\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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Given a $\omega$-compatible almost complex structure, we observe that cylinders $u: \RR_s\times S^1_t\to X$ which satisfy the $H_t$ perturbed Floer equation \begin{equation} \partial_s u + J(\partial_t u - V_{H_t})=0 \label{eqn:floerEquation} \end{equation}parameterize the negative gradient flow lines of $A_{H_t}$. For curves $\gamma_+, \gamma_-$, let $\mathcal M(\gamma_+, \gamma_-)$ denote the moduli space of solutions to Floer's equation with ends limiting to $\gamma_+, \gamma_-$. Supposing that $X$ is an exact symplectic manifold, and that our time-dependent Hamiltonian is chosen in a generically, the Floer cochains $\CF(X, H_t)$ are the graded vector space generated on the time one orbits of $H_t$. We take a slightly different convention (following Equation 5.2 of \cite{abouzaid2010geometric}) and give each orbit the grading $\deg(\gamma)=n-CZ(\gamma)$, where $CZ(\gamma)$ is the Conley-Zehnder index. The structure coefficients of the differential are given by counts of solutions to Floer's equation. The theory becomes powerful when it satisfies the following properties: \begin{itemize} \item $\CF(X, H_t)$ is a chain complex. The key step is to show that when $\dim(\mathcal M(\gamma_+, \gamma_-))=1$, there exists a compactification of this moduli space by including broken cylinders. A compactification of the space comes from applying Gromov compactness, while additional requirements on $X$ are sometimes required ensure that the only configurations which appear in the compactification are broken cylinders. In our setting, the only breaking configurations which may occur are broken cylinders, as $X$ is exact (so $\omega(\pi_2(X))=0)$. \item Given $H_{t, 0}$ and $H_{t, 1}$ two time-dependent Hamiltonians, there exists a continuation map $\CF(X, H_{t, 0})\to \CF(X, H_{t, 1})$. Furthermore, this map is a homotopy equivalence. \item Finally, we need some way to compute $\CF(X, H_{t, 1})$. One way to do this is to observe that for $C^2$ small Hamiltonians the Floer cochains agree with the Morse cochains (and only consist of constant orbits). By either using the PSS-isomorphism or by analyzing Floer trajectories, the Floer cohomology can be compared to the Morse cohomology of $X$. \end{itemize} The major difference in defining the Hamiltonian Floer cohomology for Liouville domains $X$ (as opposed to compact symplectic manifolds) comes from the proof of Gromov-compactness. The first step in the proof of Gromov-compactness is to apply Arzel\'a-Ascoli to out sequence of pseudoholomorphic maps. Because $\hat X$ is not compact, we cannot apply the Arzel\'a-Ascoli theorem to a sequence of pseudoholomorphic cylinders $u: S^1\times \RR \to \hat X$. We now give an example of where we can solve the issue of non-compactness. \begin{example} \label{exm:compactnessFromMaximumModulus} The maximum modulus principle states that if $\phi: D^2\to \CC$ is a holomorphic function from the disk to $\CC$, that the maximum of $|\phi|: D^2\to \RR_{\geq 0}$ is achieved on $\partial D^2$. Let $\hat X$ be a non-compact symplectic manifold with compatible almost complex structure $J$, along with a $J-\jmath$-holomorphic projection $W: \hat X\to \CC$. Suppose that the fibers of $W$ are compact. Pick two loops $\gamma_-, \gamma_+\subset \hat X$ and $r_0\in \RR$ large enough so that $U:=W^{-1}(\{z \st |z|\leq r\})$ contains $\gamma_-, \gamma_+$. We will prove that every pseudoholomorphic cylinder $u: S^1\times \RR\to \hat X$ with ends limiting to $\gamma_-, \gamma_+$ has image contained within the compact subset $U$. The composition $W\circ u: S^1\times \RR\to \CC$ is a holomorphic map, with ends limiting to $W(\gamma_\pm)$, and therefore satisfies the maximum modulus principle. Since the boundary is sent to $W(\gamma_\pm)$, we obtain that $|W|$ achieves a value no greater than $r_0$ on $u$; therefore $\Im(u)\subset U$. It follows that the image of $u$ is contained within a compact set. \label{exm:compactnessFromMaximumModulus} \end{example} In order to extend \cref{exm:compactnessFromMaximumModulus} to the setting of $\hat X$, we will use the maximum principle. First, we will need to assume that we have chosen our almost complex structure for $\hat X$ so that the sub-bundle spanned by the vector fields $\partial_r, R$ form an almost complex subspace. \begin{definition} \label{def:contactTypeACS} Let $\hat X$ be the completion of a Liouville domain. A choice of almost complex structure for $\hat X$ is \emph{of contact type} if \[d(\exp(r))\circ J = -\alpha.