diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpxcv" "b/data_all_eng_slimpj/shuffled/split2/finalzzpxcv" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpxcv" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\nGraph neural networks (GNNs), which have been applied to a large range of downstream tasks, have displayed superior performance on dealing with graph data within recent years, e.g., biological networks~\\cite{huang2020skipgnn} and knowledge graphs~\\cite{yu2021knowledge}. Generally, the current GNN architecture follows the message passing frameworks, where the propagation process is the key component. For example, \nGCN~\\cite{kipf2016semi} directly aggregates and propagates transformed features along the topology at each layer.\nPPNP~\\cite{klicpera2018predict} aggregates both of the transformed features and the original features at each layer.\nJKNet~\\cite{xu2018representation} selectively combines the aggregated messages from different layers via concatenation\/max-pooling\/attention operations.\n\n\n\n\n\nRecent studies~\\cite{zhu2021interpreting,ma2021unified} have proven that despite different propagation processes of various GNNs, they usually can be fundamentally unified as an optimization objective containing a feature fitting term $O_{\\text {fit }}$ and a graph regularization term $O_{\\text {reg }} $ as follows:\n\\begin{eqnarray}\n\\boldsymbol{O}=\\min _{\\mathbf{Z}}\\{\\underbrace{\\zeta\\left\\|\\mathbf{F}_{1} \\mathbf{Z}-\\mathbf{F}_{2} \\mathbf{H}\\right\\|_{F}^{2}}_{O_{\\text {fit }}}+\\underbrace{\\xi \\operatorname{tr}\\left(\\mathbf{Z}^{T} \\tilde{\\mathbf{L}} \\mathbf{Z}\\right)}_{O_{\\text {reg }}}\\},\n\\end{eqnarray}\nwhere $\\mathbf{H}$ is the original input feature and $\\mathbf{L}$ is the graph Laplacian matrix encoding the graph structure. $\\mathbf{Z}$ is the propagated representation, and $\\mathbf{F}_{1}$, $\\mathbf{F}_{2}$ are defined as arbitrary graph convolutional kernels and usually set as $\\mathbf{I}$.\nThis optimization objective reveals a mathematical guideline that essentially governs the propagation mechanism, and opens a new path to design novel GNNs. That is, such clear optimization objective is able to derive the corresponding propagation process, further making the designed GNN architecture more interpretable and reliable~\\cite{zhu2021interpreting,liu2021elastic,yang2021graph}.\nFor example,~\\cite{zhu2021interpreting} replaces $\\mathbf{F}_1$ and $\\mathbf{F}_2$ with high-pass kernel and infers new high-pass GNNs; ~\\cite{liu2021elastic} applies $l_{1}$ norm to $O_{\\text {reg }} $ term and infers Elastic GNNs. \n\nDespite the great potential of this optimization objective on designing GNNs, it is well recognized that it is only proposed for traditional homogeneous graphs, rather than the multi-relational graphs with multiple types of relations. \nHowever, in real-world applications, multi-relational graphs tend to be more general and pervasive in many areas. \nFor instance, the various types of chemical bonds in molecular graphs,\nand the diverse relationships between people in social networks.\nTherefore, it is greatly desired to design GNN models that are able to adapt to multi-relational graphs.\nSome literatures have been devoted to the multi-relational GNNs, which can be roughly categorized into feature mapping based approaches~\\cite{schlichtkrull2018modeling} and learning relation embeddings based approaches~\\cite{vashishth2019composition}.\nHowever, these methods usually design the propagation process heuristically without a clear and an explicit mathematical objective. Despite they improve the performance, they still suffer from the problems of over-parameterization~\\cite{vashishth2019composition} and over-smoothing~\\cite{oono2019graph}.\n\n\\emph{``Can we remedy the original optimization objective to design a new type of multi-relational GNNs that is more reliable with solid objective, and at the same time, alleviates the weakness of current multi-relational GNNs, e.g., over-smoothing and over-parameterization?\"}\n\nNevertheless, it is technically challenging to achieve this goal. Firstly, how to incorporate multiple relations into an optimization objective. Different relations play different roles, and we need to distinguish them in this optimization objective as well. Secondly, to satisfy the above requirements, it is inevitable that the optimization objective will become more complex, maybe with more constrains. How to derive the underlying message passing mechanism by optimizing the objective is another challenge. Thirdly, even with the message passing mechanism, \nit is highly desired that how to integrate it into deep neural networks via simple operations without introducing excessive parameters.\n\nIn this paper, we propose a novel multi-relational GNNs by designing an ensemble optimization objective. In particular, our proposed ensemble optimization objective consists of a feature fitting term and an ensemble multi-relational graph regularization (EMR) term. Then we derive an iterative optimization algorithm with this ensemble optimization objective to learn \nthe node representation and the relational coefficients as well. We further show that this iterative optimization algorithm can be formalized as an ensemble message passing layer, which has a nice relationship with multi-relational personalized PageRank and covers some existing propagation processes. Finally, we integrate the derived ensemble message passing layer into deep neural networks by decoupling the feature transformation and message passing process, and\na novel family of multi-relational GNN architectures is developed.\nOur key contributions can be summarized as follows:\n\n\\begin{itemize}\n \\item \n We make the first effort on how to derive multi-relational GNNs from the perspective of optimization framework, so as to enable the derived multi-relational GNNs more reliable. This research holds great potential for opening new path to design multi-relational GNNs.\n \\item \n \n \n We propose a new optimization objective for multi-relational graphs, and we derive a novel ensemble message passing (EnMP) layer. A new family of multi-relational GNNs is then proposed in a decoupled way.\n \n \\item\n We build the relationships between our proposed EnMP layer with multi-relational personalized PageRank, and some current message passing layers. Moreover, our proposed multi-relational GNNs can well alleviate the over-smoothing and over-parameterazion issues.\n \n \\item \n\n Extensive experiments are conducted, which comprehensively demonstrate the effectiveness of our proposed multi-relational GNNs.\n \n\\end{itemize}\n\n\n\n\n\\section{Related Work}\n\\label{sec:related}\n\\noindent\n\\textbf{Graph Neural Networks.}\nThe dominant paradigms of GNNs can be generally summarized into two branches: spectral-based GNNs~\\cite{defferrard2016convolutional,klicpera2018predict} and spatial-based GNNs~\\cite{gilmer2017neural,klicpera2018predict}. \nVarious of representative GNNs have been proposed by designing different information aggregation and update strategies along topologies, e.g.,~\\cite{gilmer2017neural,klicpera2018predict}.\nRecent works~\\cite{zhu2021interpreting,ma2021unified} have explore the intrinsically unified optimization framework behind existing GNNs. \n\n\\noindent\n\\textbf{Multi-relational Graph Neural Networks.}\nThe core idea of multi-relational GNNs~\\cite{schlichtkrull2018modeling,vashishth2019composition,thanapalasingam2021relational} is to encode relational graph structure information into low-dimensional node or relation embeddings.\nAs a representative relational GNNs, RGCN~\\cite{schlichtkrull2018modeling} designs a specific convolution for each relation, and then the convolution results under all relations are aggregated, these excess parameters generated are completely learned in an end-to-end manner.\nAnother line of literature~\\cite{ji2021heterogeneous,wang2019heterogeneous,fu2020magnn,yun2019graph} considers the heterogeneity of edges and nodes to construct meta-paths, then aggregate messages from different meta-path based neighbors.\n\n\n\n\n\\section{Proposed Method}\n\\label{sec:proposed}\n\\textbf{Notations.} Consider a multi-relational graph $\\mathbf{\\mathcal{G}}=(\\mathcal{V}, \\mathcal{E}, \\mathcal{R})$ with nodes $v_{i} \\in \\mathcal{V}$ and labeled edges (relations) $\\left(v_{i}, r, v_{j}\\right) \\in \\mathcal{E}$ , where $r \\in \\mathcal{R}$ is a relation type. \nGraph structure $\\mathcal{G}^{r}$ under relation $r$ can be described by the adjacency matrix $\\mathbf{A}^{r} \\in \\mathbb{R}^{n \\times n}$, where $\\mathbf{A}^r_{{i, j}}=1$ if there is an edge between nodes $i$ and $j$ under relation $r$, otherwise $0$. \nThe diagonal degree matrix is denoted as $\\mathbf{D}^{r}=\\operatorname{diag}\\left(d^{r}_{1}, \\cdots, d^{r}_{n}\\right)$, \nwhere $d^{r}_{j}=\\sum_{j} \\mathbf{A}^{r}_{i, j}$.\nWe use $\\tilde{\\mathbf{A}}^{r}=\\mathbf{A}^{r}+\\mathbf{I}$ to represent the adjacency matrix with added self-loop and $\\tilde{\\mathbf{D}}^{r}=\\mathbf{D}^{r}+\\mathbf{I}$. \nThen the normalized adjacency matrix is $\\hat{\\tilde{\\mathbf{A}}}^{r}=\\left(\\tilde{\\mathbf{D}}^{r}\\right)^{-1 \/ 2} \\tilde{\\mathbf{A}}^{r} \\left(\\tilde{\\mathbf{D}}^{r}\\right)^{-1 \/ 2}$. Correspondingly, $\\tilde{\\mathbf{L}}^{r}=\\mathbf{I}-\\hat{\\tilde{\\mathbf{A}}}^{r}$ is the normalized symmetric positive semi-definite graph Laplacian matrix of relation $r$.\n\n\n\n\n\n\\subsection{Ensemble Optimization Framework}\n \n\nGiven a multi-relational graph, one basic requirement is that the learned representation $\\mathbf{Z}$ should capture the homophily property in the graph with relation $r$, i.e., the representations $\\mathbf{Z}_i$ and $\\mathbf{Z}_j$ should be similar if nodes $i$ and $j$ are connected by relation $r$. We can achieve the above goal by minimizing to the following term with respect to $\\mathbf{Z}$:\n\\begin{eqnarray}\n\\operatorname{tr}\\left(\\mathbf{Z}^{T} \\tilde{\\mathbf{L}}^r \\mathbf{Z}\\right)=\\sum_{i, j}^{n} \\hat{\\tilde{\\mathbf{A}}}_{i, j}^r\\left\\|\\mathbf{Z}_{i}-\\mathbf{Z}_{j}\\right\\|^{2},\n\\end{eqnarray}\nwhere $\\hat{\\tilde{\\mathbf{A}}}_{i, j}^r$ represents the node $i$ and node $j$ are connected under relation $r$.\n\n\nWith all the $R$ relations, we need to simultaneously capture the graph signal smoothness. Moreover, consider that different relations may play different roles, we need to distinguish their importance as well, which can be modelled as an ensemble multi-relational graph regularization as follows: \n\\begin{eqnarray}\n\\mathcal{O}_{\\text {e-reg }} = \n\\left\\{\n\\begin{aligned}\n \\lambda_{1} \\sum_{r=1}^{R} \\mu_{r}\\sum_{i, j}^{n} \\hat{\\tilde{\\mathbf{A}}}_{i, j}^{r}\\left\\|\\mathbf{Z}_{i}-\\mathbf{Z}_{j}\\right\\|^{2}\n+\\lambda_{2}\\left\\|\\boldsymbol{\\mu}\\right\\|_{2}^{2},\\\\\n\\text { s.t. } \\sum_{r=1}^{R}\\mu_{r}=1, \\mu_{r} \\geq 0, \\forall r=1,2, \\ldots, R,\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nwhere $\\mathbf{\\lambda}_{1}$ and $\\mathbf{\\lambda}_{2}$ are non-negative trade-off parameters.\n$\\mu_{r} \\geq 0$ is the weight corresponding to relation $r$, and \nthe sum of weights is 1 for constraining the search space of possible graph Laplacians.\nThe regularization term $\\left\\|\\boldsymbol{\\mu}\\right\\|_{2}^{2}$ is to avoid the parameter overfitting to only one relation~\\cite{geng2012ensemble}.\n\n\n\nIn addition to the topological constraint by $\\mathcal{O}_{\\text {e-reg }}$ term, we should also build the relationship between the learned representation $\\mathbf{Z}$ with the node features $\\mathbf{H}$. Therefore, there is a feature fitting term: $\\mathcal{O}_{\\text {fit }} =\\|\\mathbf{Z-H}\\|_{F}^{2}$,\nwhich makes $\\mathbf{Z}$ encode information from the original feature $\\mathbf{H}$, so as to alleviate the over-smoothing problem.\nFinally, our proposed optimization framework for multi-relational graphs, which includes constraints on features and topology, is as follows:\n\\begin{eqnarray}\n\\label{overall}\n\\notag& \\arg&\\min_{\\mathbf{Z},\\mathbf{\\mu}}\\underbrace{\\|\\mathbf{Z-H}\\|_{F}^{2}}_{\\mathcal{O}_{\\text {fit }}}+\n\\underbrace{\\lambda_{1} \\sum_{r=1}^{R} \\mu_{r} \\mathbf{tr}\\left(\\mathbf{Z}^{\\top}\\tilde{\\mathbf{L}}^{r} \\mathbf{Z}\\right)+\\lambda_{2}\\left\\|\\boldsymbol{\\mu}\\right\\|_{2}^{2}}_{\\mathcal{O}_{\\text {e-reg }}}, \\\\\n &\\text { s.t. }& \\sum_{r=1}^{R}\\mu_{r}=1, \\mu_{r} \\geq 0, \\forall r=1,2, \\ldots, R.\n\\end{eqnarray}\n\nBy minimizing the above objective function, the optimal representation $\\mathbf{Z}$ not only captures the smoothness between nodes, but also preserves the personalized information. Moreover, the optimal relational coefficients $\\boldsymbol{\\mu}$ can be derived, reflecting the importance of different relations.\n\n\n\\subsection{Ensemble Message Passing Mechanism }\n\nIt is nontrivial to directly optimize $\\mathbf{Z}$ and $\\boldsymbol{\\mu}$ together because Eq.\\eqref{overall} is not convex w.r.t. $ (\\mathbf{Z}, \\boldsymbol{\\mu})$ jointly.\nFortunately, an iterative optimization strategy can be adopted,\ni.e., i.) first optimizing Eq.\\eqref{overall} w.r.t. $ \\boldsymbol{\\mu}$ with a fixed $\\mathbf{Z}$,\nresulting in the solution of relational coefficients $ \\boldsymbol{\\mu}$;\nii.) then solving Eq.\\eqref{overall} w.r.t. $\\mathbf{Z}$ with $ \\boldsymbol{\\mu}$ taking the value solved in the last iteration.\nWe will show that performing the above two steps corresponds to one ensemble message passing layer in our relational GNNs.\n\n\n\n\n\n\\setcounter{secnumdepth}{3}\n\\subsubsection*{Update Relational Coefficients}\n\n\n\nWe update relational parameters $ \\boldsymbol{\\mu}$ by fixing $\\mathbf{Z}$, then the objective function \\eqref{overall} w.r.t. $ \\boldsymbol{\\mu}$ is reduced to:\n\\begin{eqnarray}\n\\label{cor} \n\\notag &\\arg& \\min _{\\mu} \\sum_{r=1}^{R} \\mu_{r} s_{r} +\n\\mathbf{\\frac{\\lambda_{2}}{\\lambda_{1}}}\\left\\|\\boldsymbol{\\mu}\\right\\|_{2}^{2},\\\\\n &\\text { s.t. }& \\sum_{r=1}^{R}\\mu_{r}=1, \\boldsymbol{\\mu} \\geq 0, \\forall r=1,2, \\ldots, R,\n\\end{eqnarray}\nwhere $s_{r}=\\mathbf{tr}\\left(\\mathbf{Z}^{\\top}\\tilde{\\mathbf{L}}^{r} \\mathbf{Z}\\right)$.\n\n(1) When $\\mathbf{\\frac{\\lambda_{2}}{\\lambda_{1}}}=0$, \nthe coefficient might concentrate on one certain relation,\ni.e., $\\mu_{j}=1$ if ${s}_{j}=\\min_{r=1, \\ldots, R} {s}_{r}$, and $\\mu_{j}=0$ otherwise. \nWhen $\\mathbf{\\frac{\\lambda_{2}}{\\lambda_{1}}}=+\\infty$, each relation will be assigned equal coefficient, i.e., $\\mu_r=\\frac{1}{R}$~\\cite{geng2012ensemble}.\n\n(2) Otherwise, theoretically, Eq.\\eqref{cor} can be regarded as a convex function of $\\boldsymbol{\\mu}$ with the constraint in a standard simplex~\\cite{chen2011projection}, i.e., $\\Delta=$ $\\left\\{\\boldsymbol{\\mu} \\in \\mathbb{R}^{R}: \\sum_{r=1}^{R} \\mu_{r}=1, \\boldsymbol{\\mu} \\succcurlyeq 0\\right\\}$.\nTherefore, the mirror entropic descent algorithm (EMDA)~\\cite{beck2003mirror} can be used to optimize $\\boldsymbol{\\mu}$, where the update process is described by Algorithm~\\ref{alg1}.\nThe objective $f(\\cdot)$ should be a convex Lipschitz continuous function with Lipschitz constant $\\phi$ for a fixed given norm. Here, we derive this Lipschitz constant from $\\|\\nabla f(\\boldsymbol{\\mu})\\|_{1} \\leq \\frac{2\\lambda_{2}}{\\lambda_{1}}+\\|\\boldsymbol{s}\\|_{1}=\\phi$, where $\\boldsymbol{s}=\\left\\{s_{1}, \\ldots, s_{R}\\right\\}$. \n\n\n\n\\subsubsection*{Update Node Representation}\nThen we update node representation $\\mathbf{Z}$ with fixing $\\boldsymbol{\\mu}$, where\nthe objective function Eq. \\eqref{overall} w.r.t. $\\mathbf{Z}$ is reduced to:\n\\begin{eqnarray}\n\\label{z}\n\\arg \\min _{\\mathbf{Z} }\\|\\mathbf{Z}-\\mathbf{H}\\|_{F}^{2}+\n\\lambda_{1} \\sum_{r=1}^{R} \\mu_{r} \\mathbf{tr}\\left(\\mathbf{Z}^{\\top}\\tilde{\\mathbf{L}}^{r} \\mathbf{Z}\\right)\n.\\end{eqnarray}\nWe can set the derivative of Eq.~\\eqref{z} with respect to $\\mathbf{Z}$ to zero and get the optimal $\\mathbf{Z}$ as:\n\n\\begin{eqnarray}\n\\frac{\\partial\\left\\{\n\\|\\mathbf{Z}-\\mathbf{H}\\|_{F}^{2}+\n\\lambda_{1} \\sum_{r=1}^{R} \\mu_{r} \\mathbf{tr}\\left(\\mathbf{Z}^{\\top}\\tilde{\\mathbf{L}}^{r} \\mathbf{Z}\\right)\n\\right\\}}{\\partial \\mathbf{Z}}=0\\\\ \\quad \\Rightarrow \\quad \\mathbf{Z}-\\mathbf{H} + \\lambda_{1}\\sum_{r=1}^{R} \\mu_{r}\\tilde{\\mathbf{L}}^{r} \\mathbf{Z}=0.\n\\end{eqnarray}\nSince the eigenvalue of $\\mathbf{I}+\\lambda_{1}\\sum_{r=1}^{R} \\mu_{r}\\tilde{\\mathbf{L}}^{r}$ is positive, it has an inverse matrix, and we can obtain the closed solution as follows:\n\\begin{eqnarray}\n\\label{close}\n\\notag\\mathbf{Z}&=&\\left(\\mathbf{I}+\\lambda_{1} \\sum_{r=1}^{R} \\mu_{r}\\tilde{\\mathbf{L}}^{r} \\right)^{-1} \\mathbf{H}\\\\ &=&\\frac{1}{1+\\lambda_{1}}\\left(\\mathbf{I}-\\frac{\\lambda_{1}}{1+\\lambda_{1}} \\sum_{r=1}^{R} \\mu_{r}\\hat{\\tilde{\\mathbf{A}}}^{r}\\right)^{-1} \\mathbf{H}.\n\\end{eqnarray}\n\nHowever, obtaining the inverse of matrix will cause a\ncomputational complexity and memory requirement of $\\mathcal{O}\\left(n^{2}\\right)$, \nwhich is inoperable in large-scale graphs. \nTherefore, we can approximate Eq.\\eqref{close} using the following iterative update rule:\n\\begin{eqnarray}\n\\label{update}\n\\mathbf{Z}^{(k+1)}=\\frac{1}{\\left(1+\\lambda_{1}\\right)}\\mathbf{H}+\\frac{\\lambda_{1}}{\\left(1+\\lambda_{1}\\right)}\\left(\\sum_{r=1}^{R} \\mu_{r}^{(k)}\\hat{\\tilde{\\mathbf{A}}}^{r}\\right)\\mathbf{Z}^{(k)}.\n\\end{eqnarray}\nwhere $k$ is the iteration number.\n\n\\begin{algorithm}\n\n\t\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\t\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\t\\caption{Relational Coefficients Learning}\n\t\\label{alg1}\n\t\\begin{algorithmic}[1]\n\t \\REQUIRE \n\t Candidate Laplacians $\\left\\{\\tilde{\\mathbf{L}}^{1},\\cdots, \\tilde{\\mathbf{L}}^{R} \\right\\}$, \n\t the embedding matrix $\\mathbf{Z}$,\n\t the Lipschitz constant $\\phi$, the tradeoff parameters $\\lambda_{1},\\lambda_{2}$.\n\t \\ENSURE Relational coefficients $\\boldsymbol{\\mu}$.\n\t\t\\STATE Initialization: $\\boldsymbol{\\mu}=[\\frac{1}{R},\\frac{1}{R},\\cdots,\\frac{1}{R}] $\n\t\t\\FOR { $r=1$ to $R$}\n\t\t\\STATE $s_{r}=\\mathbf{tr}\\left(\\mathbf{Z}^{\\top}\\tilde{\\mathbf{L}}^{r} \\mathbf{Z}\\right)$\n \\REPEAT\n \\STATE $T_{t} \\leftarrow \\sqrt{\\frac{2 ln R}{t \\phi^{2}}}$,\n \\STATE $f^{\\prime}\\left(\\mu_{r}^{t}\\right) \\leftarrow \\frac{2\\lambda_{2}}{\\lambda_{1}} \\mu_{r}^{t}+s_{r}$,\n \\STATE $\\mu_{r}^{t+1} \\leftarrow \\frac{\\mu_{r}^{t} \\exp \\left[-T_{t} f^{\\prime}\\left(\\mu_{r}^{t}\\right)\\right]}{\\sum_{r=1}^{R} \\mu_{r}^{t} \\exp \\left[-T_{t} f^{\\prime}\\left(\\mu_{r}^{t}\\right)\\right]}$,\n \\UNTIL Convergence\n \\ENDFOR\n \\RETURN $\\boldsymbol{\\mu}$\n\t\\end{algorithmic} \n\\end{algorithm}\n\n\n\n\n\n\\subsubsection*{Ensemble Message Passing Layer (EnMP layer)}\nNow with the node representation $\\mathbf{Z}$ and the relation coefficient $\\boldsymbol{\\mu}$, we can propose our ensemble message passing layer, consisting of the following two steps: (1) relational coefficient learning step (RCL step), i.e., update the relational coefficients $\\boldsymbol{\\mu}$ according to Algorithm~\\ref{alg1}; (2) propagation step (Pro step), i.e., update the node representation $\\mathbf{Z}$ according to Eq.\\eqref{update}. The pseudocode of EnMP layer is shown in appendix A. We will show some properties of our proposed EnMP layer. \n\n\n\n\\begin{thm}[Relationship with Multi-Relational\/Path Personalized PageRank]\n\\label{reppr}\nGiven a realtion $r$, we have the relation based probability transition matrix $\\mathbf{A}^{r}_{rw}=\\mathbf{A}^{r}(\\mathbf{D}^{r})^{-1} $. \nThen, the single relation based PageRank\nmatrix is calculated via:\n\\begin{eqnarray}\n\\boldsymbol{\\Pi}_{\\mathrm{pr}}^{r}=\\mathbf{A}^{r}_{\\mathrm{rw}}\\boldsymbol{\\Pi}_{\\mathrm{pr}}^{r}.