sin(10*\t) }); \draw[->, blue] ({(\r+1) * cos(10*\t)},{ (\r+1) * sin(10*\t) })-- ({(0.9*\r+1) * cos(10*\t)},{( 0.9*\r +1)* sin(10*\t) }); } } \end{scope} \end{scope} \begin{scope}[shift={(0,-11.5)}] \draw (-5,5) rectangle (5,-5); \node at (3,4) {$L_1\#L_2$ in $T^*_0\mathbb R^2$}; \begin{scope} \draw (-3,0)--(3,0) (0,3)--(0,-3); \foreach \r in {.5, 1,1.5, 2,2.5, 3} { \foreach \t in {0,...,36} { \draw[->, red] ({\r * cos(10*\t)},{ \r * sin(10*\t) })-- ({(\r+.2) * cos(10*\t)},{ (\r+.2) * sin(10*\t) }); \draw[->, blue] ({\r * cos(10*\t)},{ \r * sin(10*\t) })-- ({(\r-.2) * cos(10*\t)},{ (\r-.2) * sin(10*\t) }); } } \end{scope} \end{scope} \end{scope} \end{tikzpicture}\caption{Lagrangian surgery of two sections of \(T^*\RR^2\)} \end{figure} \end{example} \section{the Lagrangian surgery exact triangle } \label{art:surgeryExactTriangle} Now that we have a geometric description of $L_1\#_\lambda L_2$, we discuss what object it represents in the Fukaya category. We restrict ourselves to $\dim_\RR(X)=2$ so that we may draw pictures. However, the pictures are only for intuition (and in fact the sketch of proof we give only work when $\dim(X)\geq 4$). Let $L_0$ be some test Lagrangian, which intersects both $L_1$ and $L_2$ as in \cref{fig:roundingCorner}. If the surgery neck is chosen to lie in a neighborhood disjoint from the intersections $L_0\cap (L_1\cup L_2)$, then these intersections are in bijection with the intersections $L_0\cap (L_1\# L_2)$. Therefore $\CF(L_0, L_1\# L_2)=\CF(L_0, L_1)[1]\oplus \CF(L_, L_2)$ as vector spaces. \begin{figure} \label{fig:roundingCorner} \centering \begin{tikzpicture}\begin{scope}[] \fill[gray!20] (-2,2) rectangle (2,-2); \fill[red!20] (0,0) .. controls (0,0.5) and (0,1) .. (0,1.5) .. controls (-1,1.5) and (-1.5,1) .. (-1.5,0) .. controls (1,0) and (0.5,0) .. (0,0); \draw (0, 2)--(0,-2); \draw (2,0) -- (-2,0); \draw (2,1.5) .. controls (1,1.5) and (0.5,1.5) .. (0,1.5) .. controls (-1,1.5) and (-1.5,1) .. (-1.5,0) .. controls (-1.5,-0.5) and (-1.5,-1.5) .. (-1.5,-2); \node[right] at (2,1.5) {$L_0$}; \node[right] at (2,0) {$L_1$}; \node[above] at (0,2) {$L_2$}; \node[below right] at (0,0) {$p_{12}$}; \node[above right] at (0,1.5) {$p_{02}$}; \node[below left] at (-1.5,0) {$p_{10}$}; \node[fill=black, circle, scale=.3] at (0,0) {}; \node[fill=black, circle, scale=.3] at (0,1.5) {}; \node[fill=black, circle, scale=.3] at (-1.5,0) {}; \node at (1,-1.5) {$X$}; \end{scope} \begin{scope}[shift={(6.5,0)}] \fill[gray!20] (-2,2) rectangle (2,-2); \draw (2,1.5) .. controls (1,1.5) and (0.5,1.5) .. (0,1.5) .. controls (-1,1.5) and (-1.5,1) .. (-1.5,0) .. controls (-1.5,-0.5) and (-1.5,-1.5) .. (-1.5,-2); \node[right] at (2,1.5) {$L_0$}; \node[right] at (2,0) {$L_1\#L_2$}; \draw[fill=blue!20] (0,1.5) .. controls (0,0.5) and (-0.5,0) .. (-1.5,0) .. controls (-1.5,1) and (-1,1.5) .. (0,1.5); \draw (0,2) .. controls (0,1.5) and (0,2) .. (0,1.5) .. controls (0,0.5) and (-0.5,0) .. (-1.5,0) .. controls (-2,0) and (-2,0) .. (-2,0); \draw (0,-2) .. controls (0,-2) and (0,-2) .. (0,-1.5) .. controls (0,-0.5) and (0.5,0) .. (1.5,0) .. controls (2,0) and (2,0) .. (2,0); \node[above right] at (0,1.5) {$p_{02}$}; \node[below left] at (-1.5,0) {$p_{10}$}; \node[fill=black, circle, scale=.3] at (0,1.5) {}; \node[fill=black, circle, scale=.3] at (-1.5,0) {}; \node at (1,-1.5) {$X$}; \end{scope} \end{tikzpicture}\caption{By rounding the corner, we can compare holomorphic triangles with holomorphic strips on the surgery.