\] \label{def:contactTypeACS} \end{definition} We will also need to assume that we have chosen our Hamiltonian so that over the symplectization it only depends on the $r$-coordinate. For such a contact type almost complex structure and Hamiltonian there exists a version of \cref{exm:compactnessFromMaximumModulus}. \begin{proposition} \label{prp:liouvilleIsGeometricallyBounded} \label{prp:liouvilleIsGeometricallyBounded} Let $H: \hat X\to \RR$ be a Hamiltonian which on the symplectization takes the form of $h(\exp(r))$. Let $\gamma_+, \gamma_-$ be time 1 orbits of $V_{H_{t}}$. For a contact type almost complex structure, every solution $u: \RR\times S^1\to \hat X$ of the Floer equation with ends limiting to $\gamma_+, \gamma_-$ has image contained in the subset $\hat X|_{\exp(r)\leq C}$, where $C$ is the maximum value of $\exp(r)$ on the orbits $\gamma_+, \gamma_-$. \end{proposition} \begin{proof}\cite{seidel2006biased} \label{prf:liouvilleIsGeometricallyBounded} Let $u: \RR\times S^1\to \RR$ be a solution to the Floer equation (\cref{eqn:floerEquation}). Let $\rho=\exp(r\circ u)$. By applying $d(\exp(r))$ to the Floer equation, and using \cref{def:contactTypeACS} we obtain : \begin{align*} 0=d(\exp(r))\circ \left(\partial_s u + J(\partial_t u - V_{H})\right)=& \partial_s(\rho) - \alpha(\partial_t u)+ \alpha(V_H) \end{align*} Because $H= h(\rho)$, the Hamiltonian vector field associated to $H$ is $h'(\rho) V_\alpha$, where $V_\alpha$ is the Reeb flow. From \cref{rem:increasingHamiltonian}, we see that $\alpha(V_H)=h'(\rho).$ \begin{align*} =& \partial_s(\rho) - \alpha(\partial_t u)+ h'(\rho) . \end{align*} Similarly, applying $\alpha$ to the Floer equation: \begin{align*} 0=\alpha \left(\partial_s u + J(\partial_t u - V_{H})\right) =& \alpha(\partial_su)+ \partial_t(\exp(\rho))+ V_{H}(\exp(\rho)) \intertext{Because $H= h(\rho)$ has the same level sets as $\rho$, $V_H(\rho)=0$. } =& \alpha(\partial_s u) + \partial_t(\rho) \end{align*} Differentiating the first line with respect to $s$, the second line with respect to $t$, and summing the lines together we obtain \begin{align*} 0=& (\partial_s^2 + \partial_t^2)\circ \rho- \partial_s\alpha(\partial_t u)+\partial_s \rho h'(\rho) +\partial_t\alpha(\partial_s u)\\ \intertext{As $[\partial_s, \partial_t]=0$, we can substitue $-\partial_t\alpha(\partial_s u)+ \partial_s\alpha (\partial_t u)= -u^*\omega(\partial_s, \partial_t)$ } =& \Delta \rho- u^*\omega(\partial_s, \partial_t)+ \rho h'(\rho)\partial_s\rho + \rho h''(\rho) \partial_s\rho\\ \intertext{By again applying Floer's equation, and using the compatibility of almost complex structure with $J$, we may substitue $u^*\omega(\partial_s, \partial_t)= u^*\omega(\partial_s, J\partial_t-X_H)=|\partial_s u^2|-dh'(\rho)\partial_s$ } =& \Delta \rho-|\partial_s|^2+\rho h''(\rho)\partial_s\rho \end{align*} We therefore obtain that $\Delta\rho+\rho\cdot h''(\rho) \partial_s\rho\geq 0$. Observe now that where $z\in S^1\times \RR$ is a proposed maximum for $\rho$ that $\partial_s\rho=0$, allowing us to write $(\partial^2_s \rho + \partial^2_t \rho)|_z \geq 0$. This implies that at least one of $\partial^2_s, \partial^2_s$ has to be non-negative --- in particular, the second derivative test does not detect the maximum. A more general argument --- the maximum principle --- states that $\rho$ achieves no local maxima; therefore $\sup_{S^1\times \RR} \rho \leq \max_{t\in S^1} \exp(r\circ \gamma_\pm)=:C.$ It follows that the image of $u$ is contained in $\hat X|_{\rho< C}$, which is a compact set. \end{proof} \printbibliography \end{document}