\n\\end{eqnarray}\nConsidering we have $R$ relations, i.e., $r=1,2, \\ldots, R$, the weights of each relation are $\\left\\{\\mu_{1}, \\ldots, \\mu_{R}\\right\\}$, according to~\\cite{lee2013pathrank,ji2021heterogeneous}, we can define the multiple relations based PageRank matrix:\n\\begin{eqnarray}\n\\boldsymbol{\\Pi}_{\\mathrm{pr}}=\\left(\\sum_{r=1}^{R} \\mu_{r}\\mathbf{A}^{r}_{\\mathrm{rw}}\\right)\n\\boldsymbol{\\Pi}_{\\mathrm{pr}}^{r}.\n\\end{eqnarray}\nAccording to~\\cite{klicpera2018predict}, the multi-relational personalized PageRank matrix can be defined:\n\\begin{eqnarray}\n\\boldsymbol{\\Pi}_{\\mathrm{ppr}}=\\alpha\\left(\\boldsymbol{I}_{n}-(1-\\alpha)\n\\left(\\sum_{r=1}^{R} \\mu_{r}\\hat{\\tilde{\\boldsymbol{A}}}^{r}\\right)\n\\right)^{-1},\n\\end{eqnarray}\nwhere $\\hat{\\tilde{\\boldsymbol{A}}}^{r}$ is a normalized adjacency matrix with self-loops, $\\boldsymbol{I}_{n}$ represents unit matrix, $\\alpha \\in(0,1]$ is teleport (or restart) probability.\nIf $\\alpha=\\frac{1}{\\left(1+\\lambda_{1}\\right)}$, \nthe closed-form solution in Eq.\\eqref{close} is to propagate features via multi-relational personalized PageRank scheme.\n\n\n \n\n\n\n\n\n\n\\end{thm}\n\\begin{thm}[Relationship with APPNP\/GCN]\nif ${\\lambda_{2}}=+\\infty$, the solution in Eq.\\eqref{cor} is $\\boldsymbol{\\mu}=[\\frac{1}{R},\\frac{1}{R},\\cdots,\\frac{1}{R}] $,\ni.e., each relaiton is assigned equal coefficient,\nthen the ensemble multi-relational graph $\\sum_{r=1}^{R} \\mu_{r}^{(k)}\\hat{\\tilde{\\mathbf{A}}}^{r}$ reduces to a normalized adjacency matrix $ \\frac{1}{R} \\sum_{r=1}^{R} \\hat{\\tilde{\\mathbf{A}}}^{r}$ averaged over all relations. The proposed message passing scheme reduces to:\n\\begin{eqnarray}\n\\mathbf{Z}^{(k+1)}=\\frac{1}{\\left(1+\\lambda_{1}\\right)}\\mathbf{H}+\\frac{\\lambda_{1}}{\\left(1+\\lambda_{1}\\right)}\\frac{1}{R} \\sum_{r=1}^{R} \\hat{\\tilde{\\mathbf{A}}}^{r}\\mathbf{Z}^{(k)}\n,\\end{eqnarray}\nif $\\lambda_{1}=\\frac{1}{\\alpha}-1$, it recovers the message passing in APPNP on the averaged relational graph:\n\\begin{eqnarray}\n\\mathbf{Z}^{(k+1)}=\\alpha\\mathbf{H}+(1-\\alpha)\\frac{1}{R} \\sum_{r=1}^{R} \\hat{\\tilde{\\mathbf{A}}}^{r}\\mathbf{Z}^{(k)},\n\\end{eqnarray}\nif $\\lambda_{1}= +\\infty $, it recovers the message passing in GCN on the averaged relational graph:\n\\begin{eqnarray}\n\\mathbf{Z}^{(k+1)}=\\frac{1}{R} \\sum_{r=1}^{R} \\hat{\\tilde{\\mathbf{A}}}^{r}\\mathbf{Z}^{(k)}.\n\\end{eqnarray}\n\n\\end{thm}\n\n\n\\begin{figure}[htp]\n\\centering\n \\includegraphics[width = 8cm,height = 2.8cm]{framework}\n \\caption{Model architecture.}\n \\label{model}\n\\end{figure}\n\n\n\n\n\\subsection{Ensemble Multi-Relational GNNs}\n\n\n\nNow we propose our ensemble multi-relational graph neural networks (EMR-GNN) with the EnMP layer. Similar as \\cite{klicpera2018predict}, we employ the decoupled style architecture, i.e., the feature transformation and the message passing layer are separated. The overall framework is shown in Figure~\\ref{model}, and the forward propagation process is as follows:\n\\begin{eqnarray}\n\\mathbf{Y}_{\\text {pre }}= g_{\\theta}\\left(\n\\mathbf{EnMP}^{(K)}\\left(f\\left(\\mathbf{X};\\mathbf{W}\\right), R, \\lambda_{1}, \\lambda_{2}\\right) \\right),\n\\end{eqnarray}\nwhere $\\mathbf{X}$ is the input feature of nodes,\nand $f\\left(\\mathbf{X};\\mathbf{W}\\right)$ denotes the MLPs or linear layers (parameterized by $ \\mathbf{W}$) which is used to feature extraction.\n$\\mathbf{EnMP}^{(K)}$ represents our designed ensemble relational message passing layer with $K$ layers, where $R$ is the number of relations, and $\\lambda_{1}, \\lambda_{2}$ are hyperparameters in our message passing layer.\n$g_{\\theta}(\\cdot)$ is MLPs as classifier with the learnable parameters $\\theta$.\nThe training loss is: $\\ell(\\mathbf{W}, \\theta) \\triangleq \\mathcal{D}\\left(\\boldsymbol{y}^{*}_{i}, \\hat{\\boldsymbol{y}_{i}}\\right),$\nwhere $\\mathcal{D}$ is a discriminator function of cross-entropy, \n$\\boldsymbol{y}^{*}_{i}$ and $ \\hat{\\boldsymbol{y}_{i}}$ are the predicted and ground-truth labels of node $i$, respectively.\nBackpropagation manner is used to optimize parameters in MLPs, i.e., $\\mathbf{W}$ and $\\theta$,\nand the parameters in our EnMP layers are optimized during the forward propagation. We can see that EMR-GNN is built on a clear optimization objective. Besides, EMR-GNN also has the following two advantages:\n\\begin{itemize}\n \n \\item As analyzed by Remark~\\ref{reppr}, our proposed EnMP can keep the original information of the nodes with a teleport (or restart) probability, thereby alleviating over-smoothing.\n \\item For each relation, there is a parameterized relation-specific weight matrix or parameterized relation encoder used in the traditional RGCN~\\cite{vashishth2019composition,schlichtkrull2018modeling}. While in our EnMP, only one learnable weight coefficient is associated with a relation, greatly alleviating the over-parameterization problem.\n \n \n \n \n\\end{itemize}\n\n\n\n\\begin{table}\n \\centering\n \\renewcommand{\\arraystretch}{1.2}\n \\small \n \\resizebox{8cm}{!}{\n \\begin{tabular}{lrrrrlr}\n \\toprule\n Datasets &Nodes &\\makecell*[r]{Node\\\\Types}& Edges&\\makecell*[c]{ Edge \\\\Types} & Target & Classes\\\\\n \\midrule\n MUTAG & 23,644 & 1 & 74,227&23&molecule &2\\\\\n BGS & 333,845& 1 & 916,199&103&rock &2\\\\\n DBLP&26,128&4&239,566&6&author&4\\\\\n ACM&10,942&4&547,872&8&paper&3\\\\\n \\bottomrule\n \\end{tabular}}\n \\caption{Statistics of multi-relational datasets.}\n \\label{sta}\n\\end{table}\n\n\n\\begin{table*}[!htb]\n\\centering\n\\renewcommand{\\arraystretch}{1.2}\n\\setlength\\tabcolsep{2.6pt}\n\n\\small\n\\begin{tabular}{l|r|r|r|r|r|r|r|r}\n\\bottomrule \nDataset &\\multicolumn{2}{c|}{DBLP} &\\multicolumn{2}{c|}{ACM}&\\multicolumn{2}{c|}{MUTAG}&\\multicolumn{2}{c}{BGS}\\\\\n\\hline\nMetric &{Acc ($\\%$)}& Recall ($\\%$) &Acc ($\\%$)& Recall ($\\%$)&Acc ($\\%$)& Recall ($\\%$)&Acc ($\\%$)&Recall ($\\%$)\\\\\n\\hline\n GCN & 90.39\u00b10.38 & 89.49\u00b10.52 &89.58\u00b11.47 &89.47\u00b11.49 &72.35\u00b12.17 &63.28\u00b12.95 &85.86\u00b11.96&80.21\u00b12.21\\\\\n \n GAT & 91.97\u00b10.40 & 91.25\u00b10.58 &88.99\u00b11.58 &88.89\u00b11.56 & 70.74\u00b12.13&63.01\u00b13.79 &88.97\u00b13.17&86.13\u00b14.96\\\\\n HAN &91.73\u00b10.61 &91.15\u00b10.72 & 88.51\u00b10.35 &88.50\u00b10.30 &- &- &-&-\\\\\n RGCN & 90.08\u00b10.60 &88.56\u00b10.76& 89.79\u00b10.62&89.71\u00b10.59 &71.32\u00b12.11 & 61.97\u00b13.52&85.17\u00b15.87& 81.58\u00b17.94\\\\\n\n e-RGCN & 91.77\u00b10.90& 91.18\u00b11.02 &83.00\u00b11.04 &84.03\u00b10.75 &69.41\u00b12.84& \\textbf{67.57\u00b18.04} &82.41\u00b11.96&84.51\u00b13.38\\\\\n\n\n \n \\hline\n EMR-GNN & \\textbf{93.54\u00b10.50} & \\textbf{92.39\u00b10.78 } & \\textbf{90.87\u00b10.11} & \\textbf{90.84\u00b10.13} & \\textbf{74.26\u00b10.78} & 64.19\u00b11.08&\\textbf{89.31\u00b14.12}&\\textbf{86.39\u00b1 5.33} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{ \nThe mean and standard deviation of classification\naccuracy and recall over 10 different runs on four datasets.}\n\\label{total}\n\\end{table*}\n\n\n\\begin{table*}[!htb]\n\\centering\n\\renewcommand{\\arraystretch}{1.2}\n\n\\small \n\n\\begin{tabular}{cccccc}\n\n\\bottomrule \n GCN&GAT& HAN &RGCN & e-RGCN &EMR-GNN\\\\\n\\hline\n$\\mathcal{O}\\left(K d^{2}\\right)$&\n$\\mathcal{O}\\left(2 K N d^{2}\\right)$&\n$\\mathcal{O}\\left(\n2K\\left(|\\mathcal{R}| N +1 \\right)d^{2} + 2 Kd\n\\right)$&\n$\\mathcal{O}\\left(\\mathcal{B} K d^{2}+\\mathcal{B} K|\\mathcal{R}|\\right)$&\n$\\mathcal{O}\\left(\\mathcal{B} (K-1) d^{2}+|\\mathcal{R}|d+\\mathcal{B} (K-1)|\\mathcal{R}|\\right)$&\n$\\mathcal{O}\\left(2 d^{2}+K|\\mathcal{R}|\\right)$\n\n\\\\\n\n\\bottomrule\n\\end{tabular}\n\\caption{\nComparison of the number of parameters.\nHere, $K$ denotes the number of layers in the model, $d$ is the embedding dimension, $\\mathcal{B}$ represents the number of bases, $|\\mathcal{R}|$ indicates the total number of relations in the graph and $N$ is the number of heads of attention-based models.}\n\\label{pa}\n\\end{table*}\n\n\n\n\\begin{figure*}[t]\n\\centering \n\\subfigure[GCN (SC=0.6196)]{\n\\label{Fig.sub.2}\n\\includegraphics[width=4cm,height = 3cm]{GCN}}\n\\subfigure[GAT (SC=0.6351)]{\n\\label{Fig.sub.2}\n\\includegraphics[width=4cm,height = 3cm]{GAT}}\n\\subfigure[RGCN (SC=0.6093)]{\n\\label{Fig.sub.2}\n\\includegraphics[width=4cm,height = 3cm]{RGCN}}\n\\subfigure[EMR-GNN (SC=0.7799)]{\n\\label{Fig.sub.1}\n\\includegraphics[width=4cm,height = 3cm]{EMR}}\n\\caption{Visualization of the learned node embeddings on DBLP dataset.}\n\\label{vis}\n\\end{figure*}\n\n\\begin{figure}[t]\n\\centering \n\\subfigure[DBLP]{\n\\label{Fig.sub.1}\n\\includegraphics[width=4.15cm,height = 4.cm]{dblp_layer}}\n\\subfigure[ACM]{\n\\label{Fig.sub.2}\n\\includegraphics[width=4.15cm,height = 4.cm]{acm_layer}}\n\\caption{ Analysis of propagation layers. Missing value in red line means CUDA is out of memory.}\n\\label{nlayer}\n\\end{figure}\n\n\\begin{figure}[htp]\n \\centering\n \\includegraphics[width=7cm,height = 5cm]{cor}\n \\caption{Accuracy under each single relation and corresponding relational coefficient.}\n \\label{coe}\n\\end{figure}\n\n\\section{Experiment}\n\n\\label{sec:experiment}\n\n\n\n\n\n\n\\subsection{Experimental Settings}\n\\paragraph{Datasets.}\nThe following four real-world heterogeneous datasets in various fields are utilized and can be divided into two categories: i) the node type and edge type are both heterogeneous (DBLP~~\\cite{fu2020magnn}, ACM~\\cite{lv2021we}).\nii) the node type is homogeneous but the edge type is heterogeneous (MUTAG~\\cite{schlichtkrull2018modeling}, BGS~\\cite{schlichtkrull2018modeling}).\nThe statistics of the datasets can be found in Table \\ref{sta}.\nThe basic information about datasets is summarized in appendix B.1.\n\n\n\\paragraph{Baselines.}\nTo test the performance of the proposed EMR-GNN, \nwe compare it with five state-of-the-art baselines.\nAmong them, GCN~\\cite{kipf2016semi} and GAT~\\cite{velivckovic2017graph} as two popular approaches are included.\nIn addition, we compare with the heterogeneous graph model HAN~\\cite{wang2019heterogeneous}, since HAN can also employ multiple relations.\nTwo models that are specially designed for multi-relational graphs are compared, i.e., RGCN~\\cite{schlichtkrull2018modeling} and e-RGCN~\\cite{thanapalasingam2021relational}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Parameter settings.}\nWe implement EMR-GNN based on Pytorch.\\footnote{\\url{https:\/\/pytorch.org\/}}\nFor $f\\left(\\mathbf{X};\\mathbf{W}\\right)$ and $g_{\\theta}(\\cdot)$,\nwe choose one layer MLP for DBLP and ACM, and linear layers for MUTAG and BGS. \nWe conduct 10 runs on all datasets with the fixed training\/validation\/test split for all experiments.\nMore implementation details can be seen in appendix B.3.\n\n\n\n\n\\subsection{Node Classification Results}\nTable \\ref{total} summarizes the performances of EMR-GNN and several baselines on semi-supervised node classification task.\nSince HAN's code uses the heterogeneity of nodes to design meta-paths, we do not reproduce the results of HAN on homogeneous dataset (MUTAG, BGS) with only one type of nodes.\nWe use accuracy (Acc) and recall metrics for evaluation, and report the mean and standard deviation of classification accuracy and recall.\nWe have the following observations:\n(1) Compared with all baselines, the proposed EMR-GNN generally achieves the best performance across all datasets on seven of the eight metrics, which demonstrates the effectiveness of our proposed model. \ne-RGCN has a higher recall but a lower accuracy on MUTAG, which may be caused by overfitting.\n(2) Meanwhile, the number of parameters of our model and other baselines are shown in Table \\ref{pa}. We can see that \nEMR-GNN is more parameter efficient than all baselines, \n i.e., $\\mathcal{O}\\left(2 d^{2}+K|\\mathcal{R}|\\right)$,\nbut achieves maximum relative improvements of $4.14\\%$ than RGCN on BGS.\nIt means that EMR-GNN largely overcomes the over-parameterization in previous multi-relational GNNs.\n\n\\subsection{Model Analysis}\n\\paragraph{Alleviating over-smoothing problem.}\nAs mentioned before, EMR-GNN is able to alleviate over-smoothing issue.\nHere, we take one typical single-relation GCN (GAT) and one representative multi-relational GCN (RGCN) as baselines to test their performance with different propagation depths, where the results are shown in\nFigure.\\ref{nlayer}. We have the following observations:\n(1) Our model significantly alleviates the over-smoothing problem, \nsince there is generally no performance degradation when the depth increases.\nThis benefits from the adjustable factor $\\mathbf{\\lambda_{1}}$ in EMR-GNN, which flexible controls the influence of node feature information.\nIn contrast, the performance of RGCN and GAT drops seriously with increasing depth, \nimplying that these models suffer from the over-smoothing problem.\n(2) RGCN needs huge storage cost, making it difficult to stack multiple layers.\nCuda out of memory occurs when the propagation depth increases, i.e., DBLP for more than 16 layers, and ACM can merely stack 8 layers. This is not available for capturing long-range dependencies.\n\n\n\n\n\n \n\n\n\\begin{table}[!htb]\n\n\\renewcommand{\\arraystretch}{1.2}\n\n\\small \n\\begin{tabular}{l|l|rrrr}\n\\bottomrule \nDatasets &Method &\\multicolumn{4}{c}{Size of training set}\\\\\n\\hline\n & & $10\\%$ & $15\\%$& $20\\%$&$25\\%$\\\\\n\\hline\n DBLP &RGCN & 0.5381& 0.6388& 0.7515&0.7721\\\\\n\n & EMR-GNN & 0.8768& 0.9109 & 0.9128& 0.9364 \\\\\n \\hline\n ACM &RGCN & 0.7492 &0.8136 &0.8278&0.8344 \\\\\n\n & EMR-GNN & 0.8489 & 0.8654 & 0.8739 & 0.8753 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Classification accuracy w.r.t. different training set.}\n\\label{size}\n\\end{table}\n\\paragraph{Alleviating over-parameterization problem.}\nTo further illustrate the advantages of alleviating over-parameterization,\nwe verify EMR-GNN with small-scale training samples.\nWe conduct experiments on EMR-GNN and RGCN using two datasets. We only select a small part of nodes from original training samples as the new training samples. \nAs can be seen in Table \\ref{size}, EMR-GNN consistently outperforms RGCN with different training sample ratios, which again validates the superiority of the proposed method.\nOne reason is that a limited number of parameters in EMR-GNN can be fully trained with few samples.\nIn contrast, RGCN with excess parameters requires large-scale training samples as the number of relations increases.\nThe time complexity is analyzed in appendix C.\n\n\n\\noindent\n\\paragraph{Analysis of relational coefficients.}\nBesides the performance, \nwe further show that EMR-GNN can produce reasonable relational coefficients.\nTo verify the ability of relational coefficients learning, \ntaking ACM dataset as example, we evaluate the classification performance under each single relation.\nThe classification accuracy and the corresponding relational coefficient value are reported in Figure \\ref{coe}.\nWe can see that basically, the relation which achieves better accuracy is associated with a larger coefficient. Moreover, we compute the pearson correlation coefficient between the accuracy of a single relation and its relational coefficient, which is 0.7918, well demonstrating that they are positively correlated.\n\n\n\n\n\n\n\n\\noindent\n\\paragraph{Visualization.}\nFor a more intuitive comparison, we conduct the task of visualization on\nDBLP dataset.\nWe plot the output embedding on the last layer of EMR-GNN and three baselines (GCN, GAT and RGCN) using t-SNE~\\cite{van2008visualizing}. \nAll nodes in Figure \\ref{vis} are colored by the ground truth labels.\nIt can be observed that EMR-GNN performs best, since the significant boundaries between nodes of different colors, and the relatively dense distribution of nodes with the same color.\nHowever, the nodes with different labels of GCN and RGCN are mixed together.\nIn addition, we also calculate the silhouette coefficients (SC) of the classification results of different models, and EMR-GNN achieves the best score, furthering indicating that the learned representations of EMR-GNN have a clearer structure.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this work, \nwe study how to design multi-relational graph neural networks from the perspective of optimization objective. We propose an ensemble optimization framework, and derive a novel ensemble message passing layer. Then we present the ensemble multi-relational GNNs (EMR-GNN), which has nice \nrelationship with multi-relational\/path personalized PageRank and can recover some popular GNNs.\nEMR-GNN not only is designed with clear objective function, but also can well alleviate over-smoothing and over-parameterization issues.\nExtensive experiments demonstrate the superior performance of EMR-GNN over the several state-of-the-arts. \n\n\n\n\n\\clearpage\n\n\\section*{Acknowledgements}\nThe research was supported in part by the National Natural Science Foundation of China (Nos. 61802025, 61872836, U1936104) and Meituan.\n\n\\input{ijcai22-multiauthor.bbl}\n\n\n\\end{sloppypar}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\\subsection{Background}\nThe cost incurred while traversing a vertex via two consecutive edges of different colors is called \\emph{traversal cost} and it depends only on the colors of the traversed edges. This cost leads to two different global cost measures that appeared in the literature under the names of \\emph{reload cost} and \\emph{changeover cost}.\n\nThe reload cost concept is defined in \\cite{wirth2001reload} and it received attention only recently, although it has numerous applications. For instance, a color may represent the mode of transportation in an intermodal cargo transportation network. The traversal cost corresponds to the cost of transferring cargo from one carrier to another. Another application is in energy distribution networks, where the energy transfer from one carrier to another one, such as the conversion of natural gas from liquid to gas state, results in loss of energy. In telecommunications, traversal costs arise in numerous settings. For instance, routing in a heterogeneous network requires switching among different technologies such as cables, fibers, and satellite links. This switching cost can be modeled by traversal costs. Even within the same technology, switching between different providers, for instance switching between different commercial satellite providers in satellite networks, leads to a switching cost. All applications hitherto mentioned can be modeled using traversal costs where an edge-colored graph is given as input, and this is the focus of the works in the literature, e.g. \\cite{wirth2001reload, galbiati2008complexity, GGM14mincyclecover, gourves2010dmtcs, AGM11minreloadpathstoursflows, gamvros2012reload, GGM11minchangeover, GSVZ14}. However, problems of finding a proper edge coloring so as to minimize the reload (or changeover) cost have important applications as well. For instance, recently, cognitive radio networks (CRN) have gained increasing attention in the communication networks research community. Unlike other wireless technologies, CRNs are envisioned to operate in a wide range of frequencies. Therefore, switching from one frequency band to another frequency band in a CRN has a significant cost in terms of delay and power consumption \\cite{gozupek2013spectrum}. An optimal allocation of frequencies to the wireless links in CRNs so that the switching cost is minimized, corresponds to a proper edge coloring minimizing the traversal cost. In this work we focus on this type of problems.