} \end{figure} The intuition from \cite{fukaya2007lagrangian} is that there is a bijection between certain holomorphic triangles with boundary on $L_0, L_1, L_2$ which passes through the intersection point $p_{12}$, and holomorphic strips with boundary on $L_0$ and $L_1\# L_2$. Since holomorphic triangles contribute to the $\emprod^3$ structure coefficients, and strips to the differential, it is reasonable to hope that we can state a relation between $L_1, L_2,$ and $L_1\#L_2$ as objects of the Fukaya category. First, we observe that the intersection point $p_{12}$ determines a morphism in $\hom(L_2, L_1)$. Since we've assumed that $L_1$ and $L_2$ intersect at only one point, we know that $\emprod^1(p_{12})=0$. We can therefore form the twisted complex $\cone(p_{12})$. We now provide justification for why this is isomorphic to $L_1\# L_2$. We have already observed that for our test Lagrangian $L_0$ we had an isomorphism of vector spaces between $\hom(L_0, L_1)\oplus \hom(L_0, L_2)$ and $\hom(L_0, L_1\#_\lambda L_2)$. The differential on $\hom(L_0, L_1\#_\lambda L_2)$ comes from counting holomorphic strips, which we break into two types: those which avoid a neighborhood of the surgery neck, and those which pass through the surgery neck. \begin{proposition} \label{prp:stripsInLagrangianSurgery} Let $\dim_\RR(X)\geq 4$, and let $L_1, L_2$ be exact Lagrangian submanifolds which intersect at a single point $p_{12}$. then we can choose an almost complex structure $J$ so that whenever $p_{01}, q_{01}\in L_0\cap L_1$ are intersections, and $u: [0,1]\times \RR\to X$ is a $J$ holomorphic strip, then the boundary of $u$ is disjoint from a small neighborhood of $L_1\cap L_2$. In particular, $u$ gives a $J$-holomorphic strip with boundary on $L_0, L_1\#_\lambda L_2$. \end{proposition} The more difficult portion is to understand the strips which pass through the neck. \begin{theorem}\cite{fukaya2007lagrangian} \label{thm:roundingCorner} Let $\dim(X)\geq 4$, and $L_1, L_2$ be exact Lagrangian submanifolds which intersect transversely at a single point $p_{12}$. Let $L_0$ be another exact Lagrangian submanifold which intersects $L_1, L_2$ transversely. Then for sufficiently small surgery necks, there exists a choices of almost complex structure on $X$ for which we have a bijection between \begin{itemize} \item $J$-holomorphic strips with boundary on $L_0, L_1\# L_2$ which pass through the surgery neck; \item $J$-holomorphic triangles with boundary on $L_0, L_1, L_2$. \end{itemize} \end{theorem} In fact, \cite{fukaya2007lagrangian} proves the above statement in much greater generality than we state here. The condition that $\dim(X)\geq 4$ can already be seen in in \cref{fig:roundingCorner}. Observe that if we have a pseudoholomorphic triangle whose boundary passes through the point $p_{12}$ in the wrong way, that there is no corresponding pseudoholomorphic strip with boundary on $L_0$ and $L_1\# L_2$. Ignoring the potential complications in the definition of the Fukaya category, we obtain: Let $\dim(X)\geq 4$, and $L_1, L_2$ be exact Lagrangian submanifolds which intersect transversely at a single point $p_{12}$. Then $L_1\#_{\lambda} L_2$ is isomorphic to the twisted complex $(L_1[1]\oplus L_2, \emprod^2(T^{-\lambda} p_{12}-))$ in $\Fuk(X)$.\printbibliography \end{document}