\n\nThe reload cost refers to the sum of the traversal costs of a set of paths in a graph, whereas the changeover cost does not depend on the amount of commodity (or number of paths) traversing a vertex. The works \\cite{wirth2001reload,galbiati2008complexity} consider the problem of finding a spanning tree having minimum diameter with respect to reload cost. The paper \\cite{GGM14mincyclecover} considers the \\emph{minimum reload cost cycle cover} problem, which is to find a set of vertex disjoint cycles spanning all vertices, with minimum reload cost. In \\cite{gourves2010dmtcs}, the authors study the problem of finding a path, trail, or walk of minimum reload cost between two given vertices. They consider (a)symmetric reload costs and reload costs with(out) triangle inequality.\nThe paper \\cite{gamvros2012reload} studies the problem of finding a spanning tree that minimizes the sum of reload costs over the paths\nbetween \\emph{all} pairs of vertices. The paper \\cite{AGM11minreloadpathstoursflows} presents various path, tour, and flow problems related to reload costs.\nOne of the problems studied in that work is the \\emph{minimum reload cost path-tree}, which\nis to find a spanning tree that minimizes the reload cost from a given root vertex to all other vertices.\nThe paper \\cite{GGM11minchangeover} studies a closely related, yet different problem, called\n\\emph{minimum changeover cost arborescence} in which the goal is to find a spanning tree that minimizes the changeover cost from a given root vertex to all other vertices.\nThe work in \\cite{GSVZ14} considers the same problem and derives several inapproximability results, as well as polynomial-time algorithms for some special cases. In \\cite{GSSZ14-ChangeoverTreewidth} these special cases are extended to show that the problem is polynomial-time solvable in bounded treewidth graphs.\n\n\\subsection{Our Contribution and Research Directions}\nIn this paper, we consider a different problem and focus on proper edge coloring of a given graph such that the reload\/changeover cost is minimized. To the best of our knowledge, this paper is the first study of this family of problems. Specifically, we formulate two pairs of problems. In the first pair of problems, given a set of paths comprising a graph, the goal is to find a proper edge coloring of the graph so that the reload (resp. changeover) cost is minimized. In the second pair of problems, given a graph and a root vertex, the goal is to find proper edge coloring and a spanning tree of the graph so that the reload (resp. changeover) cost from the root vertex to all other vertices is minimized. We present several hardness results as well as polynomial-time solvable special cases of these problems.\n\nSpecifically, we show that the first pair of problems are hard to approximate in general, within any polynomial-time computable function of the input length, and this result is valid when the traversal costs are unbounded. When the traversal costs are bounded, we prove that the problems remain $\\nph$ even when the underlying network is a star. However, we prove that this pair of problems are polynomial-time solvable in trees when the degrees of the vertices are bounded.\n\nWe then prove that the second pair of problems are $\\apxh$ under bounded traversal costs in directed graphs. On the positive side, for these problems, we present a polynomial-time algorithm on trees (where both the degrees, and traversal costs are unbounded). We extend this algorithm to graphs where the difference between the number of edges and the number of vertices is constant, i.e. graphs that are in some sense close to trees. However, this extension does not cover cactus graphs, which have treewidth $2$. To solve the problem for other special graph classes such as cactus graphs or bounded treewidth graphs is one possible research direction. Another interesting research direction is to analyze these problems from the perspective of parameterized complexity where a few important and practical parameters are the treewidth of the graph, the number of colors, and the ratio of the largest traversal cost to the smallest non-zero traversal cost.\n\n\\section{Introduction}\\label{sec:Intro}\n\\input{introduction}\n\n\\section{Preliminaries}\n\\label{sec:Preliminaries}\n\\input{preliminaries}\n\n\\section{Hardness Results}\\label{sec:Herdness}\n\\input{hardness}\n\n\\section{Polynomial-time Solvable Cases}\\label{sec:Algorithms}\n\\input{algorithms}\n\n\n\n\\small\n\\input{Main.bbl}\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe spread of conspiracy theories online is not a new problem.\nFor instance, conspiracy theories related to the 9\/11 attack (e.g., the attack was, in fact, a controlled demolition) and the Sandy Hook shooting (e.g., the shooting was staged) were extensively disseminated on the Web.\nOver the past few years, however, we witness an explosion in the spread and popularity of conspiracy theories on the Web.\nMore worrying is that newer conspiracy theories seem to have a more negative impact on the online and offline world. \nPeople get radicalized online from the continuous exposure to conspiratorial content and then perpetrate violent acts in the real world.\nFor instance, the Pizzagate conspiracy theory was the driving factor for a shooting at a pizzeria in Washington DC in 2016~\\cite{pizzagate_shooting}.\nTaken all together, there is a pressing need to understand how these conspiracy theories spread online and how users are radicalized from the exposure to conspiratorial content.\n\nOne conspiracy theory that is considered very persuasive and attracted high engagement from people is the QAnon conspiracy theory.\nThe theory alleges that a secret cabal of people run a child sex-trafficking ring and is working against Donald Trump (as the US president by the time the theory emerged).\nOver the years, the conspiracy theory attracted many new followers across the globe and essentially evolved into a cult. \nWorryingly, the followers of the QAnon conspiracy theory started making threats or participate in violent real-world incidents (e.g., Capitol attack in 2021~\\cite{qanon_capitol}), hence highlighting the impact that the conspiracy theory has in the real world~\\cite{qanon_violence}.\n\nMotivated by the negative impact that QAnon has in the real world, mainstream platforms like Facebook, Twitter, and YouTube, started moderating and removing QAnon-related content.\nDue to this, QAnon supporters sought new online ``homes'' in less-moderated platforms and migrated to other platforms like Parler and Telegram. \nAt the same time, the QAnon conspiracy theory became a global phenomenon; the QAnon conspiracy theory started accumulating new followers across the globe, particularly in European countries like Germany and Spain~\\cite{qanon_europe}.\nOverall, it is imperative to understand how QAnon evolved and became a global phenomenon over time.\nTo do this, we use Telegram as the source of our study for two reasons. \nFirst, anecdotal evidence suggests that QAnon followers migrated to Telegram after bans on other platforms~\\cite{telegram_qanon}.\nSecond, Telegram is a rapidly growing platform with worldwide coverage, hence it is the ideal platform for effectively studying the QAnon conspiracy theory across the globe.\n\n\n\\descr{Research Questions.} We focus on answering the following research questions:\n\\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt]\n\t\\item \\textbf{RQ1:} How does the QAnon community evolve on Telegram over time and across languages? How toxic is the QAnon community on Telegram?\n\t\\item \\textbf{RQ2:} How popular is QAnon content on Telegram?\n\t\\item \\textbf{RQ3:} What are the main topics of discussion on QAnon-related groups\/channels on Telegram? Are there differences across languages?\n\\end{itemize}\n\nTo answer the above-mentioned research questions, we perform a large-scale data collection and analysis of QAnon-related groups\/channels on Telegram.\nOverall, we collect 4.5M messages shared in 161 Telegram groups\/channels between September 2017 and March 2021.\nUsing Google's Perspective API, we investigate the toxicity of QAnon content on Telegram and assess whether the movement is becoming more toxic over time and whether there are substantial differences across languages.\nAlso, using a multilingual BERT-based topic modeling approach, we study the QAnon discourse across multiple countries\/languages.\n\n\n\n\n\\descr{Main findings.} Our study provides some key findings:\n\n\\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt]\n\t\\item We find that the QAnon movement increased substantially during 2020 and 2021 on Telegram, which is likely due to people migrating from other mainstream platforms that take moderation actions against QAnon content. Furthermore, by comparing content across languages, we find that German content overshadowed English in popularity during 2020 and 2021 (\\textbf{RQ1}).\n\t\\item By analyzing the toxicity of QAnon-related messages, we find that toxicity is rising over time (almost 2x times more toxic messages in March 2021 compared to September 2019), while at the same time, we find substantial differences across languages, with German and Portuguese being the most toxic languages. (\\textbf{RQ1}).\n\t\\item Our analysis indicates that QAnon content on Telegram is popular and can be viewed by many users. Specifically, we find one order of magnitude more views and forwards on QAnon content than a baseline dataset containing a set of political oriented groups. Also, we find that over time QAnon content is becoming even more popular and reaches an increasing number of Telegram users (\\textbf{RQ2}).\n\t\\item Our discourse analysis highlights that QAnon has evolved into discussing various topics of interest within far-right movements across the globe. We find several topics of discussions like world politics, conspiracy theories, COVID-19, and the anti-vaccination movement (\\textbf{RQ3}).\n\\end{itemize}\n\n\\noindent Overall, our analysis portraits a nuanced overview of the QAnon movement across multiple countries.\nOur observations suggest that the QAnon movement is adapting and expanding from the US to other countries, usually embodied in far-right movements.\nAs a consequence, efforts to reduce redundant work from fact checkers around the world, such as the \\#CoronaVirusFacts led by the International Fact-Checking Network~\\cite{covid_poynter}, may be relevant to debunk misinformation associated with QAnon globally. \n\n\n\n\\section{Background \\& Related Work}\nTo the best of our knowledge, our effort is the first to explore the spread of QAnon movement over Telegram. Thus, next we briefly we provide background information and review previous work related to QAnon and Telegram. \n\n\\descr{QAnon.} QAnon is a conspiracy theory alleging that a secret group of people (i.e., a cabal consisting of Democratic politicians, government officials, and Hollywood actors) were running a global child sex trafficking ring and were plotting against former US president Donald Trump~\\cite{qanon_guardian}.\nThe conspiracy theory started in October 2017 with a post on 4chan by a user named ``Q,'' who claimed that he was an American government official with classified information about plots against then-President Donald Trump.\nSubsequently, ``Q'' continued disseminating cryptic messages about the QAnon conspiracy theory (called ``Q drops'') mainly on 8chan.\nThe QAnon conspiracy theory accumulated many followers on fringe Web communities like 4chan\/8chan and mainstream ones like Facebook~\\cite{qanon_fb_presence} and Twitter, especially after then-president Donald Trump retweeted QAnon-related content~\\cite{qanon_twitter_trump}.\nQAnon followers usually use their motto ``Where We Go One, We Go All'' (or simply wwg1wga) to tag content related to the QAnon conspiracy theory on the Web.\n\nOver the past few years, followers of the QAnon conspiracy theory made violent threats or were linked with several incidents of real-world violence~\\cite{qanon_violence}, with the Federal Bureau of Investigation (FBI) labeling it as a potential domestic terrorist threat~\\cite{qanon_fbi_domestic}. \nThe last straw of these incidents happened on January 6th, 2021, when supporters of the QAnon conspiracy theory attacked the US capitol in an attempt to overturn Donald Trump's defeat in the 2020 US elections by disrupting the Congress that was in the process of formalizing Joe Biden's victory~\\cite{qanon_capitol}.\nDue to these threats and violent incidents, mainstream platforms like Facebook~\\cite{facebook_suspensions}, Twitter~\\cite{twitter_suspensions}, Reddit~\\cite{reddit_suspensions}, and YouTube~\\cite{youtube_suspensions} started monitoring content related to QAnon and actively removing groups, subreddits, and users that are related to the QAnon conspiracy theory.\nNaturally, following these content moderation interventions, supporters of the QAnon conspiracy theory flocked to other fringe Web communities, with lax moderation, like Parler~\\cite{aliapoulios2021early} and Gab~\\cite{lima2018inside,zannettou2018gab}, or messaging platforms like Telegram~\\cite{telegram_qanon}.\n\nDespite that the idea of the QAnon conspiracy theory is US-centric, the conspiracy theory recently started becoming a global phenomenon, and in particular it became popular among people with far-right ideology.\nDuring 2020, the QAnon theory spread to Europe~\\cite{qanon_europe}.\nThe conspiracy theory is nowadays shared among people from Spain, Italy, United Kingdom, and Germany, one of the most popular ``representatives'' in Europe~\\cite{qanon_germany}.\n\nPrevious work investigates several aspects of the QAnon conspiracy theory.\n\\cite{papasavva2020qoincidence} analyze content toxicity and narratives in a Qanon community on Voat, finding that discussions in popular communities on Voat are more toxic than in QAnon communities.\n\\cite{aliapoulios2021early} provide a dataset of 183M Parler posts, and they highlight that QAnon is one of the dominant topics on Parler.\n\\cite{miller2021characterizing} investigates a sample of QAnon-related comments on YouTube, highlighting the international nature of the movement.\n\\cite{garry2021qanon} explore QAnon supporters' behavior in spreading disinformation on Gab and Telegram, finding that the dissemination of disinformation is one of the main reasons for the dramatic growth of QAnon conspiracy.\n\\cite{hannah2021qanon} also investigate the reasons for the growth of QAnon, finding that sharing and discussing Q drops is one of the main reasons.\n\\cite{chandler2020we} investigates how QAnon followers are influenced by Q drops, finding that Q drops focus on the perceived allies or enemies of QAnon. \n\\cite{planck2020we} compares the QAnon community's rhetoric with a mainstream conservative community on Twitter, finding that tweets posted by QAnon supporters are more violent.\n\\cite{aliapoulios2021gospel} investigate a dataset of 4.9K canonical Q drops from six aggregation sites. \nThey identify inconsistencies among the drops shared across aggregation sites and demonstrate that the drops have multiple authors.\nFinally,~\\cite{ferrara2020characterizing} investigate 240M election-related tweets \nfinding that 13\\% of users spreading political conspiracies (including QAnon) are bots.\n\n\n\\descr{Telegram.} Telegram is a popular messaging platform, with over 500M monthly active users~\\cite{telegram_users}. \nUsers can create public and private chat rooms called \\emph{channels} or \\emph{groups}.\nChannels support few-to-many communication, where only the creator and few administrators can post messages, while groups support many-to-many communication (all members can post messages).\nOn Telegram, groups and channels can have a large number of members, with a limit of 200,000 members for groups and an unlimited number of members in channels, hence Telegram is useful for disseminating information to a large group of people.\nTelegram users can share messages in groups\/channels, with Telegram supporting text, images, videos, audios, stickers, etc.\nAlso, users can forward messages between groups\/channels, with Telegram showing an indication in its user interface that the message is forwarded and the source channel\/group.\n\nDue to its privacy policy and encrypted nature (i.e., ``all data is stored heavily encrypted''), Telegram attracted the interest of dangerous organizations like terrorists~\\cite{Telegram-Terrorists} and far-right groups~\\cite{Telegram-White-Supremacists}.\nGiven this history and use of Telegram, in this work, we study the QAnon conspiracy theory through the lens of the Telegram platform. \nAlso, we select Telegram as it is popular across the globe, hence assisting us in studying the globalization of the QAnon conspiracy theory.\n\n\\section{Dataset}\\label{sec:dataset}\n\n\n\\begin{table}[]\n\\small\n\\centering\n\\begin{tabular}{lllrr}\n\\hline\n\\textbf{Dataset} & \\textbf{Source} & \\textbf{\\#Groups} & \\multicolumn{1}{l}{\\textbf{\\#Senders}} & \\multicolumn{1}{l}{\\textbf{\\#Messages}} \\\\ \\hline\n\\multirow{3}{*}{\\textbf{QAnon}} & \\textbf{Twitter\/FB} & 77 & 92,475 & 3,564,381 \\\\\n & \\textbf{Forwarded} & 84 & 84 & 942,998 \\\\ \\cline{2-5} \n & \\textbf{Total} & 161 & 92,559 & 4,507,379 \\\\ \\hline\n\\textbf{Baseline} & \\textbf{Twitter\/FB} & 869 & 200,591 & 9,358,946 \\\\ \\hline\n\\end{tabular}%\n\\caption{Overview of our Telegram dataset.}\n\\label{tab:dataset}\n\\end{table}\n\nAn inherent challenge that exists when studying phenomena through platforms like Telegram is to discover groups\/channels related to the topic of interest.\nTo overcome this challenge and discover groups\/channels related to QAnon, we follow and build on top of the methodology by~\\cite{hoseini2020demystifying}.\nSpecifically, we: \n1)~search on Twitter and Facebook for URLs to Telegram groups\/channels;\n2)~collect metadata for each group\/channel;\n3)~select groups\/channels based on QAnon-related keywords.\n4)~manually assess the relevance of the selected groups\/channels;\n5)~join and collect all messages from all the discovered QAnon groups\/channels;\nand 6)~expand our QAnon groups\/channels based on forwarded messages shared in already discovered QAnon groups\/channels and repeat Step 5.\nBelow, we elaborate on each step.\n\n\\descr{1. Discovering groups\/channels.} We use Twitter and Facebook to discover Telegram groups\/channels. \nFor Twitter, we use the Search and Streaming API to collect tweets that include Telegram URLs, following the methodology by~\\cite{hoseini2020demystifying}, while for Facebook, we use the Crowdtangle API to obtain posts including Telegram URLs~\\cite{crowdtangle}.\nFor both data sources, we perform queries with three URL patterns obtained from~\\cite{hoseini2020demystifying}: \\emph{t.me}, \\emph{telegram.me}, and \\emph{telegram.org}.\nWe collect Twitter and Facebook posts, including URLs from the above patterns between April 8, 2020, and October 10, 2020, ultimately collecting a set of 5,488,596 tweets and 14,004,394 Facebook posts that include a set of 922,289 unique Telegram URLs.\nNote that the Crowdtangle API tracks and provides data only from publicly available Groups and Pages (i.e., does not include posts from user timelines).\n\n\n\\descr{2. Collecting group\/channel metadata.} Having discovered a set of Telegram URLs, we then use Telegram's Web client and obtain basic group\/channel metadata from the URLs. These include:\na) Name of the group\/channel;\nb) Description of the group\/channel;\nc) Number of members; and\nd) the type of the Telegram URL (i.e., if it corresponds to a channel or a group).\n\n\n\n\\descr{3. Selecting QAnon groups\/channels.} The next step is to narrow down the set of groups\/channels to the ones that mention QAnon.\nTo do this, we search for the appearance of QAnon-related keywords on Twitter\/Facebook posts that shared Telegram URLs or on the group\/channel metadata obtained from Step 2. \nWe use two QAnon-related keywords: \\emph{qanon} and \\emph{wwg1wga}.\nThe former refers to the conspiracy theory itself, while the latter is the QAnon movement's motto that refers to ``Where We Go One We Go All.''\nWe select these specific keywords mainly because they are prevalent and used extensively by members of the QAnon movement.\nOverall, we find 204 Telegram groups\/channels that include the above keywords on their group\/channel metadata or any collected Twitter\/Facebook posts.\n\n\n\\descr{4. Validating QAnon groups\/channels.} \nThen, we validate that the selected groups\/channels are related to QAnon and remove any groups\/channels that are not directly related (i.e., mentioning QAnon only once because of mentions in the news). \nTo do this, an author of this study, who has previous experience on the QAnon conspiracy theory, manually annotated the 204 groups\/channels obtained from Step 3. \nSpecifically, the annotator viewed each group\/channel via Telegram's Web client and spent 5-10 minutes reading content shared in the group\/channel and checking the group\/channel metadata to decide on whether the group\/channel is related to the QAnon conspiracy theory. \nNote that since many groups\/channels are in languages other than English, the annotator used Google's translate functionality to translate content in English.\nOverall, we annotate all 204 groups\/channels and find 77 QAnon groups\/channels.\n\n\\descr{5. Joining and collecting messages in QAnon groups\/channels.} \nThe next step in our data collection methodology is to join the QAnon groups\/channels and collect all their messages.\nWe join all QAnon groups\/channels, and then we use Telegram's API~\\cite{Telegram-API} to collect all the messages shared within these groups.\nNote that we only join and collect data from public groups\/channels.\nInitially, we collect 3.5M messages shared in 77 QAnon groups\/channels from 92K senders between September 7, 2017, and March 9, 2021 (see Table~\\ref{tab:dataset}). \n\n\n\n\\descr{6. Expanding QAnon groups\/channels.} During our manual validation of the QAnon groups\/channels, we observed many messages shared in QAnon groups\/channels that are forwarded messages from other groups\/channels.\nAiming to expand our set of QAnon groups\/channels, we extract all groups\/channels that forwarded messages in the 77 already discovered QAnon groups\/channels and manually validate (see Step 4) the top 200 groups\/channels in terms of the number of forwarded messages.\nNote that we only validate the top 200, as manually checking and validating the groups\/channels is time-consuming.\nUsing this approach, we discover an additional 84 QAnon groups\/channels.\nThen, we repeat Step 5 for the newly discovered groups and collect all their messages.\nOverall, by combining the initial dataset and the one after expanding the QAnon groups\/channels, we obtain a set of 4.5M messages from 92K senders in 161 QAnon groups\/channels between September 7, 2017, and March 9, 2021 (see Table~\\ref{tab:dataset}).\n\n\n\n\\descr{Baseline dataset.} We also collect a baseline dataset for comparing it with our QAnon dataset. \nTo collect our baseline dataset, we follow Steps 1, 2, 3, and 5 as described above, with the only difference that we use a different set of keywords for selecting the groups\/channels (note that we do not validate and manually check the groups\/channels because they are not focusing on a specific topic).\nSpecifically, we use a set of keywords obtained from First Draft~\\cite{firstdraft} that includes 133 keywords\/phrases about important events that happened in 2020. \nFor example, our keywords set includes words related to the 2020 US elections and the COVID-19 pandemic.\nOverall, we join 869 channels and collect 9,358,946 messages shared between September 27, 2015 and March 9, 2021 (see Table~\\ref{tab:dataset}).\n\n\n \n\n\n\n\\descr{Ethical considerations.} \nBefore collecting any data, we obtained approval from our institution's ethical review board. \nAlso, we stress that: \na) we work entirely with publicly available data; \nb) we do not make any attempt to de-anonymize users; and\nc) we do not make any attempt to track users across platforms.\nOverall, we follow standard ethical guidelines~\\cite{rivers2014ethical} throughout our data collection and analysis. \n\n\n\\begin{figure*}[t!]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.3145\\textwidth]{active_groups_per_week}\\label{fig:active_groups}}\n\\subfigure[]{\\includegraphics[width=0.341\\textwidth]{messages_per_week}\\label{fig:messages_per_week}}\n\\subfigure[]{\\includegraphics[width=0.3345\\textwidth]{senders_per_week}\\label{fig:senders_per_week}}\n\\vspace{-4mm}\n\\caption{Activity within the QAnon groups\/channels over time, in terms of active groups\/channels, number of messages, and number of senders per week.}\n\\label{fig:temporal_activity}\n\\vspace{-4mm}\n\\end{figure*}\n\n\\section{Methods}\\label{sec:methods}\n\nIn this section, we present our methods for analyzing multilingual content posted in QAnon-related groups\/channels.\n\n\\descr{Topic Modeling.} One of the goals of this study is to analyze the QAnon-related discourse across languages.\nTo achieve this, we use a Bidirectional Encoder Representations from Transformers (BERT)-based topic modeling methodology by~\\cite{grootendorst2020bertopic}.\nSpecifically, we use a pre-trained multilingual BERT model (stsb-xlm-r-multilingual) from~\\cite{reimers2020making} to embed documents from multiple languages to the same high-dimensional vector space.\nWe select this specific model mainly because it supports 50+ languages and performs well in semantic similarity tasks.\nThen, we use the Uniform Manifold Approximation and Projection (UMAP) approach proposed by~\\cite{mcinnes2018umap} to reduce the dimensionality of the extracted embeddings.\nThis is an important step, as it allows us to reduce the dimensionality of the embeddings, hence increasing the performance and scalability of the next step (i.e., clustering).\nThen, we group the reduced embeddings using the HDBSCAN algorithm~\\cite{mcinnes2017hdbscan}.\nFinally, using Term Frequency\u2013Inverse Document Frequency (TF-IDF) on the clustered documents, we generate topic representations (i.e., a set of terms describing each topic). \n\n\\descr{Toxicity Assessment.}\nAlso, we aim to quantify how extreme QAnon Telegram content is and whether there are changes in content's toxicity over time.\nTo assess this, we use Google's Perspective API~\\cite{jigsaw2018perspective} to annotate each message in our dataset with a score that reflects on how rude or disrespectful a comment is.\nSpecifically, following previous work~\\cite{ribeiro2020does}, we use the SEVERE\\_TOXICITY model provided by the Perspective API and treat each message that has a score of 0.8 or more as toxic. \nWe elect to use Perspective API for annotating content as toxic, mainly because it offers production-ready models that support multiple languages; as of May 2021, the Perspective API supports English, Spanish, French, German, Portuguese, Italian, and Russian.\nTherefore, we can only assess the toxicity of messages posted in any of the seven languages above, which corresponds to 65\\% of the messages in our dataset.\nThe rest of the messages do not include any text (20\\% are sharing only audio, video, or images) or are in other languages (15\\%) that the Perspective API does not support.\nNote that the use of the Perspective API to assess the toxicity of content is likely to introduce some false positives or biases~\\cite{davidson2019racial}. \nPrevious work~\\cite{gehman2020realtoxicityprompts}, has validated the performance of the Perspective API, however, it focuses mainly on the English annotations.\nTherefore, although the models are in production, it is likely that there are biases in the Perspective API across languages.\n\n\n\\section{Results}\\label{sec:results}\n\n\n\\subsection{Activity in QAnon Groups\/Channels}\n\n\nWe start our analysis by looking into the general activity in QAnon groups\/channels.\nFig.~\\ref{fig:temporal_activity} shows the number of active groups, number of messages, and number of senders per week in our dataset.\nOverall, we observe a substantial increase in the number of messages and senders after October 2020. \nThis is likely because Facebook~\\cite{facebook_suspensions} removed accounts and groups related to QAnon from their platforms during October 2020, hence users likely migrated to alternative platforms like Telegram.\nAlso, we observe a peak in activity during early 2021, which coincides with the attack in the US capitol by QAnon supporters. \nThis initial analysis indicates that the QAnon conspiracy theory is growing rapidly on Telegram in terms of the number of groups\/channels, the number of messages, and the number of users sharing messages.\n\nNext, we analyze the languages of QAnon content on Telegram.\nFig.~\\ref{fig:top-languages} shows the percentage of groups\/channels that have most of their messages in German, English, Spanish, Italian, and Portuguese (top five languages in our dataset).\nInterestingly, we observe that the most popular language is German, with 58\\% of the groups\/channel sharing messages mainly in German.\nThe second most popular language is English with 24\\% of the groups\/channels, followed by Spanish with 7\\%, Italian with 4\\%, and Portuguese with 2\\%.\nOverall, these results highlight the multilingual aspect of the spread of conspiracy theories on platforms with a worldwide user base like Telegram.\n\nNext, we look into how the popularity of these languages changed over time to understand how the QAnon conspiracy theory became a global phenomenon on Telegram.\nFig.~\\ref{fig:languages_over_time} shows the evolution of the languages over time on QAnon-related groups\/channels (for readability purposes, we limit this analysis after September 2019).\nSpecifically, Fig.~\\ref{fig:languages_over_time_abs} shows the absolute number of messages in each language per week, while Fig.~\\ref{fig:languages_over_time_stack} shows the percentage of messages in each language weekly. \nWe observe that English was the most popular language between September 2019 and December 2019, with over half of the QAnon-related messages posted in English, with German having a substantial percentage (around 40\\% of all weekly QAnon related messages).\nThen, between February 2020 and April 2020, we observe a substantial increase in the popularity of the Portuguese language, which became the most popular language for this period, overshadowing both English and German.\nThis period coincides with the beginning of the COVID-19 pandemic in Brazil when the virus was first confirmed to have spread to Brazil on February 2020~\\cite{brazilian_1stcovid}.\nFinally, after June 2020, we find that German is consistently the most popular language in our dataset, followed by English and Spanish\/Portuguese having stable popularity.\n\n\n\n\\begin{figure}[tp]\n\t\\centering\n\t\\includegraphics[width=0.7\\columnwidth]{top-languages}\n\t\\caption{Percentage of groups\/channels for each of the top languages (based on the language with the largest number of messages in each group\/channel).} \n\\label{fig:top-languages}\t\n\\end{figure} \n\n\n\\begin{figure}[t!]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.99\\linewidth]{figs\/language_week_2019.pdf}\\label{fig:languages_over_time_abs}}\n\\subfigure[ ]{\\includegraphics[width=0.99\\linewidth]{figs\/language_percentage_stack_2019.pdf}\\label{fig:languages_over_time_stack}}\n\\vspace{-4mm}\n\\caption{Distribution of messages across languages on a weekly basis: (a) shows the number of messages for each language; and (b) the percentage of each language over the total number of messages.}\n\\label{fig:languages_over_time}\n\\vspace{-4mm}\n\\end{figure}\n\n\n\n\\subsection{Toxicity in QAnon Groups\/Channels}\n\n\n\nHere, we investigate the toxicity of content shared within QAnon-related groups\/channels on Telegram. \nThe QAnon movement has links with events of real-world violence, hence it is important to analyze the toxicity of QAnon discussions on messaging platforms like Telegram.\nWe aim to uncover whether QAnon discussions on Telegram are more toxic than other discussions or QAnon discussions on other platforms and how toxicity changes over time in Telegram (i.e., are QAnon-related discussions becoming more toxic over time). \nTo do this, as mentioned in Methods Section, we leverage Google's Perspective API that provides a signal (i.e., a score) of how toxic a message is.\n\nFirst, we look into how toxic are QAnon discussions on Telegram by comparing it with our baseline dataset and the Voat dataset obtained from~\\cite{papasavva2020qoincidence}.\nFig.~\\ref{fig:cdf_sever_toxicity} shows the Cumulative Distribution Function (CDF) of the toxicity scores obtained from the SEVERE\\_TOXICITY model in Perspective API.\nOverall, we observe that QAnon discussions on Telegram tend to be more toxic compared to our baseline Telegram dataset (median score is {0.07} for QAnon discussions and {0.03} for baseline dataset). \nThis finding is in contrast with the findings from~\\cite{papasavva2020qoincidence} that found that QAnon discussions on Voat were less toxic compared to other popular content on Voat.\nThis difference in our findings and the findings from~\\cite{papasavva2020qoincidence} is likely due to the fundamental differences that exist between the two platforms, while Voat is a fringe Web community mainly discussing conspiracy theories, Telegram is a more general-purpose and mainstream platform that includes less toxic non-QAnon-related discussions.\nBy comparing QAnon discussions across platforms (i.e., Telegram and Voat), we find comparable toxicity levels with QAnon discussions on Voat being very similar compared to Telegram (median toxicity score of {0.06} for Voat and {0.07} for Telegram).\nTo assess whether the distributions have statistically significant differences, we run a two-sample Kolmogorov-Smirnov test, finding that the distributions between Voat and Telegram are statistically significant ($p<0.01$).\n\n\n\\begin{figure}[t!]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.7\\linewidth]{figs\/cdf_comparison_SEVERE_TOXICITY}\\label{fig:cdf_sever_toxicity}}\n\\subfigure[]{\\includegraphics[width=0.7\\columnwidth]{figs\/cdf_languages_SEVERE_TOXICITY.pdf}\\label{fig:cdf_language_toxicity}}\n\\vspace{-4mm}\n\\caption{CDF of the toxicity scores obtained from Google's Perspective API. We compare toxicity scores in various datasets and across languages.}\n\\label{fig:toxicity}\n\\vspace{-4mm}\n\\end{figure}\n\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=\\columnwidth]{figs\/toxicity_hate_perc_day_2019SEVERE_TOXICITY.pdf}\\label{fig:toxicity_over_time_all}}\n\t\\subfigure[]{\\includegraphics[width=\\columnwidth]{figs\/toxicity_language_week_2019SEVERE_TOXICITY.pdf}\\label{fig:toxicity_over_time_language}}\n\t\\vspace{-4mm}\n\t\\caption{Percentage of messages that are toxic per week for: (a) all messages; and (b) for each of the top five languages in our dataset (i.e., percentage of messages that are toxic in specific language per week). } \n\t\\label{fig:toxicity-per-week}\n\t\\vspace{-4mm}\n\\end{figure} \n\n\nPreviously, we observed that the QAnon movement has different popularity across languages over time, hence we aim to quantify whether toxicity in QAnon discussion varies across languages.\nFig.~\\ref{fig:cdf_language_toxicity} shows the CDF of the SEVERE\\_TOXICITY scores for all messages in the five most popular languages in our dataset.\nOverall, we find that languages like German, Spanish, and Portuguese tend to be more toxic in QAnon discussions on Telegram compared to English and Italian.\nSpecifically, we find a median toxicity score of {0.09}, {0.07}, {0.14} for \nGerman, Spanish, and Portuguese, respectively, while for English and Italian, we find a median score of {0.03}.\nLooking into the proportion of messages that are toxic (i.e., SEVERE\\_TOXICITY score $>=0.8$, see Methods Section), we find {2.44\\%}, {0.54\\%}, {1.29\\%}, {3.57\\%}, {0.26\\%} toxic posts, \nfor German, English, Spanish, Portuguese, and Italian, respectively. \nTaken altogether, these results show substantial differences, in terms of toxicity, in QAnon discussions across various languages.\nAlso, by combining the findings from the previous section, our results indicate that the QAnon movement has become more popular in other languages over time (e.g., German), and at the same time, the discussion in these languages is more toxic.\nThese findings prompt the need to study such emerging issues by considering the multilingual aspect that exists in the spread of conspiracy theories that become a global phenomenon.\n\nFinally, we look into how the toxicity of discussion changes over time.\nFig.~\\ref{fig:toxicity_over_time_all} shows the weekly percentage of messages that are toxic for all the messages, while Fig.~\\ref{fig:toxicity_over_time_language} shows the weekly percentage of toxic messages in a specific language for the five most popular languages in our dataset.\nOur results show an overall increase in toxicity over time (see Fig.~\\ref{fig:toxicity_over_time_all}).\nSpecifically, between September 2019 and December 2019, approximately 1\\% of all the messages were toxic (per week). In contrast, by the end of our dataset, the same percentage rises to over 2\\% of all the messages.\nBy looking into the toxic posts broken down to specific languages (see Fig.~\\ref{fig:toxicity_over_time_language}), we observe an overall increase of toxicity by the end of our dataset, especially for German and English. \nFor Portuguese, we observe a substantial increase in toxicity during early 2020, which coincides with the rapid increase in the overall number of messages posted in Portuguese (see Fig.~\\ref{fig:languages_over_time_stack}).\nBy looking at the messages, we can observe those toxic messages are related to the COVID-19 pandemic in Brazil, including anti-vaccine conspiracies.\nAlso, we find politics-related messages that are attacking two Brazilian ex-ministers that left the government during this period.\n\n\\subsection{Popularity and Message Dissemination Across QAnon Groups\/Channels}\n\nThis section discusses message forwarding across QAnon groups\/channels and how popular is QAnon content on Telegram. \nAlso, we explore how QAnon structures its network within Telegram groups\/channels.\n\n\\begin{figure}[tp]\n\t\\centering\n\t\\includegraphics[width=0.7\\columnwidth]{figs\/cdf_forwarded.pdf}\n\t\\caption{CDF of the percentage of messages that are forwarded per group\/channel.} \n\\label{fig:cdf_outter_source}\t\n\\end{figure} \n\n\n\n\n\n\nFirst, we evaluate the percentage of messages from each group\/channel that are forwarded messages; i.e., messages that originate from another group\/channel in Telegram. \nFig.~\\ref{fig:cdf_outter_source} shows the CDF of the percentage of forwarded messages per group\/channel.\nWe observe that QAnon groups\/channels tend to have more messages forwarded.\nIn 24\\% of QAnon groups\/channels, more than half of their messages are forwarded messages, while just 5\\% of the groups\/channels in the baseline have the same percentage of forwarded content.\nAlso, 48\\% of the baseline groups\/channels are rarely have forwarded messages (i.e., less than 1\\% of all messages), while for QAnon groups\/channels, only 8\\% of them have similar behavior.\n\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{forward-graph-white.png}\n\t\\vspace{-4mm}\n\t\\caption{Forwarding graph representation of the QAnon groups\/channels. Note that we omit the labels on small nodes for visualization purposes. } \n\t\\vspace{-4mm}\n\\label{fig:graph}\t\n\\end{figure}\n\nTo better understand how messages are shared between QAnon groups\/channels we visualize message forwarding graphs.\nFig.~\\ref{fig:graph} shows a forwarding graph, where each node is a group\/channel and an edge exists between them if they forwarded messages from one to another.\nThe size of the nodes is based on the number of messages that a node forwarded to other groups\/channels and we use a visualization algorithm that takes into account the edge weights (in this case number of forwards), hence nodes with larger number of forwards are laid out closer in the graph space.\nWe also perform community detection using the Louvain method and color the nodes accordingly.\nWe observe that the graph consists of one huge connected component (90\\% of the nodes), indicating that the QAnon community is a tightly-knit community that disseminates messages across multiple groups\/channels.\nLooking at the nodes with the biggest size, we observe groups\/channels like ``Qlobal-Change'' that disseminates a large number of messages to other groups and in particular the nodes in the orange community (e.g., ``Trump Comes Back,'' ``QANON Memes Trump's Army,'' etc.)\nAnother community of interest is the pink one, which is seemingly an international set of groups with the groups ``Veikko Stoltzer aka the Son of Q'' and ``ATTILA HILDMANN'' forwarding a large number of messages to other international groups.\nAlso, the blue community at the bottom consists of groups\/channel dedicated in QAnon research with groups\/channels like ``1Research7Intel'' and ``INTO THE LIGHT NETWORK'' disseminating a lot of messages to other QAnon groups\/channels.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.85\\columnwidth]{languages_graph.pdf}\n\t\\caption{Forwarding graph with groups\/channels aggregated based on their most popular language. The direction of the edge is based on the color of the source node and the percentage of messages is based on the overall number of forwarded messages from the source groups\/channels.} \n\t\\label{fig:graph-langs}\t\n\\end{figure}\n\n\nWe also visualize the forwarding graph by aggregating groups\/channels based on their most popular language (Fig.~\\ref{fig:graph-langs}), which allows us to quantify the interaction of QAnon groups based on their language.\nWe find that for all languages except English, the majority of forwarded messages (over 94\\%) are within groups of the same language.\nInterestingly for English groups\/channels, 69\\% of the forwarded messages are towards primarily German-speaking groups\/channels.\nOther non-negligible interactions between languages are Spanish to Portuguese (3.8\\%), English to Spanish (3.6\\%), and Portuguese to German (3.4\\%).\nOverall, these findings highlight interaction across languages within the QAnon community on Telegram, with the stronger connection being from English to German groups\/channels.\n\n\n\n\nNext, we analyze the total number of views and forwards for the messages in our dataset, as they indicate how popular the QAnon content is across the \\emph{entire} Telegram network (not limited to the set of groups\/channels we collected).\nNote that the Telegram API does not provide metrics about views and forwards for all messages; the metrics are available only for messages posted on channels, hence, here we report results for 52.31\\% of the messages from our QAnon dataset and 38.61\\% of the baseline messages.\nLooking at Fig.~\\ref{fig:cdf_popularity}, we observe that messages in QAnon groups\/channels have a substantially larger number of views and forwards compared to the baseline dataset.\nSpecifically, we find a median of 1,986 views and 6 forwards for QAnon messages, while for the baseline dataset, we find a median of 236 views and 0 forwards.\nThese results suggest that QAnon content on Telegram is more popular than the baseline dataset as it has at least an order of magnitude more number of views and forwards across the entire Telegram network.\nWe also look into how the popularity of content changes over time; Fig.~\\ref{fig:views_fwd_day} shows the mean number of views and forwards of the messages over time. \nWe find a quick increase in both views and forwards for messages from the QAnon dataset in early April 2020 and still increasing in 2021, ending the period with a weekly mean of more than 100 forwards per message and more than 10,000 views per message (compared to the initial 10 forwards and 1,000 views per message in the 09\/2019). \nThis shows the increasing popularity of QAnon content over time and even surpassing the baseline dataset in one order of magnitude for both views and forwards at the beginning of 2021.\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{figs\/cdf_views_forwards.pdf}\n\\caption{CDF of the number of views and forwards (only for messages posted in channels).}\n\\label{fig:cdf_popularity}\n\\vspace{-4mm}\n\\end{figure}\n\n\n\\begin{figure}[tp]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{figs\/means_per_week_2019.pdf}\n\t\\caption{Mean number of views and forwards per week (only for messages posted in channels). } \n\t\\label{fig:views_fwd_day}\t\n\t\\vspace{-4mm}\n\\end{figure} \n\n\n\\subsection{QAnon Discourse Across Languages}\n\n\n\n\n\n\n\n\\begin{table*}[]\n\\centering\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{@{}llrr@{}}\n\\toprule\n\\textbf{Topic} & \\textbf{Example Terms} & \\textbf{\\#Messages} & \\textbf{\\begin{tabular}[c]{@{}r@{}}\\%Toxic \\\\ Messages\\end{tabular}} \\\\ \\midrule\nCOVID-19\/Vaccines & vacina,vaccine,impfen,impfungen,geimpft,vacinas,impfstoff,mandatory,vaccinations,nebenwirkungen & 63,508 & 1.98\\% \\\\\nTrump\/US elections & bless,gewinnt,gewonnen,win,won,supporters,trust,gewinnen,trump,siegen,presidents & 44,567 & 1.11\\% \\\\\nBrazilian Politics & brasileiro,paulo,rio,brasil,brasileira,jair,bolsonaro,governador,ministro,doria, & 40,622 & 3.50\\% \\\\\nGerman Politics\/Hitler & hitler,nazi,nazis,adolf,hitlers,alois,tochter,nazista,ss,nazareth,rothschild & 27,428 & 4.99\\% \\\\\nMoney\/Crypto & money,bitcoin,dinheiro,banken,geld,bank,schulden,gamestop,steuern,aktien & 22,973 & 1.95\\% \\\\\nDeplatforming & instagram,account,gesperrt,suspended,deleted,accounts,banned,whatsapp,fb,facebook & 18,700 & 1.01\\% \\\\\nReligion & jesus,cristo,bibel,senhor,deus,palavra,gottes,christus,bible,religion & 18,173 & 1.52\\% \\\\\nQDrops\/Promoting QAnon & qs,qanons,qdrops,qmap,anon,clock,anons,posts,buchstabe,qmor,qq,missle,qposts,qr,publicaes & 18,047 & 0.51\\% \\\\\nShootings\/Guns & suspect,shooting,shot,woman,charged,murder,police,cops,fatally,killed & 10,219 & 1.58\\% \\\\\nItalian Politics & italien,italia,italy,conte,italienische,italienischen,italian,leonardo,salvini,italiano & 8,784 & 1.19\\% \\\\\nJews & jews,jewish,juden,israel,judeus,rabino,judaism,zionisten,zionist,holocaust & 6,054 & 3.74\\% \\\\\nSatanists & satanisten,satan,satanic,satans,satanische,satanist,devil,satanismus,teufel,demnio,satanischen,diabo & 4,688 & 9.94\\% \\\\\nPlandemic & kits,positivo,plandemic,vacinada,test,hoax,cov,bito,protocola,covid1984lgen & 4,148 & 0.67\\% \\\\\nPizzagate & pizza,pizzagate,anrufer,sir,gate,podesta,caller,comet,google,conspiracy & 3,974 & 2.39\\% \\\\\nObamagate & obamagate,michelle,obama,barack,2013,obamacare,hussein,prisoner,michae,geburtsurkunde & 1,688 & 1.89\\% \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\vspace{-2mm}\n\\caption{Main topics extracted from our multilingual BERT topic modeling. We report the main theme of the topic, example terms that describe the topic, the number of messages that are mapped to each topic, and the percentage of those messages that are toxic.}\n\\label{tab:main-topics}\n\\vspace{-4mm}\n\\end{table*}\n\n\n\n\n\n\n\nThus far, we have looked into the evolution, toxicity, and popularity of QAnon content on Telegram, without analyzing the content itself. \nTherefore, here, we analyze the content of the messages shared within QAnon groups\/channels with a focus on understanding the main topics of discussions and identifying differences across languages. \nTo do this, as described in Methods Section, we employ a multilingual BERT-based topic modeling approach that enables us to generate topics across languages.\n\nFirst, we preprocess all messages by removing emojis and URLs from the text and filtering out messages with an empty body (i.e., messages sharing only URLs, emojis, videos, or images).\nAfter our preprocessing step, we end up with a set of 3M messages, which is the input for our topic modeling approach.\nThen, using the topic modeling approach described in Methods Section, we extract a set of 2,023 topics. \nNote that 58\\% of the messages are not mapped to any topic and are considered noise by the clustering algorithm.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\\includegraphics[width=\\linewidth]{figs\/topics_languages.pdf}\\label{fig:topics_languages}}\n\\subfigure[]{\\includegraphics[width=\\linewidth]{figs\/topics_languages2.pdf}\\label{fig:topics_languages2}}\n\\vspace{-4mm}\n\\caption{Distribution of topic-specific messages across languages.}\n\\label{fig:topics_languages_all}\n\\vspace{-4mm}\n\\end{figure}\n\nOut of the 2K topics, we notice that many of them are related to the same high-level topic.\nTherefore, we qualitatively analyze the top 100 topics in terms of frequency of messages to extract high-level topics.\nThen, based on the keywords related to the high-level topics, we group together topics that are part of the same high-level topic.\nTable~\\ref{tab:main-topics} reports the extracted high-level topics from our analysis along with example terms (across languages), the number of messages that are mapped to each topic, and the percentage of those messages that are toxic, whereas Fig.~\\ref{fig:topics_languages_all} shows the popularity of each topic across languages.\n\nWe make several interesting observations.\nFirst, we find that the most popular high-level topic is related to the COVID-19 pandemic and discussions about the vaccines (63K messages).\nLooking at the terms of this topic, we find the word ``nebenwirkungen'' (German) that translates to ``side effects'', hence Telegram users on QAnon groups were discussing possible side-effects from vaccines.\nOverall, this topic is primarily popular among German groups and to a lesser extent to English and Portuguese ones (see Fig.~\\ref{fig:topics_languages}).\nSecond, we find several topics about politics and real-world events, like the 2020 US elections, and German, Italian, and Portuguese politics.\nThis indicates that supporters of the QAnon conspiracy theory discuss political issues across multiple countries.\nFor German politics, we find several topics related to the Alt-Right and, in particular, discussions about Nazis and Hitler.\nFor all political topics, except Italian politics, their popularity across languages is based on the country of interest (see Fig.~\\ref{fig:topics_languages_all}).\n\nOther topics are discussions about money, cryptocurrency, and stocks (22K messages), deplatforming (18K messages), and religion (18k).\nWe also find specific topics that discuss Q drops and promoting QAnon content (18K), as well as discussions about fringe topics like various other conspiracy theories like the Plandemic (4K), Pizzagate (3.9K), and Obamagate (1.6K).\nAlso, similarly to~\\cite{planck2020we}, we find discussions related to Satanism (4.6K messages), likely because QAnon supporters accuse the ``deep state'' as being Satanists.\nInterestingly, this topic has the largest percentage of toxic messages, with 9.9\\% of all messages mapped to this topic being toxic.\n\nAdditionally, we find a topic related to the Jewish community (6K messages), with 3.7\\% of the messages in this topic being toxic.\nLooking into the terms, we find the term ``holocaust,'' which indicates that QAnon supporters discuss the holocaust and the implications to the Jewish community.\nThe relatively high percentage of toxic posts on this topic likely indicates that QAnon supporters share antisemitic content.\n\n\n\\section{Conclusion}\n\nIn this work, we performed the first multilingual analysis of QAnon content on Telegram.\nWe joined 161 groups\/channels on Telegram and collected a total of 4.5M messages shared over almost 4 years.\nUsing Perspective API and multilingual topic modeling, we shed light into how the QAnon conspiracy theory evolved and became a global phenomenon through Telegram.\nAmong other things, we found that the QAnon movement on Telegram is extremely popular in Europe, with German overshadowing English during 2020 and 2021.\nAlso, we find that QAnon discussions in other languages like German and Portuguese tend to be more toxic compared to English.\nThis indicates that not only the QAnon movement is becoming popular on other languages, it also becomes more toxic, similar to how viruses are mutating and become more lethal.\nAdditionally, we find that QAnon content is more popular on Telegram compared to a baseline of politics-oriented groups\/channels, and that QAnon supporters discuss a lot of topics ranging from world politics, to COVID-19 vaccines, and various conspiracy theories. \nTaken altogether, our analysis highlights that the QAnon movement is adapting and becoming a political strategy that is embodied in far-right movements across the globe.\nTo conclude, our work highlights the need for the creation of organizations that aim to fact check and tackle the spread of QAnon-related false information across languages and countries (e.g., efforts similar to the \\#CoronaVirusFacts Alliance focusing on the COVID-19 pandemic~\\cite{covid_poynter}).\n\n\n\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\'{E}tale maps between Quot schemes}\\label{quotmess}\nLet $\\varphi\\colon X\\rightarrow X'$ be a morphism of varieties, with $X'$ proper. Let $F'$ be a coherent sheaf on $X'$, set $F = \\varphi^\\ast F'$ and\n\\be\\label{quots123}\nQ = \\Quot_X(F,n),\\qquad Q' = \\Quot_{X'}(F',n).\n\\end{equation}\nIf we have a scheme $S$, we denote by $F_S$ the pullback of $F$ along the projection $X\\times S\\rightarrow X$. For instance, we let\n\\[\nF_Q\\twoheadrightarrow \\mathscr T\n\\]\ndenote the universal quotient, living over $X\\times Q$.\n\n\\begin{lemma}\\label{lemma:surj726}\nLet $\\varphi\\colon X\\rightarrow X'$ be an \\'etale map of quasi-projective varieties, $F'$ a coherent sheaf on $X'$ and let $F = \\varphi^\\ast F'$. Let $[\\theta\\colon F\\twoheadrightarrow T]\\in Q$ be a point such that $\\varphi$ is injective on $\\Supp T$. Then\n\\be\\label{surj837}\nF'\\rightarrow \\varphi_\\ast\\varphi^\\ast F'\\rightarrow \\varphi_\\ast T\n\\end{equation}\nstays surjective.\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nFirst of all, to check surjectivity of \\eqref{surj837}, we may replace $X'$ by any open neighborhood of the support of $\\varphi_\\ast T$, for example the image of $\\varphi$ itself. So now $\\varphi$ is faithfully flat, hence $\\varphi^\\ast$ is a faithful functor, in particular it reflects epimorphisms. It is easy to see that we may also replace $X$ with any open neighborhood $V$ of the support $B = \\Supp T$.\n\\begin{comment}\n\\small{\n\\textcolor{red}{Let us check that we may replace $X$ with any open neighborhood $V$ of the support $B = \\Supp T$. So let $V$ be such an open neighborhood. Denote by $\\ell\\colon B\\rightarrow V$ the closed immersion and by $i\\colon V\\rightarrow X$ the open immersion. Observe that the sheaf $i^\\ast T$ is supported on $B$, so we have $\\ell_\\ast\\ell^\\ast (i^\\ast T) = i^\\ast T$ canonically. Now let us assume we have proved the statement when $V$ is taken in place of $X$. First of all this means that we view $[\\theta]$ as the point $[i^\\ast F\\twoheadrightarrow i^\\ast T]$ of the Quot scheme $\\Quot_{V}(i^\\ast F,n)$, which is open in the initial one. Let $\\overline \\varphi\\colon V \\rightarrow X'$ be the composition $\\varphi\\circ i$. Then what we know is that\n\\[\nF'\\rightarrow \\overline \\varphi_\\ast \\overline \\varphi^\\ast F'\\rightarrow \\overline \\varphi_\\ast i^\\ast T\n\\]\nis onto.\nHowever, this can be rewritten as the top row of the diagram}\n\\[\n\\begin{tikzcd}\nF'\\arrow[bend left,two heads]{rr}\\arrow{r}\\arrow[equal]{d} & \\varphi_\\ast(i_\\ast i^\\ast F)\\arrow{r} & \\varphi_\\ast(i_\\ast i^\\ast T) \\\\\nF'\\arrow{r} & \\varphi_\\ast F\\arrow{r}\\arrow{u} & \\varphi_\\ast T\\isoarrow{u}\n\\end{tikzcd}\n\\]\n\\textcolor{red}{where we know that the rightmost vertical map is an iso (reason: we know $\\ell_\\ast\\ell^\\ast (i^\\ast T) = i^\\ast T$, and if $j\\colon B\\rightarrow X$ is the original closed immersion, we have $T = j_\\ast j^\\ast T = i_\\ast \\ell_\\ast \\ell^\\ast (i^\\ast T) = i_\\ast i^\\ast T$). Because the upper row is surjective, this proves the map we are interested in (the bottom row) is onto, so we can do the reduction.\n\\end{comment}\nSince $\\varphi$ is \\'etale, it is in particular quasi-finite and unramified: the fibres $X_p$, for $p\\in X'$, are finite, reduced of the same length. Then\n\\[\nA=\\coprod_{b\\in B}X_{\\varphi(b)}\\setminus \\Set{b}\\subset X\n\\]\nis a closed subset, and we can consider the open neighborhood\n\\[\nB\\subset V = X\\setminus A\\subset X\n\\]\nof the support of $T$. Since $\\varphi$ is injective on $B$, and we have just removed the points in $\\varphi^{-1}(\\varphi(B))$ that are not in $B$, the map $\\varphi$ is now an immersion around $B$, so after replacing $X$ by $V$ we observe that the canonical map\n\\be\\label{can191}\n\\varphi^\\ast\\varphi_\\ast T\\rightarrow T\n\\end{equation}\nis an isomorphism.\nLet us pullback \\eqref{surj837} along $\\varphi$, to get\n\\be\\label{pullbackl173}\n\\rho\\colon F\\rightarrow \\varphi^\\ast\\varphi_\\ast T.\n\\end{equation}\nIf we compose $\\rho$ with \\eqref{can191} we get back $\\theta\\colon F\\twoheadrightarrow T$, the original surjection. But \\eqref{can191} is an isomorphism, thus $\\rho$ is surjective. Since $\\varphi^\\ast$ reflects epimorphisms, \\eqref{surj837} is also surjective, as claimed.\n\\end{proof}\n\n\\begin{prop}\\label{prop:comparison}\nLet $\\varphi\\colon X\\rightarrow X'$ be a morphism of varieties, with $X'$ proper. Let $Q$ and $Q'$ be Quot schemes as in \\eqref{quots123}. \nFix a point $\\theta = [F\\twoheadrightarrow T]\\in Q$ such that $\\varphi$ is \\'etale around $B = \\Supp T$ and $\\varphi|_{B}$ is injective. Then there is an open neighborhood $\\theta\\in U\\subset Q$ admitting an \\'etale map $\\Phi\\colon U\\rightarrow Q'$.\n\\end{prop}\n\n\n\\begin{proof}\nFirst of all, we may replace $X$ by any open neighborhood $V$ of $B$. We may choose $V$ affine, so we may assume $\\varphi$ is affine and \\'etale. In the diagram\n\\[\n\\begin{tikzcd}\nX\\times \\theta \\MySymb{dr} \\arrow[hook]{r}{i} \\arrow[swap]{d}{\\varphi} \n& X\\times Q \\MySymb{dr} \\arrow{r}{p}\\arrow{d}{\\widetilde\\varphi} & X\\arrow{d}{\\varphi}\\\\\nX'\\times \\theta \\arrow[swap,hook]{r}{j} \n& X'\\times Q\\arrow[swap]{r}{\\rho} & X'\n\\end{tikzcd}\n\\]\nthe map $\\widetilde\\varphi$ is now affine, so $j^\\ast \\widetilde\\varphi_\\ast\\,\\widetilde{\\rightarrow}\\,\\varphi_\\ast i^\\ast$, and similarly we have $\\rho^\\ast \\varphi_\\ast\\,\\widetilde{\\rightarrow}\\,\\widetilde\\varphi_\\ast p^\\ast$ by flat base change. Let us look at the canonical map\n\\[\n\\alpha\\colon \\rho^\\ast F'\\rightarrow \\rho^\\ast \\varphi_\\ast F\\,\\widetilde{\\rightarrow}\\, \\widetilde\\varphi_\\ast F_Q\\twoheadrightarrow \\widetilde\\varphi_\\ast \\mathscr T.\n\\]\nWe know by Lemma \\ref{lemma:surj726} that restricting $\\alpha$ to $\\theta\\in Q$ (that is, applying $j^\\ast$) we get a surjection $F'\\rightarrow \\varphi_\\ast F\\twoheadrightarrow \\varphi_\\ast T$. Since $\\varphi$ is \\'etale and $\\varphi|_B$ is injective, this gives a well-defined point\n\\[\n\\varphi_\\ast\\theta = [F'\\twoheadrightarrow \\varphi_\\ast T] \\in Q'.\n\\]\nNow we extend the association $\\theta\\mapsto \\varphi_\\ast\\theta$ to a morphism $\\Phi\\colon U\\rightarrow Q'$ for suitable $U\\subset Q$.\nNote that $\\widetilde\\varphi_\\ast \\mathscr T$, the target of $\\alpha$, is coherent (reason: $\\Supp \\mathscr T\\rightarrow Q$ proper and factors through the separated projection $X'\\times Q\\rightarrow Q$, so that $\\widetilde\\varphi\\colon \\Supp \\mathscr T\\rightarrow X'\\times Q$ is proper; but $\\mathscr T$ is the pushforward of a coherent sheaf on its support). Then the cokernel $\\mathscr K$ of $\\alpha$ is also coherent, so that $\\Supp \\mathscr K\\subset X'\\times Q$ is closed. Since $X'$ is proper, we have that the projection $\\pi\\colon X'\\times Q\\rightarrow Q$ is closed, so the image of the support of $\\mathscr K$ is closed. Let \n\\[\nU = Q\\setminus \\pi(\\Supp \\mathscr K)\\subset Q\n\\]\nbe the open complement (non-empty because $\\theta$ belongs there by assumption).\nNow consider the cartesian square\n\\[\n\\begin{tikzcd}\nX\\times U \\MySymb{dr} \\arrow[hook]{r}{} \\arrow[swap]{d}{\\varphi_U} \n& X\\times Q\\arrow{d}{\\widetilde\\varphi} \\\\\nX'\\times U \\arrow[swap,hook]{r}{} \n& X'\\times Q\n\\end{tikzcd}\n\\]\nand observe that by construction, $\\alpha$ restricts to a surjection\n\\[\n\\alpha\\big|_{X'\\times U}\\colon \\rho_U^\\ast F' \\twoheadrightarrow \\varphi_{U\\ast}\\mathscr T\\big|_{X\\times U}\n\\]\nwhere $\\rho_U$ is the projection $X'\\times U\\rightarrow X'$. But the target $\\varphi_{U\\ast}\\mathscr T\\big|_{X\\times U}$ is flat over $U$ (reason: $\\widetilde\\varphi_\\ast\\mathscr T$ is flat over $Q$ because $\\mathscr T$ is; but $\\varphi_{U\\ast}\\mathscr T\\big|_{X\\times U}$ is isomorphic to the pullback of $\\widetilde\\varphi_\\ast\\mathscr T$ along the open immersion $X'\\times U\\rightarrow X'\\times Q$, therefore it is flat over $U$). We have constructed a morphism\n\\[\n\\Phi\\colon U\\rightarrow Q'.\n\\]\nNow we show it is \\'etale by using the infinitesimal criterion. First of all, as in the proof of Lemma \\ref{lemma:surj726}, we may shrink $X$ further in such a way that, for every $x\\in B$, the fibre $X_{\\varphi(x)}$ consists of the single point $x$. This implies that the canonical map $\\varphi^\\ast\\varphi_\\ast T\\rightarrow T$ is an isomorphism; moreover, this condition only depends on the set-theoretic support of $T$, so it is preserved in infinitesimal neighborhoods; in particular we have\n\\be\\label{fat}\n\\varphi^\\ast\\varphi_\\ast\\mathscr F\\,\\widetilde{\\rightarrow}\\,\\mathscr F\n\\end{equation}\nfor all infinitesimal deformations $F_S\\twoheadrightarrow \\mathscr F$ of $\\theta$ parametrized by a fat point $S$.\n\nLet $\\iota\\colon S\\rightarrow \\overline S$ be a square zero extension of fat points, and consider a commutative diagram\n\\[\n\\begin{tikzcd}\nS\\arrow[hook,swap]{d}{\\iota}\\arrow{r}{g} & U\\arrow{d}{\\Phi} \\\\\n\\overline S\\arrow[swap]{r}{h}\\arrow[dotted]{ur}[description]{v} & Q'\n\\end{tikzcd}\n\\]\nwhere we need to find a unique $v$ making the two triangles commutative. This will correspond to a family\n\\[\nF_{\\overline S}\\twoheadrightarrow \\mathscr V\n\\]\nthat we have to find.\nTo fix notation, consider the fibre diagram\n\\be\\label{fibre618}\n\\begin{tikzcd}\nX\\times S \\MySymb{dr} \\arrow[hook]{r}{\\iota_X}\\arrow[swap]{d}{\\varphi_S} & X\\times \\overline S\\arrow{d}{\\varphi_{\\overline S}} \\\\\nX'\\times S\\arrow[hook,swap]{r}{\\iota_{X'}} & X'\\times \\overline S\n\\end{tikzcd}\n\\end{equation}\nand denote by $F_S\\twoheadrightarrow \\mathscr G$ the family corresponding to $g$ (restricting to $\\theta$ over the closed point) and by\n$F'_{\\overline S}\\twoheadrightarrow \\mathscr H$ is the family corresponding to $h$. The condition $\\varphi\\circ g = h\\circ \\iota$ means that we have a diagram of sheaves\n\\be\\label{diag22232}\n\\begin{tikzcd}\n\\iota_{X'}^\\ast F'_{\\overline S} \\arrow[equal]{d} \\arrow[two heads]{r} & \\iota_{X'}^\\ast\\mathscr H\\isoarrow{d} \\\\\nF'_S \\arrow[two heads]{r} & \\varphi_{S\\ast}\\mathscr G\n\\end{tikzcd}\n\\end{equation}\nover $X'\\times S$. The conditions $\\varphi_{\\overline S\\ast}\\mathscr V=\\mathscr H$ (translating $\\varphi\\circ v = h$) and \n$\\varphi^\\ast_{\\overline S}\\varphi_{\\overline S\\ast}\\mathscr V = \\mathscr V$, coming from \\eqref{fat}, together determine for us the family\n\\[\n\\mathscr V = \\varphi_{\\overline S}^\\ast \\mathscr H,\n\\]\nwhich we consider together with the natural surjection $\\varphi^\\ast(h)\\colon F_S\\twoheadrightarrow \\mathscr V$. Note that \n\\begin{align*}\n\\iota_X^\\ast\\mathscr V \n&= \\iota_X^\\ast \\varphi_{\\overline S}^\\ast\\mathscr H & \\\\\n&= \\varphi_S^\\ast \\iota_{X'}^\\ast \\mathscr H & \\textrm{by }\\eqref{fibre618} \\\\\n&= \\varphi_S^\\ast \\varphi_{S\\ast}\\mathscr G & \\textrm{by } \\eqref{diag22232} \\\\\n&= \\mathscr G & \\textrm{by } \\eqref{fat}\n\\end{align*}\nproves that $v\\circ \\iota = g$, finishing the proof.\n\\end{proof}\n\n\nFinally, we extend the statement to $X'$ quasi-projective.\n\n\n\n\\begin{prop}\\label{prop:etaleV}\nLet $\\varphi\\colon X\\rightarrow X'$ be an \\'etale map of quasi-projective varieties, $F'$ a coherent sheaf on $X'$, and let $F = \\varphi^\\ast F'$. Let $V\\subset Q$ be the open subset whose points correspond to quotients $F\\twoheadrightarrow T$ such that $\\varphi|_{\\Supp T}$ is injective. Then there is an \\'etale morphism $\\Phi\\colon V\\rightarrow Q'$.\n\\end{prop}\n\n\n\\begin{proof}\nFirst complete $X'$ to a proper scheme $Y$. Let $i\\colon X'\\rightarrow Y$ be the open immersion and note that $i^\\ast i_\\ast F' = F'$ canonically. Combining Proposition \\ref{prop:comparison} and Lemma \\ref{lemma:surj726} we get an \\'etale map $\\Phi\\colon V\\rightarrow \\Quot_{Y}(i_\\ast F',n)$. But the support of any quotient sheaf $i_\\ast \\varphi_\\ast T$ lies in the open part $X'\\subset Y$, so $\\Phi$ actually factors through $\\Quot_{X'}(F',n) = Q'$.\n\\end{proof}\n\n\n\n\n\n\\section{Relative Quot schemes and PT pairs}\\label{UniversalEmbedding}\nLet $X$ be a Calabi--Yau $3$-fold and fix the Chern character $\\alpha = (r,D,-\\beta,-m)$ with $r \\geq 1$.\nLet $\\omega\\in H^2(X,\\mathbb Z)$ be an ample class on $X$ satisfying the usual coprimality condition\n\\be\\label{coprimeagain}\n\\gcd(r,D\\cdot \\omega^2) = 1.\n\\end{equation}\n\nIn this section, we upgrade the `fibrewise' closed immersion\n\\[\n\\psi_{\\F,n}\\colon \\Quot_X(\\lExt^1(\\F,\\mathscr O_X),n)\\hookrightarrow M_{\\PT}(r,D,\\ch_2(\\F),\\ch_3(\\F)-n)\n\\]\nof Proposition \\ref{PTembedding}, where $\\F$ is a DT and PT object of $\\ch(\\F) = \\alpha$, to a \\emph{universal} closed immersion $\\psi_{\\alpha,n}$ by allowing the sheaf $\\F$ to vary in the moduli space of DT and PT objects.\nThe domain of $\\psi_{\\alpha,n}$ is a certain relative Quot scheme.\nThe morphism $\\psi_{\\alpha,n}$ is not an isomorphism; indeed, it even fails to be surjective on $\\mathbb C$-valued points. However, the coproduct $\\psi_{r,D,\\beta} = \\coprod_{n \\in \\mathbb Z} \\psi_{\\alpha,n}$ (not depending on $\\ch_3$ anymore) is a geometric bijection onto\n\\be\\label{eq:MPT_rDbeta}\nM_{\\PT}(r,D,-\\beta) = \\coprod_{n \\in \\mathbb Z} M_{\\PT}(r,D,-\\beta,-m-n).\n\\end{equation}\nThis may be seen as a new stratification of $M_{\\PT}(r,D,-\\beta)$ by relative Quot schemes. \n\nWe first show that there exists a universal sheaf on $X\\times M_{\\DT}(\\alpha)$. \nAccording to \\cite[\\textsection~4.6]{modulisheaves}, quasi-universal families exist on every moduli space of stable sheaves on a smooth projective variety.\nA universal family exists if the following numerical criterion is fulfilled.\n\\begin{theorem}[{\\cite[Thm.~4.6.5]{modulisheaves}}]\\label{thm:HLcriterion}\nLet $Z$ be a smooth projective variety, let $\\mathsf{c}$ be a class in the numerical Grothendieck group $N(Z)$, and let $\\set{B_1,\\ldots,B_\\ell}$ be a collection of coherent sheaves.\nIf\n\\[\n\\gcd(\\chi(\\mathsf{c} \\otimes B_1), \\ldots, \\chi(\\mathsf{c} \\otimes B_\\ell)) = 1,\n\\]\nthen there exists a universal family on $M(\\mathsf{c})^{s} \\times Z$.\n\\end{theorem}\n\nWe obtain the following corollary.\n\\begin{corollary}\\label{cor98274}\nThere exists a universal family $\\mathscr{F}_{\\DT}(\\alpha)$ on $M_{\\DT}(\\alpha) \\times X$.\n\\end{corollary}\nNote that $c_1 \\colon \\Pic(X) \\to H^2(X,\\mathbb Z)$ is an isomorphism since $H^i(X,\\mathscr O_X) = 0$ for $i=1,2$. We let $L$ be a line bundle represented by $\\omega\\in H^2(X,\\mathbb Z)$. Since $X$ is Calabi--Yau, its Todd class is\n\\[\n\\Td(X) = (1,0,c_2(X)\/12,0).\n\\]\n\n\\begin{proof}\nWe apply the above criterion.\nTake the sheaves $B_k = L^{\\otimes k}$ for $k = 0,1,2$ and $B_3 = \\mathscr O_x$, where $x\\in X$ is a point.\nVia the Hirzebruch--Riemann--Roch Theorem, and using the above expression for the Todd class of $X$, we compute\n\\begin{equation*}\n \\chi(\\alpha \\otimes L^{\\otimes k}) \n =\\frac{rk^3}{6}\\omega^3 + \\frac{k^2}{2}D\\cdot \\omega^2 + \\frac{rk}{12}c_2(X)\\cdot \\omega - k \\beta\\cdot \\omega + \\frac{1}{12}D\\cdot c_2(X) - m\n\\end{equation*}\nand $\\chi(\\alpha \\otimes \\mathscr O_x) = r$.\nWriting $P_{\\alpha}(k) = \\chi(\\alpha \\otimes L^{\\otimes k})$, it is not hard to see that\n\\[\nP_{\\alpha}(2) - 2P_{\\alpha}(1) + P_{\\alpha}(0) = r \\omega^3 + D \\cdot \\omega^2.\n\\]\nBy the coprimality assumption \\eqref{coprimeagain}, we find\n\\begin{equation*}\n\\begin{split}\n \\gcd(\\chi(\\alpha \\otimes \\mathscr O_x),P_{\\alpha}(2),P_{\\alpha}(1),P_{\\alpha}(0))\n &= \\gcd(r,P_{\\alpha}(2)-2P_{\\alpha}(1)+P_{\\alpha}(0),P_{\\alpha}(1),P_{\\alpha}(0)) \\\\\n &= \\gcd(r,r \\omega^3 + D \\cdot \\omega^2,P_{\\alpha}(1),P_{\\alpha}(0)) \\\\\n &=1.\n\\end{split}\n\\end{equation*}\nThe result now follows from Theorem \\ref{thm:HLcriterion}.\n\\end{proof}\n\n\nNext, we describe the domain of the morphism $\\psi_{\\alpha,n}$, which is a relative Quot scheme.\nInside the $\\alpha$-component of Lieblich's moduli stack $\\mathscr M_X(\\alpha)\\subset \\mathscr M_X$, consider the intersection\n\\[\n\\mathcal M(\\alpha)=\\mathcal M_{\\DT}(\\alpha)\\cap \\mathcal M_{\\PT}(\\alpha),\n\\]\nand let $M(\\alpha)$ be its coarse moduli space, consisting of $\\mu_\\omega$-stable sheaves of homological dimension at most one.\nLet \n\\[\n\\mathscr F_\\alpha = \\mathscr{F}_{\\DT}(\\alpha)\\big|_{X\\times M(\\alpha)}\n\\]\nbe the restriction of the universal sheaf constructed in Corollary \\ref{cor98274}; it is flat over $M(\\alpha)$.\nIf $\\iota_m\\colon X\\times\\set{m}\\hookrightarrow X\\times M(\\alpha)$ denotes the natural closed immersion, it follows that\n\\be\\label{Vanish321}\n\\lExt^2\\left(\\iota_m^\\ast\\mathscr F_\\alpha,\\mathscr O_{X}\\right) = 0\n\\end{equation}\nfor every $m\\in M(\\alpha)$.\nThen \\cite[Thm.~1.10]{PicScheme} implies that $\\lExt^2\\left(\\mathscr F_\\alpha,\\mathscr O_{X\\times M(\\alpha)}\\right) = 0$ (using that $\\mathscr F_\\alpha$ is $M(\\alpha)$-flat) and \\cite[Thm.~1.9]{PicScheme} implies that in the previous degree the base change map to the fibre is an isomorphism\n\\be\\label{BCiso}\n\\iota_m^\\ast\\lExt^1\\left(\\mathscr F_\\alpha,\\mathscr O_{X\\times M(\\alpha)}\\right)\\,\\widetilde{\\rightarrow}\\,\\lExt^1\\left(\\iota_m^\\ast\\mathscr F_\\alpha,\\mathscr O_{X}\\right).\n\\end{equation}\n\nWe consider the relative Quot scheme\n\\[\nQ(\\alpha,n) \\defeq \\Quot_{X\\times M(\\alpha)}\\left(\\lExt^1\\left(\\mathscr F_\\alpha,\\mathscr O_{X\\times M(\\alpha)}\\right),n\\right).\n\\]\nIt is a projective $M(\\alpha)$-scheme that represents the functor $(\\mathsf{Sch}\/M(\\alpha))^{\\mathrm{op}}\\rightarrow\\mathsf{Sets}$ which sends a morphism $f\\colon S\\rightarrow M(\\alpha)$ to the set of equivalence classes of surjections\n\\[\n\\pi_f^\\ast \\lExt^1\\left(\\mathscr F_\\alpha,\\mathscr O_{X\\times M(\\alpha)}\\right)\\twoheadrightarrow \\mathscr Q.\n\\]\nHere $\\mathscr Q$ is an $S$-flat family of length $n$ sheaves, $\\pi_f = \\id_X \\times f$ fits in the cartesian square\n\\[\n\\begin{tikzcd}\nX \\times S\\MySymb{dr} \\arrow{r}{\\pi_f}\\arrow[swap]{d}{\\pr_2} & X\\times M(\\alpha) \\arrow{d}{\\pr_2} \\\\\nS\\arrow[swap]{r}{f} & M(\\alpha)\n\\end{tikzcd}\n\\]\nand two surjections are equivalent if they have the same kernel.\n\nIf $m = f(s)$, and $i_s\\colon X\\times\\set{s}\\hookrightarrow X\\times S$ and $\\iota_m\\colon X\\times\\set{m}\\hookrightarrow X\\times M(\\alpha)$ denote the natural closed immersions, then we have a canonical identification\n\\be\\label{idFibers}\n\\begin{tikzcd}\nX\\times\\set{s}\\arrow[hook]{d}{i_s}\\arrow{r}{\\sim} & X\\times \\set{m}\\arrow[hook]{d}{\\iota_m} \\\\\nX\\times S\\arrow{r}{\\pi_f} & X\\times M(\\alpha)\n\\end{tikzcd}\n\\end{equation}\nwhich we rephrase as the condition\n\\be\\label{composition2874}\n\\iota_m = \\pi_f\\circ i_s. \n\\end{equation}\n\\begin{remark}\nSince $\\mathscr{F}_{\\alpha}\\in \\Coh(X \\times M(\\alpha))$ is flat over $M(\\alpha)$, we have that ${\\textbf L} \\iota_m^*\\mathscr{F}_{\\alpha}$ is a coherent sheaf on $X$ for all $m \\in M(\\alpha)$.\nIn particular, ${\\textbf L}^k \\iota_m^*\\mathscr{F}_{\\alpha} = 0$ for all $k > 0$, and so ${\\textbf L} \\iota_m^*\\mathscr{F}_{\\alpha} = \\iota_m^*\\mathscr{F}_{\\alpha}$.\nUsing \\eqref{composition2874}, we obtain\n\\[\n\\iota_m^*\\mathscr{F}_{\\alpha} = {\\textbf L}\\iota_m^*\\mathscr{F}_{\\alpha} \\cong {\\textbf L}i_s^* \\bigl({\\textbf L}\\pi_f^*\\mathscr{F}_{\\alpha}\\bigr).\n\\]\nBy \\cite[Lem.~4.3]{Bri2}, we conversely deduce that ${\\textbf L}\\pi_f^*\\mathscr F_{\\alpha}$ is a sheaf on $X \\times S$ flat over $S$.\nIn particular, $\\pi_f^*\\mathscr F_{\\alpha} = {\\textbf L} \\pi_f^*\\mathscr F_{\\alpha}$ since $\\mathscr F_{\\alpha}$ is a sheaf and hence ${\\textbf L} \\pi_f^*\\mathscr F_{\\alpha} \\in D^{\\leq 0}(X \\times S)$.\n\\end{remark}\n\n\n\\begin{remark}\\label{rem:family_version_constant_family}\nIf the morphism $f \\colon S \\to M(\\alpha)$ is constant, corresponding to a single sheaf $\\F$ parametrised by $S$ in a constant family, then $\\pi_f$ factors through the flat projection\n\\[\nX \\times S \\to X \\times \\{\\mathcal{F}\\} \\subset X \\times M(\\alpha).\n\\]\nWe are reduced to the situation of Proposition~\\ref{PTembedding} and $Q(\\alpha,n)(f) = \\Quot_X(\\lExt^1_X(\\F,\\mathscr O_X),n)(S)$. \n\\end{remark}\n\n\n\nWe now prove the following generalisation of Proposition \\ref{PTembedding}.\n\\begin{prop}\\label{prop:UnivPTimmersion}\nFor every $n\\geq 0$ there is a closed immersion\n\\[\n\\psi_{\\alpha,n}\\colon Q(\\alpha,n)\\hookrightarrow M_{\\PT}(r,D,-\\beta,-m-n).\n\\]\n\\end{prop}\n\n\n\\begin{proof}\nFix a morphism $f\\colon S\\rightarrow M(\\alpha)$ and a quotient \n\\[\nq\\colon \\pi_f^\\ast \\lExt^1\\left(\\mathscr F_\\alpha,\\mathscr O_{X\\times M(\\alpha)}\\right)\\twoheadrightarrow \\mathscr Q.\n\\]\nNote that the restriction of $q$ to the slice $X\\times \\set{s}$ is canonically identified with a surjection\n\\be\\label{map:qs}\nq_s\\colon \\lExt^1(\\iota_m^\\ast\\mathscr F_\\alpha,\\mathscr O_X)\\twoheadrightarrow \\mathscr Q_s\n\\end{equation}\nvia the identification \\eqref{idFibers} and the base change isomorphism \\eqref{BCiso}.\n\nWe claim that there exists a canonical morphism\n\\[\nu \\colon \\left(\\pi_f^*\\mathscr{F}_{\\alpha}\\right)^{\\vee} \\to \\pi_f^*\\lExt^1\\left(\\mathscr{F}_{\\alpha},\\mathscr O_{X \\times M(\\alpha)}\\right)[-1].\n\\]\nIndeed, it is the composition of the canonical morphisms\n\\[\n\\left(\\pi_f^*\\mathscr{F}_{\\alpha}\\right)^{\\vee} = \\bigl({\\textbf L} \\pi_f^*\\mathscr{F}_{\\alpha}\\bigr)^{\\vee} \\cong {\\textbf L} \\pi_f^* \\mathscr{F}_{\\alpha}^{\\vee} \\to \\pi_f^*\\lExt^1\\left(\\mathscr F_{\\alpha},\\mathscr O_{X \\times M(\\alpha)}\\right)[-1].\n\\]\nThe second isomorphism follows because the derived pullback commutes with the derived dualising functor.\nThe last morphism exists by the following argument.\n\nThe vanishing of \\eqref{Vanish321} implies $\\mathscr{F}_{\\alpha} \\in D^{[0,1]}(X \\times M(\\alpha))$.\nThere is a canonical triangle\n\\[\n{\\textbf L} \\pi_f^*H^0(\\mathscr{F}_{\\alpha}^{\\vee}) \\to {\\textbf L} \\pi_f^*\\mathscr{F}_{\\alpha}^{\\vee} \\to {\\textbf L} \\pi_f^*\\lExt^1\\left(\\mathscr{F}_{\\alpha},\\mathscr O_{X \\times M(\\alpha)}\\right)[-1]\n\\]\nsince ${\\textbf L} \\pi_f^*$ is an exact functor.\nTaking cohomology yields the isomorphism\n\\[\nH^1\\bigl({\\textbf L} \\pi_f^*\\mathscr{F}_{\\alpha}^{\\vee}\\bigr) \\cong \\pi_f^*\\lExt^1\\left(\\mathscr{F}_{\\alpha},\\mathscr O_{X \\times M(\\alpha)}\\right)\n\\]\nsince ${\\textbf L} \\pi_f^*$ is a left derived functor and so ${\\textbf L}\\pi_f^*\\F_{\\alpha} \\in D^{\\leq 1}(X \\times S)$.\nThis completes the argument.\n\n\n\n\nWe now define $\\psi_{\\alpha,n}$.\nAs before, we shift the morphism $q$ and precompose by $u$ to obtain\n\\[\n\\bar{q} = q[-1] \\circ u \\colon \\left(\\pi_f^* \\mathscr{F}_{\\alpha}\\right)^{\\vee} \\to \\mathscr{Q}[-1].\n\\]\nTaking cohomology of the canonical isomorphism $\\mathbf R\\lHom(E,F) \\cong \\mathbf R\\lHom(F^{\\vee},E^{\\vee})$ yields\n\\[\n\\Hom_{X \\times S}\\bigl((\\pi_f^* \\mathscr{F}_{\\alpha})^{\\vee}, \\mathscr{Q}[-1]\\bigr) \\cong \\Ext^1_{X \\times S}\\bigl(\\mathscr{Q}^{\\mathrm D}[-1],\\pi_f^* \\mathscr{F}_{\\alpha}\\bigr),\n\\]\nwhere we have appealed to Lemma \\ref{TopExt}, just as in the proof of Proposition \\ref{PTembedding}.\nWe identify $\\bar{q}$ with the corresponding extension under this identification, to obtain a triangle\n\\begin{equation}\\label{JSfamily}\n\\pi_f^*\\mathscr{F}_{\\alpha} \\to J^{\\bullet} \\to \\mathscr{Q}^{\\mathrm D}[-1]\n\\end{equation}\nof perfect complexes on $X \\times S$.\nWe claim that $J^{\\bullet}$ defines an $S$-family of PT pairs.\n\nClearly, the derived fibre $J^{\\bullet}_s = {\\textbf L} i_s^*J^{\\bullet}$ of each closed point $s \\in S$ defines a pre-PT pair based at the sheaf corresponding to $m = f(s)\\in M(\\alpha)$.\nTo see that $J^{\\bullet}_s$ is a PT pair, recall that for a perfect complex the operations of taking the derived fibre and taking the derived dual commute.\nBy taking the derived fibre of the triangle \\eqref{JSfamily}, we obtain the triangle\n\\[\n{\\textbf L}i_s^* \\pi_f^*\\mathscr{F}_{\\alpha} \\to J^{\\bullet}_s \\to \\mathscr{Q}^{\\mathrm D}_s[-1]\n\\]\nin $\\Perf(X)$.\nUsing the fact that ${\\textbf L}i_s^*\\pi_f^*\\mathscr F_{\\alpha} = \\iota_m^*\\mathscr F_{\\alpha}$ and applying the derived dualising functor ${\\textbf R}\\lHom_X(-,\\mathscr O_X)$, we obtain the triangle\n\\[\n\\mathscr{Q}_s[-2] \\to J_s^{\\bullet \\vee} \\to \\bigl(\\iota_m^*\\mathscr{F}_{\\alpha}\\bigr)^{\\vee}.\n\\]\nFurther taking cohomology yields the exact sequence\n\\[\n\\cdots \\to \\lExt^1_{X_s}\\bigl(\\iota_m^*\\mathscr{F}_{\\alpha},\\mathscr O_{X_s}\\bigr) \\xrightarrow{q_s} \\mathscr{Q}_s \\to \\lExt^2_{X_s}(J^{\\bullet}_s,\\mathscr O_{X_s}) \\to 0\n\\]\nwhere the last $0$ is $\\lExt^2(\\iota_m^\\ast\\mathscr F_\\alpha,\\mathscr O_{X})$ as in \\eqref{Vanish321} and $q_s$ is precisely the map \\eqref{map:qs} obtained by restricting the original surjection $q$. As such, it is surjective, therefore $\\lExt^2(J^{\\bullet}_s,\\mathscr O_{X_s}) = 0$, proving that $J^\\bullet$ defines a family of PT pairs by Lemma~\\ref{lem:PrePT_is_PT}. \nWe conclude that \n\\[\n\\psi_{\\alpha,n}\\colon q\\mapsto J^\\bullet\n\\]\ndefines a morphism. The properness of $\\psi_{\\alpha,n}$ follows from the valuative criterion, along with the fact that $\\psi_{\\alpha,n}$ restricted to a fibre $\\tau^{-1}(\\F)$ of the structure morphism $\\tau\\colon Q(\\alpha,n)\\rightarrow M(\\alpha)$ is precisely the closed immersion $\\psi_{\\F,n}$ of Proposition \\ref{PTembedding}; see Remark~\\ref{rem:family_version_constant_family}.\nFinally, the same argument used in Proposition \\ref{PTembedding} shows injectivity on all valued points, thus proving that $\\psi_{\\alpha,n}$ is a closed immersion as claimed.\n\\end{proof}\n\n\nSet $Q(r,D,-\\beta) = \\coprod_{n \\in \\mathbb Z} Q(\\alpha,n)$ and recall the morphism $\\psi_{r,D,\\beta}$ from equation~\\eqref{eq:MPT_rDbeta}.\n\\begin{corollary}\nThe morphism $\\psi_{r,D,\\beta} \\colon Q(r,D,-\\beta) \\to M_{\\PT}(r,D,-\\beta)$ is a geometric bijection.\n\\end{corollary}\nThis may be seen as a new stratification of $M_{\\PT}(r,D,-\\beta)$ by relative Quot schemes.\n\\begin{proof}\nLet $\\F \\to J^{\\bullet} \\to Q^{\\mathrm D}[-1]$ be a $\\mathbb C$-valued point of $M_{\\PT}(r,D,-\\beta)$.\nWrite $\\ch_3(\\F) = -m$ and $\\ell(Q^{\\mathrm D}) = n$, so that $\\ch_3(J^{\\bullet}) = -m-n$.\nSet $\\alpha = (r,D,-\\beta,-m)$ and let $f \\colon \\Spec(\\mathbb C) \\to M(\\alpha)$ be the morphism corresponding to $\\F$.\nApplying $\\lHom(-,\\mathscr O_X)$ to the triangle of $J^{\\bullet}$ yields a surjection $q \\colon \\lExt^1_X(\\F,\\mathscr O_X) \\twoheadrightarrow Q$, which is a $\\mathbb C$-valued point of $Q(\\alpha,n)$.\n\\end{proof}\n\n\n\n\n\\section{Introduction}\n\nLet $X$ be a smooth projective Calabi--Yau $3$-fold over $\\mathbb C$.\nDonaldson--Thomas (DT) invariants, introduced in \\cite{ThomasThesis}, are virtual counts of stable objects in the bounded derived category $D^b(X)$ of $X$.\nParticularly well-studied examples of such stable objects are \\emph{ideal sheaves} (rank one torsion free sheaves with trivial determinant) of curves in $X$ and the \\emph{stable pairs} of Pandharipande--Thomas (PT) \\cite{PT}.\nThese objects are related by wall-crossing phenomena (cf.~Section \\ref{sec:previouswork}), giving rise to the famous DT\/PT correspondence \\cite{Bri,Toda1,Toda0}.\n\nRecently Toda \\cite{Toda2} generalised the classical (rank one) DT\/PT correspondence to arbitrary rank.\nLet $\\omega$ be an ample class on $X$, and let $(r,D) \\in H^0(X) \\oplus H^2(X)$ be a pair such that $r\\geq 1$ and $\\gcd(r,D \\cdot \\omega^2)=1$.\nThe higher rank analogues of ideal sheaves are $\\omega$-slope stable torsion free sheaves of Chern character $\\alpha = (r,D,-\\beta,-m)$, for given $(\\beta,m) \\in H^4(X) \\oplus H^6(X)$.\nThe higher rank analogues of stable pairs are certain two-term complexes $J^\\bullet \\in D^b(X)$ of class $\\alpha$, first described by Lo (see Section \\ref{sec:Moduli}), called \\emph{PT pairs} \\cite{Lo}.\nThe virtual counts of these objects can be computed as Behrend's virtual Euler characteristic of their moduli space.\n\nToda's higher rank wall-crossing formula \\cite[Thm.~1.2]{Toda2} is the equality\n\\be\\label{Todaformula}\n\\mathsf{DT}_{r,D,\\beta}(q) = \\mathsf M((-1)^rq)^{r\\chi(X)}\\cdot \\mathsf{PT}_{r,D,\\beta}(q)\n\\end{equation}\nof generating series.\nHere $\\mathsf{DT}_{r,D,\\beta}$ is the generating function of DT invariants of Chern characters of the form $(r,D,-\\beta,-m)$, with $m \\in H^6(X)$ varying, and similarly for the series $\\mathsf{PT}_{r,D,\\beta}$. \nThe ``difference'' between the DT and PT generating functions is measured by a \\emph{wall-crossing factor}, expressed in terms of the MacMahon function \n\\[\n\\mathsf M(q) = \\prod_{m\\geq 1}(1-q^m)^{-m},\n\\]\nthe generating function of plane partitions of natural numbers.\n\nIn \\cite{Gholampour2017}, Gholampour--Kool proved a formally similar relation in the following situation.\nFix a torsion free sheaf $\\mathcal F$ of rank $r$ and homological dimension at most one on a smooth projective $3$-fold $X$, not necessarily Calabi--Yau.\nThen \\cite[Thm.~1.1]{Gholampour2017} states the equality\n\\be\\label{GKformula}\n\\sum_{n\\geq 0}\\chi(\\Quot_X(\\mathcal F,n))q^n = \\mathsf M(q)^{r\\chi(X)}\\cdot \\sum_{n\\geq 0}\\chi(\\Quot_X(\\lExt^1(\\mathcal F,\\mathscr O_X),n))q^n,\n\\end{equation}\nwhere $\\chi$ is the topological Euler characteristic and $\\Quot_X(E,n)$ is the Quot scheme of length $n$ quotients of the coherent sheaf $E$.\n\nIn this paper, we explain the formal similarity between equations~\\eqref{Todaformula} and \\eqref{GKformula}, answering a question raised in \\cite[Sec.~1]{Gholampour2017}. Our method exhibits formula \\eqref{GKformula} as an Euler characteristic shadow of a \\emph{local} version of~\\eqref{Todaformula} based at the sheaf $\\mathcal{F}$.\nMoreover, we explicitly compute (cf.~Corollary \\ref{cor:locfree8237}) the $\\mathcal{F}$-local DT generating series when $\\mathcal{F}$ is a locally free sheaf: this can be seen (cf.~Remark \\ref{rmk:comparison}) as the higher rank analogue of the point contribution to rank one DT theory, originally the first of the three MNOP conjectures, cf.~\\cite[Conj.~1]{MNOP1}.\n\n\\subsection{Main results}\nWe give some details and state our results in order of appearance. In the final part of the Introduction (Section \\ref{sec:previouswork}) we give a short outline of related work on wall-crossing and Quot schemes in Enumerative Geometry. \n\nTo any complex scheme $Y$ of finite type, Behrend \\cite{Beh} associates a canonical constructible function $\\nu_Y\\colon Y(\\mathbb C)\\rightarrow \\mathbb Z$.\nWe recall in Section~\\ref{behrendstuff} the properties we will need.\nThe \\emph{virtual Euler characteristic} of $Y$ is the ``motivic integral'' of $\\nu_Y$, namely\n\\be\\label{virtualchi}\n\\widetilde\\chi(Y) \\defeq \\chi(Y,\\nu_Y) \\defeq \\sum_{k\\in \\mathbb Z}k\\cdot \\chi(\\nu_Y^{-1}(k)) \\in \\mathbb Z.\n\\end{equation}\nOur first result, proven in Section~\\ref{sec:thmA}, is the following virtual refinement of Gholampour--Kool's formula \\eqref{GKformula} in the locally free case.\n\\begin{thm}\\label{Thm:LocallyFree}\nLet $X$ be a smooth $3$-fold, $\\mathcal F$ a locally free sheaf of rank $r$ on $X$. Then\n\\begin{equation}\\label{us}\n\\sum_{n\\geq 0}\\widetilde\\chi(\\Quot_X(\\mathcal F,n)) q^n = \\mathsf M((-1)^rq)^{r\\chi(X)}.\n\\end{equation}\n\\end{thm}\nThe case $r = 1$, corresponding to the Hilbert scheme parametrising zero-dimensional subschemes of $X$, has been proven by Behrend--Fantechi in \\cite{BFHilb}.\nWe establish \\eqref{us} by generalising their approach (technical details are in Appendix \\ref{quotmess}). See also \\cite{LEPA,JLI} for different proofs for $r=1$. Note that no Calabi--Yau or projectivity assumptions on $X$ are required.\n\n\\begin{remark}\nIf $\\mathcal F$ is a locally free sheaf, a local model for $\\Quot_X(\\mathcal F,n)$ is the Quot scheme $\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)$. We show that the latter is a \\emph{critical locus} (Theorem \\ref{thm:criticallocus}), so that in particular it carries a symmetric perfect obstruction theory in the sense of \\cite{BFHilb}. This motivates the interest in the virtual Euler characteristic computed in Theorem \\ref{Thm:LocallyFree}. However, even for \\emph{reflexive} sheaves $\\mathcal F$ over Calabi--Yau $3$-folds, we do not know when $\\Quot_X(\\mathcal F,n)$ carries a symmetric perfect obstruction theory.\n\\end{remark}\n\nWe now describe the local higher rank DT\/PT correspondence.\nFor a given $\\omega$-slope stable sheaf $\\F$ of homological dimension at most one, we embed the Quot schemes\n\\[\n\\phi_{\\F}\\colon\\Quot_X(\\mathcal F)\\hookrightarrow M_{\\DT}(r,D),\\qquad \\psi_{\\F}\\colon \\Quot_X(\\lExt^1(\\mathcal F,\\mathscr O_X))\\hookrightarrow M_{\\PT}(r,D)\n\\]\nin suitable DT and PT moduli spaces via closed immersions (cf.~Propositions~\\ref{DTembedding} and~\\ref{PTembedding}).\nThe former, $\\phi_{\\F}$, consists of taking the kernel of a surjection $\\mathcal{F} \\twoheadrightarrow Q$.\nThe latter, $\\psi_{\\F}$, might be of independent interest, so we describe its action on $\\mathbb C$-valued points here.\n\nLet $t \\colon \\lExt^1(\\F,\\mathscr O_X) \\twoheadrightarrow Q$ be a zero-dimensional quotient.\nSince $\\F$ is of homological dimension at most one, there is a natural morphism\n\\[\n\\bar{t} \\colon \\F^{\\vee} \\to \\lExt^1(\\F,\\mathscr O_X)[-1] \\xrightarrow{t[-1]} Q[-1],\n\\]\nwhere $(-)^{\\vee} = {\\textbf R}\\lHom(-,\\mathscr O_X)$ is the derived dualising functor.\nThis functor is an involutive anti-equivalence of $D^b(X)$, hence dualising again yields a canonical isomorphism\n\\[\n\\Hom(\\F^{\\vee},Q[-1]) \\cong \\Ext^1(Q^{\\mathrm D}[-1],\\F),\n\\]\nwhere $Q^{\\mathrm D} = \\lExt^3(Q,\\mathscr O_X)$.\nWe define $\\psi_{\\F}$ by sending $t$ to the corresponding extension\n\\[\n\\F \\to J^{\\bullet} \\to Q^{\\mathrm D}[-1]\n\\]\nin $D^b(X)$.\nWe prove in Proposition~\\ref{PTembedding} that this defines a higher rank PT pair and, moreover, that the association $t\\mapsto J^\\bullet$ extends to a morphism that is a closed immersion.\n\nWe define $\\F$-\\emph{local DT} and \\emph{PT invariants} by restricting the Behrend weights on the full DT and PT moduli spaces via these closed immersions.\\footnote{In general, these immersion are not open, so the restriction of the Behrend weight of the full moduli space does \\emph{not} in general agree with the intrinsic Behrend weight of the Quot schemes.}\nWe collect these in a generating function\n\\[\n\\mathsf{DT}_{\\mathcal F}(q) = \\sum_{n\\geq 0}\\chi\\left(\\Quot_X(\\mathcal F,n),\\nu_{\\DT}\\right) q^n\n\\]\non the DT side, and a similar generating function $\\mathsf{PT}_{\\F}(q)$ on the PT side.\n\nWe prove the following relation (Theorem \\ref{thm:DTPT}) after establishing a key identity in a certain motivic Hall algebra, and applying the Behrend weighted integration morphism.\n\\begin{thm}\\label{thm2}\nLet $X$ be a smooth projective Calabi--Yau $3$-fold, and let $\\F$ be an $\\omega$-stable torsion free sheaf on $X$ of rank $r$ and homological dimension at most one.\nThen\n\\be\\label{dtptF}\n\\mathsf{DT}_{\\mathcal F}(q) = \\mathsf M((-1)^rq)^{r\\chi(X)} \\cdot \\mathsf{PT}_{\\mathcal F}(q).\n\\end{equation}\n\\end{thm}\nApplying the unweighted integration morphism instead, taking Euler characteristics, we recover the main result of \\cite{Gholampour2017} in the special case where $\\F$ is slope-stable.\n\\begin{thm}\\label{thm3}\nLet $X$ be a smooth projective $3$-fold, and let $\\F$ be an $\\omega$-stable torsion free sheaf on $X$ of rank $r$ and homological dimension at most one.\nThen Gholampour--Kool's formula \\eqref{GKformula} holds.\n\\end{thm}\nSince the formula of Gholampour--Kool descends from the same Hall algebra identity giving rise to \\eqref{dtptF}, one can interpret it as an Euler characteristic shadow of the $\\F$-local higher rank DT\/PT correspondence, in the case that $\\F$ is $\\omega$-slope stable.\n\nIn the final section, we consider some special cases by imposing further restrictions on $\\F$.\nIn particular, the following result is the `intersection' between Theorems \\ref{Thm:LocallyFree} and \\ref{thm2}.\n\\begin{corollary}\\label{cor:locfree8237}\nWith the assumptions of Theorem~\\ref{thm2}, if $\\F$ is locally free then \n\\[\n\\mathsf{DT}_{\\F}(q) = \\mathsf M((-1)^rq)^{r\\chi(X)}.\n\\]\n\\end{corollary}\n\nWe show in Proposition~\\ref{prop:PT_tensor_by_line_bundle} the invariance property $\\mathsf{PT}_{\\F \\otimes L}(q) = \\mathsf{PT}_{\\F}(q)$, where $L$ is a line bundle on $X$. Finally, Corollary \\ref{cor292} shows that $\\mathsf{PT}_{\\F}$ is a polynomial if $\\F$ is reflexive.\n\n\\subsection{Previous work on wall-crossing}\\label{sec:previouswork}\nBoth the DT invariant and the Euler characteristic\n\\[\n\\widetilde\\chi(M_X(\\alpha))\\in \\mathbb Z,\\quad \\chi(M_X(\\alpha))\\in \\mathbb Z\n\\]\ncan be seen as ways to size, or ``count points'' in the moduli space\n$M_X(\\alpha)$ of stable torsion free sheaves of class $\\alpha$.\nUnlike the virtual invariants, the Euler characteristic is not \\emph{deformation invariant}.\nIndeed, deformation invariance of $\\widetilde\\chi(M_X(\\alpha))$ is a consequence of the virtual class technology available for $M_X(\\alpha)$.\nNonetheless, the virtual and the naive invariants share a common behavior: both satisfy \\emph{wall-crossing formulas}.\nThese are relations describing the transformation that the invariants undergo when one deforms not the complex structure of $X$, but rather the \\emph{stability condition} defining the moduli space.\n\nThe first calculations in DT theory, and their original motivation, involved ideal sheaves, namely the rank one DT objects; see for instance \\cite{MNOP1,BFHilb}.\n\nThe DT\/PT correspondence was first phrased rigorously as a wall-crossing phenomenon by Bayer \\cite{Bayer1} using polynomial stability conditions. It was established in the rank one case by Bridgeland \\cite{Bri} and Toda \\cite{Toda1} for the virtual invariants, and previously by Toda \\cite{Toda0} for Euler characteristics.\n\nSimilar wall-crossing formulas hold for the Quot schemes\n\\[\n\\Quot_X(\\mathscr I_C,n)\\subset \\Hilb(X),\n\\]\nwhere $C\\subset X$ is a fixed Cohen--Macaulay curve and $X$ need not be Calabi--Yau.\nSee work of Stoppa--Thomas \\cite{ST} for the naive invariants, and \\cite{LocalDT} for the virtual ones (where also projectivity is not needed, but $C$ is required to be smooth).\n\nIn \\cite{Ob1,Ricolfi2018}, a cycle-local DT\/PT correspondence is proved in rank one.\nCycle-local DT invariants gather contributions of those ideals $\\mathscr I_{Z}\\subset \\mathscr O_X$ for which the cycle $[Z]$ is equal to a fixed element of the Chow variety of $X$.\nSpecialising to the case $\\F = \\mathscr I_C$, where $C \\subset X$ is a Cohen--Macaulay curve, our Theorem~\\ref{thm2} refines this cycle-local DT\/PT correspondence to an $\\mathscr I_C$-local correspondence around the fixed subscheme $C\\subset X$ (see Remark \\ref{rmk:cyclelocal}).\n\nIn the recent work \\cite{Lo2018}, Lo approaches the problem of relating $\\Quot_X(\\lExt^1(\\F,\\mathscr O_X))$ to the moduli space $M_{\\PT}(r,D)$ from a categorical point of view, constructing a functor `the other way around', i.e.~from PT pairs to quotients.\nIt would be interesting to find out the precise relationship between Lo's functor and our closed immersion. \n\n\n\\subsection*{Acknowledgements}\nWe thank Arend Bayer, Martijn Kool, Georg Oberdieck, and Richard Thomas for helpful comments and suggestions during the completion of this work.\nThis project was set up during a visit of A.R. to the University of Edinburgh, and it was completed during a visit of S.B. to the Max-Planck Institute in Bonn.\nWe thank both institutions for their hospitality and excellent working conditions.\nS.B. was supported by the ERC Starting Grant no.~337039 \\emph{WallXBirGeom}.\n\n\\subsection*{Conventions}\nWe work over $\\mathbb C$.\nAll rings, schemes, and stacks will be assumed to be \\emph{locally of finite type} over $\\mathbb C$, unless specified otherwise. All categories and functors will be $\\mathbb C$-linear. A Calabi--Yau $3$-fold is a smooth projective $3$-fold $X$ such that $K_X=0$ and $H^1(X,\\mathscr O_X)=0$.\nIf $M$ is a scheme, we write $D^b(M)$ for the \\emph{bounded coherent} derived category of $M$ and $\\Perf(M)\\subset D^b(M)$ for the category of perfect complexes.\nWe write $\\Coh_0(M)$ (resp.~$\\Coh_{\\leq 1}(M)$) for the full subcategory of coherent sheaves on $M$ of dimension zero (resp.~of dimension at most $1$). The \\emph{homological dimension} of a coherent sheaf $\\F$ on a smooth projective variety is the minimal length of a locally free resolution of $\\F$. For a sheaf $E\\in \\Coh(X)$ of codimension $c$, we set $E^{\\mathrm D} = \\lExt^c(E,\\mathscr O_X)$.\\footnote{This differs slightly from the notation used in \\cite{modulisheaves}, where $E^{\\mathrm D}$ denotes $\\lExt^c(E,\\omega_X)$.}\nWe write $\\Quot_X(E,n)$ for the Quot scheme of length $n$ quotients of $E$, and we set \n$\\Quot_X(E)=\\coprod_{n\\geq 0}\\Quot_X(E,n)$.\nWe refer the reader to \\cite{modulisheaves} for generalities on moduli spaces of sheaves and to \\cite[Sec.~1]{PicScheme} for base change theory for local ext.\n\n\n\n\n\n\n\n\\section{Quotients of a free sheaf on affine 3-space}\nB.~Szendr\\H{o}i proved in \\cite[Theorem 1.3.1]{MR2403807} that the Hilbert scheme of points $\\Hilb^n\\mathbb A^3$ is a global critical locus. The goal of this section is to prove that the same is true, more generally, for the Quot scheme\n\\[\n\\Quot_{\\mathbb A^3}(\\mathscr O^r,n),\n\\]\nfor all $r\\geq 1$ and $n\\geq 0$. \nIn other words, we will show that the Quot scheme can be written as the scheme-theoretic zero locus of an exact one-form $\\dd f$, where $f$ is a regular function defined on a smooth scheme. In particular, this proves that $\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)$ carries a symmetric perfect obstruction theory, defined by the Hessian of $f$. By inspecting the virtual \\emph{motivic} refinements of these critical loci, we deduce the formula\n\\[\n\\sum_{n\\geq 0}\\widetilde\\chi(\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)) q^n = \\mathsf M((-1)^rq)^r.\n\\]\nIn Section~\\ref{sec:thmA}, we generalise this formula to arbitrary locally free sheaves on smooth quasi-projective $3$-folds, thus establishing Theorem~\\ref{Thm:LocallyFree}.\n\n\n\n\n\\subsection{Quiver representations}\n\nLet $Q$ be a quiver, i.e.~a finite directed graph. We denote by $Q_0$ and $Q_1$ the sets of vertices and edges of $Q$ respectively.\nA representation $M$ of $Q$ is the datum of a finite dimensional vector space $V_i$ for every $i\\in Q_0$, and a linear map $V_i\\rightarrow V_j$ for every edge $i\\rightarrow j$ in $Q_1$.\nThe \\emph{dimension vector} of $M$ is\n\\[\n\\underline{\\dim}\\, M = (\\dim V_i)\\in \\mathbb N^{Q_0}.\n\\]\nIt is well known that the representations of $Q$ form an abelian category, that is moreover equivalent to the category of left modules over the path algebra $\\mathbb C Q$ of the quiver.\n\nFollowing \\cite{King}, we recall the notion of (semi)stability of a representation of $Q$.\n\n\\begin{definition}\\label{centralcharge}\nA \\emph{central charge} is a group homomorphism $Z\\colon \\mathbb Z^{Q_0}\\rightarrow \\mathbb C$ such that the image of $\\mathbb N^{Q_0}\\setminus 0$ lies inside $\\mathbb H_+ = \\set{re^{i\\pi\\varphi}|r>0,\\,0<\\varphi\\leq 1}$. For every $\\alpha\\in \\mathbb Z^{Q_0}$, we denote by $\\varphi(\\alpha)$ the real number $\\varphi$ such that $Z(\\alpha) = re^{i\\pi\\varphi}$. It is called the \\emph{phase} of $\\alpha$.\n\\end{definition}\n\nNote that every vector $\\theta\\in \\mathbb R^{Q_0}$ induces a central charge $Z_{\\theta}$ given by\n\\[\nZ_{\\theta}(\\alpha) = -\\theta\\cdot \\alpha + i|\\alpha|,\n\\]\nwhere $|\\alpha| = \\sum_i\\alpha_i$. We denote by $\\varphi_\\theta$ the induced phase function, and we put\n\\[\n\\varphi_\\theta(M) = \\varphi_\\theta(\\underline{\\dim}\\,M)\n\\]\nfor every $Q$-representation $M$.\n\n\\begin{definition}\\label{stablereps}\nFix $\\theta\\in \\mathbb R^{Q_0}$. Then a representation $M$ of $Q$ is called \\emph{$\\theta$-semistable} if \n\\[\n\\varphi_\\theta(A)\\leq \\varphi_\\theta(M)\n\\]\nfor every nonzero proper subrepresentation $A\\subset M$. When strict inequality holds, we say that $M$ is \\emph{$\\theta$-stable}. Vectors $\\theta\\in \\mathbb R^{Q_0}$ are referred to as \\emph{stability parameters}.\n\\end{definition}\n\nFix a stability parameter $\\theta\\in \\mathbb R^{Q_0}$. To each $\\alpha\\in\\mathbb N^{Q_0}\\setminus 0$ one can associate its \\emph{slope} (with respect to $\\theta$), namely the rational number\n\\[\n\\mu_\\theta(\\alpha) = \\frac{\\theta\\cdot\\alpha}{|\\alpha|}\\in\\mathbb Q.\n\\]\nIt is easy to see that $\\varphi_\\theta(\\alpha)<\\varphi_\\theta(\\beta)$ if and only if $\\mu_\\theta(\\alpha)<\\mu_\\theta(\\beta)$. So, after setting $\\mu_\\theta(M) = \\mu_\\theta(\\underline{\\dim}\\,M)$, one can check stability using slopes instead of phases.\n\n\n\\subsection{Framed representations}\nLet $Q$ be a quiver with a distinguished vertex $0\\in Q_0$, and let $r$ be a positive integer. Consider the quiver $\\widetilde Q$ obtained by adding one vertex $\\infty$ to the original vertices in $Q_0$ and $r$ edges $\\infty\\rightarrow 0$.\nIf $r = 1$, this construction is typically referred to as a \\emph{framing} of $Q$. \n\nA representation $\\widetilde M$ of $\\widetilde Q$ can be uniquely written as a pair $(M,v)$, where $M$ is a representation of $Q$ and $v = (v_1,\\dots,v_r)$ is an $r$-tuple of linear maps $v_i\\colon V_\\infty\\rightarrow V_0$. We always assume our framed representations to satisfy $\\dim V_\\infty = 1$, so that\n\\[\n\\underline{\\dim}\\,\\widetilde M=(1,\\underline{\\dim}\\,M).\n\\]\nThe vector space $V_{\\infty}$ will be left implicit.\n\n\\begin{definition}\\label{framedstability}\nLet $\\theta\\in \\mathbb R^{Q_0}$ be a stability parameter.\nA representation $(M,v)$ of $\\widetilde Q$ with $\\dim V_\\infty = 1$ is said to be \\emph{$\\theta$-(semi)stable} if it is $(\\theta_\\infty,\\theta)$-(semi)stable in the sense of Definition \\ref{stablereps}, where $\\theta_\\infty = -\\theta\\cdot \\underline{\\dim}\\,M$.\n\\end{definition}\n\n\n\nThe space of all representations of $Q$, of a given dimension vector $\\alpha\\in \\mathbb N^{Q_0}$, is the affine space\n\\[\n\\Rep_\\alpha(Q) = \\prod_{i\\rightarrow j}\\Hom_\\mathbb C(\\mathbb C^{\\alpha_i},\\mathbb C^{\\alpha_j}).\n\\]\nOn this space there is an action of the gauge group $\\GL_\\alpha = \\prod_{i\\in Q_0} \\GL_{\\alpha_i}$ by simultaneous conjugation. The quotient stack $\\Rep_\\alpha(Q)\/\\GL_\\alpha$ parametrises isomorphism classes of representations of $Q$ with dimension vector $\\alpha$. Imposing suitable stability conditions, one can restrict to $\\GL_\\alpha$-invariant open subschemes of $\\Rep_\\alpha(Q)$ such that the induced action is free, so that the quotient is a smooth \\emph{scheme}. We will do this in the next subsection for framed representations of the three loop quiver\n\n\\subsection{The three loop quiver}\nConsider the quiver $\\mathsf L_3$ with one vertex and three loops, labelled $x$, $y$ and $z$. A representation of $\\mathsf L_3$ is the datum of a left module over the free algebra\n\\[\n\\mathbb C\\langle x,y,z\\rangle\n\\]\non three generators, which is the path algebra of $\\mathsf L_3$.\nWe now add $r$ framings at the vertex, thus forming the quiver $\\widetilde{\\mathsf L}_{3}$ (see Fig.~\\ref{L3quiver}).\n\n\\begin{figure}[ht]\n\\begin{tikzcd}\n{\\small{\\infty}}\\,\\,\\bullet \n\\arrow[bend left]{rr}[description]{v_1}\n\\arrow[bend right]{rr}[description]{v_r} & \n{\\small{\\vdots}} &\n\\bullet\n\\arrow[loop above]{}{x}\n\\arrow[loop right]{}{y}\n\\arrow[loop below]{}{z}\n\\end{tikzcd}\n\\caption{The quiver $\\widetilde{\\mathsf L}_3$ with its $r$ framings.}\\label{L3quiver}\n\\end{figure}\n\n\n\nLetting $V_n$ be a fixed $n$-dimensional vector space, representations of $\\widetilde{\\mathsf L}_{3}$ with dimension vector $(1,n)$ form the affine space\n\\[\n\\mathcal R_{n,r} = \\End(V_n)^3\\times V_n^r,\n\\]\nof dimension $3n^2+rn$. Consider the open subset\n\\be\\label{unr}\nU_{n,r}\\subset \\mathcal R_{n,r}\n\\end{equation}\nparametrising tuples $(A,B,C,v_1,\\dots,v_r)$ such that the vectors $v_1,\\dots,v_r$ span the underlying (unframed) representation $(A,B,C)\\in \\Rep_n(\\mathsf L_3)$. Equivalently, the $r$ vectors $v_i$ span $V_n$ as a $\\mathbb C\\langle x,y,z\\rangle$-module. \n\n\nConsider the action of $\\GL_n$ on $\\mathcal R_{n,r}$ given by \n\\[\ng\\cdot (A,B,C,v_1,\\dots,v_r) = (A^g,B^g,C^g,g v_1,\\dots,g v_r),\n\\]\nwhere $A^g$ denotes the conjugation $gAg^{-1}$ by $g\\in \\GL_n$. This action is free on $U_{n,r}$.\nThus the GIT quotient $U_{n,r}\/\\GL_n$ with respect to the character $\\det \\colon \\GL_n \\to \\mathbb C^{\\times}$ is a smooth quasi-projective variety.\n\nWe now show that the open set $U_{n,r}$ parametrises stable framed representations of $\\mathsf L_3$, with stability understood in the sense of Definition \\ref{framedstability}.\n\nIf $r=1$, then for a point $(A,B,C,v)\\in U_{n,1}$ one usually says that $v$ is a \\emph{cyclic vector}. For $r>1$, we say that $v_1,\\dots,v_r$ \\emph{jointly generate} a representation $M=(A,B,C)\\in \\Rep_n(\\mathsf L_3)=\\End(V_n)^3$ if $(M,v_1,\\dots,v_r)\\in U_{n,r}$. \n\n\\begin{prop}\nLet $\\widetilde M = (M,v_1,\\dots,v_r)$ be a representation of the framed quiver\\, $\\widetilde{\\mathsf L}_{3}$ depicted in Figure \\ref{L3quiver}.\n\tChoose a vector $\\theta = (\\theta_1,\\theta_2)$ with $\\theta_1 \\geq \\theta_2$. Then $\\widetilde M$ is $\\theta$-stable if and only if $v_1,\\ldots,v_r$ jointly generate $M$.\n\\end{prop}\n\n\\begin{proof}\nSuppose that $v_1,\\ldots,v_r$ jointly generate a proper subrepresentation $0\\neq W\\subsetneq M$.\nWe obtain a subrepresentation $\\widetilde N = (W,v_1,\\dots,v_r)\\subset \\widetilde M$ of dimension vector $(1,d)$ with $0 < d = \\dim W < n$.\nWe claim that $\\widetilde N$ destabilises $\\widetilde M$. \nIndeed, the inequality\n\\begin{equation*}\n\t\\mu_{\\theta}(\\widetilde N) = \\dfrac{\\theta_1 + d\\theta_2}{1+d} \\geq \\dfrac{\\theta_1 + n\\theta_2}{1+n} = \\mu_{\\theta}(\\widetilde M)\n\\end{equation*}\nholds if and only if $\\theta_1 \\geq \\theta_2$, which holds by assumption.\nSince stability and semistability coincide for this choice of dimension vector, we conclude that $\\widetilde M$ is unstable.\n\nFor the converse, suppose that $v_1,\\ldots,v_r$ jointly generate $M$.\tIf $\\widetilde M$ is not stable, there exists a non-trivial proper destabilising subrepresentation $0 \\neq \\widetilde N \\subsetneq \\widetilde M$ of dimension vector $(d_1,d_2)$ with $0 \\leq d_1 \\leq 1$ and $0 \\leq d_2 \\leq n$. There are two cases to consider.\n\t\\begin{enumerate}\n\t\t\\item If $d_1 = 1$, it follows that $d_2 = n$ since $v_1,\\ldots,v_r$ jointly generate $M$.\n\t\t\t\n\t\t\tBut then $\\widetilde N = \\widetilde M$, which is a contradiction.\n\t\t\\item If $d_1 = 0$ then $d_2 > 0$, and we directly compute\n\t\t\t\\begin{equation*}\n\t\t\t\t\\mu_{\\theta}(\\widetilde N) = \\dfrac{d_2\\theta_2}{d_2} = \\theta_2\n\t\t\t\t\t= \\dfrac{(1+n)\\theta_2}{1+n} \\leq \\dfrac{\\theta_1+n\\theta_2}{1+n} = \\mu_{\\theta}(\\widetilde M)\n\t\t\t\\end{equation*}\n\t\t\tbecause $\\theta_2 \\leq \\theta_1$.\n\t\t\tBut this contradicts the fact that $\\widetilde N$ destabilises $\\widetilde M$.\n\t\\end{enumerate}\n\tIt follows that $\\widetilde M$ is $\\theta$-stable.\n\tThis completes the proof.\n\\end{proof}\n\n\n\\subsection{The non-commutative Quot scheme}\n\nIn this section, we write\n\\[\nR=\\mathbb C\\langle x,y,z\\rangle\n\\]\nfor the free (non-commutative) $\\mathbb C$-algebra on three generators, and for a complex scheme $B$, we denote by $R_B$ the sheaf of $\\mathscr O_B$-algebras associated to the presheaf $R\\otimes_\\mathbb C\\mathscr O_B=\\mathscr O_B\\braket{x,y,z}$.\nWe consider the functor \n\\begin{equation}\\label{eq:NonCommutative_Quot_Functor}\n\\mathfrak Q_{n,r}\\colon\\mathsf{Sch}_{\\mathbb C}^{\\textrm{op}}\\rightarrow \\mathsf{Sets}\n\\end{equation}\nsending a $\\mathbb C$-scheme $B$ to the set of isomorphism classes of triples $(M,p,\\beta)$, where\n\\begin{enumerate}\n\\item $M$ is a left $R_B$-module, locally free of rank $n$ over $\\mathscr O_B$,\n\\item $p\\colon R_B^r\\twoheadrightarrow M$ is an $R_B$-linear epimorphism, and\n\\item $\\beta\\subset \\Gamma(B,M)$ is a basis of $M$ as an $\\mathscr O_B$-module.\n\\end{enumerate}\nTwo triples $(M,p,\\beta)$ and $(M',p',\\beta')$ are considered isomorphic if there is a commutative diagram\n\\be\\label{diag:oblin}\n\\begin{tikzcd}\nR_B^r\\arrow[two heads]{r}{p} \\arrow[equal]{d} & M\\arrow{d}{\\varphi} \\\\\nR_B^r\\arrow[two heads]{r}{p'} & M'\n\\end{tikzcd}\n\\end{equation}\nwith $\\varphi$ an $\\mathscr O_B$-linear isomorphism transforming $\\beta$ into $\\beta'$. We denote by $\\braket{M,p,\\beta}$ the corresponding isomorphism class.\n\nOne can also define a functor \n\\[\n\\overline{\\mathfrak Q}_{n,r}\\colon\\mathsf{Sch}_{\\mathbb C}^{\\textrm{op}}\\rightarrow \\mathsf{Sets}\n\\]\nby letting $\\overline{\\mathfrak Q}_{n,r}(B)$ be the set of isomorphism classes of pairs $(M,p)$ just as above, but where no basis of $M$ is chosen. Here, as before, we declare that $(M,p)\\sim (M',p')$ when there is a commutative diagram as in \\eqref{diag:oblin}. \n\nNotice that, by considering the kernel of the surjection, an equivalence class $\\braket{M,p}$ uniquely determines a left $R_B$-module $N\\subset R_B^r$ (such that the quotient $R_B^r\/N$ is a locally free $\\mathscr O_B$-module). The proof of the following result is along the same lines of \\cite[Sec.~2]{NCHilb1}. \n\n\\begin{theorem}\\label{ncquot}\nThe smooth quasi-affine scheme $U_{n,r}$ defined in \\eqref{unr} represents the functor $\\mathfrak Q_{n,r}$, whereas the GIT quotient $U_{n,r}\/\\GL_n$ represents the functor $\\overline{\\mathfrak Q}_{n,r}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $V=\\mathbb C^n$ with its standard basis $e_1,\\dots,e_n$. Consider the free module $M_0=V\\otimes_\\mathbb C\\mathscr O_{\\mathcal R_{n,r}}$ with basis\n$\\beta_0=\\set{e_j\\otimes 1:1\\leq j\\leq n}$. Let $(X_{ij},Y_{ij},Z_{ij},u_{1}^k,\\ldots,u_{r}^k)$ be the coordinates on the affine space $\\mathcal R_{n,r}$. Here $1\\leq i,j\\leq n$ correspond to matrix entries and $1\\leq k\\leq n$ to vector components.\nThen $M_0$ has distinguished elements\n\\[\n\\mathsf v_\\ell=\\sum_{j} e_j\\otimes u_{\\ell},\\qquad 1\\leq \\ell \\leq r.\n\\]\nFor each $\\ell = 1,\\ldots,r$, consider the $R$-linear map $\\theta_\\ell\\colon R_{\\mathcal R_{n,r}} \\rightarrow M_0$ given by $\\theta_\\ell(\\mathbb 1)=\\mathsf v_\\ell$. Then we can construct the map $\\theta_0 = \\oplus_\\ell \\theta_\\ell\\colon R_{\\mathcal R_{n,r}}^r\\rightarrow M_0$.\nRestricting the triple $(M_0,\\theta_0,\\beta_0)$ to $U_{n,r}\\subset \\mathcal R_{n,r}$ gives a morphism of functors \n\\[\nU_{n,r}\\rightarrow \\mathfrak Q_{n,r},\n\\]\nwhose inverse is constructed as follows.\nLet $B$ be a scheme, set again $V=\\mathbb C^n$ and fix a $B$-valued point $\\braket{M,\\theta,\\beta}\\in \\mathfrak Q_{n,r}(B)$. The $R$-action on $\\beta\\subset \\Gamma(B,M)$ determines three endomorphisms $(X,Y,Z)\\colon B\\rightarrow \\End(V)^3$. On the other hand, decomposing the map $\\theta\\colon R_B^r\\twoheadrightarrow M$ into $r$ maps $\\theta_\\ell\\colon R_B\\rightarrow M$ and taking $v_\\ell = \\theta_\\ell(\\mathbb 1)$ determines a morphism $(v_1,\\ldots,v_r)\\colon B\\rightarrow V^r$. We have thus constructed a morphism $f\\colon B\\rightarrow \\mathcal R_{n,r}$. The surjectivity of $\\theta$ says that $f$ factors through $U_{n,r}$. Therefore $U_{n,r}$ represents $\\mathfrak Q_{n,r}$.\n\nNext, let $\\pi\\colon U_{n,r}\\rightarrow \\overline U_{n,r} = U_{n,r}\/\\GL_n$ be the quotient map, which we know is a principal $\\GL_n$-bundle.\nThis implies that $\\pi^\\ast\\colon \\QCoh(\\overline U_{n,r})\\,\\widetilde{\\rightarrow}\\,\\QCoh_{\\GL_n}(U_{n,r})$ is an equivalence of categories, preserving locally free sheaves~\\cite[Prop.~$4.5$]{NCHilb1}. Consider the universal triple $\\braket{M_0,\\theta_0,\\beta_0}$ defined above. Since $M_0$ is a $\\GL_n$-equivariant vector bundle on $U_{n,r}$, it follows that, up to isomorphism, there is a unique locally free sheaf $\\mathscr M$ on $\\overline U_{n,r}$ such that $\\pi^\\ast \\mathscr M\\isom M_0$. In fact, $\\mathscr M\\isom (\\pi_\\ast M_0)^{\\GL_n}\\subset \\pi_\\ast M_0$, the subsheaf of $\\GL_n$-invariant sections. The $r$ sections $\\mathsf v_\\ell$, being $\\GL_n$-invariant, descend to sections of $\\mathscr M$, still denoted $\\mathsf v_\\ell$. These generate $\\mathscr M$ as an $R_{\\overline U_{n,r}}$-module, so we get a surjection $\\vartheta\\colon R_{\\overline U_{n,r}}^r\\twoheadrightarrow \\mathscr M$ sending $\\mathbb 1 \\mapsto \\mathsf v_\\ell$ in the $\\ell$-th component.\nIn particular, the pair $\\braket{\\mathscr M,\\vartheta}$ defines a morphism of functors \n\\[\n\\overline U_{n,r}\\rightarrow \\overline{\\mathfrak Q}_{n,r}.\n\\]\nWe need to construct its inverse. Let $B$ be a scheme and fix a $B$-valued point $\\braket{N,\\theta}\\in \\overline{\\mathfrak Q}_{n,r}(B)$. Let $(B_i:i\\in I)$ be an open cover of $B$ such that $N_i=N|_{B_i}$ is free of rank $n$ over $\\mathscr O_{B_i}$. Decompose $\\theta = \\theta_1\\oplus \\cdots\\oplus \\theta_r$ into $r$ maps $\\theta_\\ell\\colon R_B\\rightarrow N$. Choose a basis $\\beta_i\\subset \\Gamma(B_i,N_i)$ and let $v_{\\ell,i}=\\theta_\\ell(\\mathbb 1)|_{B_i}$ be the restriction of $\\theta_\\ell(\\mathbb 1)\\in \\Gamma(B,N)$ to $B_i\\subset B$.\nAs usual, the tuple $(v_{\\ell,i}:1\\leq \\ell\\leq r)$ defines a linear surjection $\\theta_i\\colon R_{B_i}^r\\twoheadrightarrow N_i$. Each triple $\\braket{N_i,\\theta_i,\\beta_i}$ then defines a point $\\psi_i\\colon B_i\\rightarrow U_{n,r}$, and for all indices $i$ and $j$ there is a matrix $g\\in \\GL_n(\\mathscr O_{B_{ij}})$ sending $\\beta_i$ to $\\beta_j$. In other words, $g$ defines a map $g\\colon B_{ij}\\rightarrow \\GL_n$ such that $g\\cdot\\psi_i=\\psi_j$. Then $\\pi\\circ \\psi_i$ and $\\pi\\circ \\psi_j$ agree on $B_{ij}$, and this determines a unique map to the quotient $f\\colon B\\rightarrow \\overline U_{n,r}$, satisfying $(N,\\theta)\\sim f^\\ast(\\mathscr M,\\vartheta)$. This shows that $\\overline U_{n,r}$ represents $\\overline{\\mathfrak Q}_{n,r}$.\n\\end{proof}\n\n\n\n\nAs a consequence of this result, the $B$-valued points of the quotient $U_{n,r}\/\\GL_n$ can be identified with left $R_B$-submodules\n\\[\nN\\subset R_B^r\n\\]\nwith the property that the quotient is locally free of rank $n$ over $\\mathscr O_B$. Because it represents the functor of quotients $R^r\\twoheadrightarrow V_n$, and $R$ is a non-commutative $\\mathbb C$-algebra, we refer to $U_{n,r}\/\\GL_n$ as a \\emph{non-commutative Quot scheme}, and we introduce the notation\n\\begin{equation*}\n\\Quot^n_r = U_{n,r}\/\\GL_n.\n\\end{equation*}\nNote that $\\Quot^n_1$ is the \\emph{non-commutative Hilbert scheme}.\nFinally, the GIT construction implies that $\\Quot^n_r$ is a smooth quasi-projective variety of dimension $2n^2+rn$.\n\n\n\n\\subsection{The potential}\nOn the three-loop quiver, consider the potential \n\\[\nW = x[y,z]\\in \\mathbb C\\mathsf L_3,\n\\]\nwhere $[-,-]$ denotes the commutator.\nThe associated trace map $U_{n,r}\\rightarrow \\mathbb A^1$, defined by $(A,B,C,v_1,\\ldots,v_r)\\mapsto \\Tr A[B,C]$, is $\\GL_n$-invariant.\\footnote{The vectors $v_i$ are not involved in the definition of the map. They are only needed to define its domain.}\nThus, it descends to a regular function on the quotient,\n\\begin{equation}\nf_n\\colon \\Quot^n_r \\rightarrow \\mathbb A^1.\n\\end{equation}\n\nWe now show that $\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)$ is the scheme-theoretic critical locus of $f_n$.\n\n\\begin{theorem}\\label{thm:criticallocus}\nThere is a closed immersion\n\\[\n\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)\\hookrightarrow \\Quot^n_r\n\\]\ncut out scheme-theoretically by the exact one-form $\\dd f_n$.\n\\end{theorem}\n\n\\begin{proof}\nLet $B$ be a scheme. Observe that there is an inclusion of sets\n\\[\n\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)(B)\\subset \\Quot^n_r(B).\n\\]\nA $B$-valued point $[N]$ of the non-commutative Quot scheme defines a $B$-valued point of the commutative Quot scheme if and only if the $R$-action on the corresponding $R_B$-module $N$ descends to a $\\mathbb C[x,y,z]$-action. This happens precisely when the actions of $x$, $y$ and $z$ on $N$ commute with each other. \nLet then $Z\\subset \\Quot^n_r$ be the image of the zero locus \n\\[\n\\Set{(A,B,C,v_1,\\ldots,v_r) \\mid [A,B]=[A,C]=[B,C]=0}\\subset U_{n,r}\n\\]\nunder the quotient map.\nThen $[N]$ belongs to $\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)(B)$ if and only if the corresponding morphism $B\\rightarrow \\Quot^n_r$ factors through $Z$. But $Z$ agrees, as a scheme, with the critical locus of $f_n$, by \\cite[Prop.~$3.8$]{Seg}.\n\\end{proof}\n\n\nIt follows that $\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)$ has a symmetric obstruction theory determined by the Hessian of $f_n$. We refer to \\cite{BFHilb} for more details on symmetric obstruction theories.\n\n\nEvery scheme $Z$ which can be written as $\\set{\\dd f = 0}$ for $f$ a regular function on a smooth scheme $U$ has a \\emph{virtual motive}, depending on the pair $(U,f)$. According to \\cite{BBS}, this is a class \n\\[\n[Z]_{\\vir} \\in \\mathcal M=K^{\\hat\\mu}(\\mathsf{Var}\/\\mathbb C)\\left[\\mathbb{L}^{-1\/2}\\right]\n\\]\nin the $\\hat\\mu$-equivariant ring of motivic weights, satisfying the property $\\chi [Z]_{\\vir} = \\widetilde\\chi(Z)$, where $\\widetilde\\chi(-)$ denotes the virtual Euler characteristic, as in \\eqref{virtualchi}. Here $\\mathbb{L} = [\\mathbb A^1]$ is the Lefschetz motive, and the Euler characteristic homomorphism $\\chi\\colon K(\\mathsf{Var}\/\\mathbb C)\\rightarrow \\mathbb Z$ from the classical Grothendieck ring of varieties is extended to the ring $\\mathcal M$ by sending $\\mathbb{L}^{1\/2}$ to $-1$.\nThe class \n\\[\n[Z]_{\\vir} = -\\mathbb{L}^{-(\\dim U)\/2}\\cdot [\\phi_f] \\in \\mathcal M\n\\]\nis constructed by means of the motivic vanishing cycle class $[\\phi_f]$ introduced by Denef--Loeser \\cite{DenefLoeser1}. We refer to \\cite{BBS} for more details.\n\nTheorem \\ref{thm:criticallocus} then produces a virtual motive $[\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)]_{\\vir}\\in\\mathcal M$ in the ring of equivariant motivic weights, by means of the pair $(\\Quot^n_r,f_n)$. Form the generating function\n\\[\n\\mathsf Z_r(t) = \\sum_{n\\geq 0}\\, \\left[\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)\\right]_{\\vir}\\cdot t^n\\in \\mathcal M\\llbracket t\\rrbracket.\n\\]\nFollowing the calculation carried out in \\cite{BBS} for $r=1$, one can prove the following.\n\n\\begin{prop}[{\\cite[Prop.~2.3.6]{ThesisR}}]\nOne has the relation\n\\be\\label{id:mot}\n\\mathsf Z_{r}(t) = \\prod_{m=1}^\\infty \\prod_{k = 0}^{rm-1}\\left(1-\\mathbb{L}^{2+k-rm\/2}t^m\\right)^{-1}.\n\\end{equation}\n\\end{prop}\n\n\\begin{corollary}\nOne has the relation\n\\[\n\\sum_{n\\geq 0}\\widetilde\\chi(\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)) t^n = \\mathsf M((-1)^rt)^r.\n\\]\n\\end{corollary}\n\n\\begin{proof}\nThis follows by applying $\\chi$ to \\eqref{id:mot}, and using that $\\chi(\\mathbb{L}^{-1\/2}) = -1$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Quotients of a locally free sheaf on an arbitrary 3-fold}\\label{sec:thmA}\n\nThe goal of this section is to prove Theorem \\ref{Thm:LocallyFree}. We follow the cut-and-paste technique of Behrend--Fantechi \\cite{BFHilb}, also used in \\cite{LocalDT}.\n\nLet $X$ be a smooth quasi-projective $3$-fold and let $\\F$ be a locally free sheaf of rank $r$. We will show that\n\\be\\label{signedchi}\n\\chi(\\Quot_X(\\F,n),\\nu) = (-1)^{rn}\\chi(\\Quot_X(\\F,n)).\n\\end{equation}\nThis was proved for $\\F = \\mathscr O_X$ in \\cite{BFHilb,LEPA,JLI} and for the torsion free sheaf $\\F=\\mathscr I_C$, where $C\\subset X$ is a smooth curve, in \\cite{LocalDT}.\nCombined with the Euler characteristic calculation \\cite[Thm.~1.1]{Gholampour2017} recalled in \\eqref{GKformula}, formula \\eqref{signedchi} proves Theorem \\ref{Thm:LocallyFree}.\n\n\\subsection{The virtual Euler characteristic}\\label{behrendstuff}\nLet $X$ be a complex scheme, and let $\\nu_X\\colon X(\\mathbb C)\\rightarrow \\mathbb Z$ be its Behrend function \\cite{Beh}. The virtual (or weighted) Euler characteristic of $X$, as recalled in \\eqref{virtualchi}, is the integer\n\\begin{equation*\n\\widetilde\\chi(X) \\defeq \\chi(X,\\nu_X) \\defeq \\sum_{k\\in \\mathbb Z}k\\cdot \\chi(\\nu_X^{-1}(k)).\n\\end{equation*}\nGiven a morphism $f\\colon Z\\rightarrow X$, one defines the \\emph{relative} virtual Euler characteristic by $\\widetilde\\chi(Z,X) = \\chi(Z,f^\\ast \\nu_X)$.\nWe now recall the properties of $\\nu$ and $\\widetilde\\chi$ we will need later. \n\\begin{enumerate}\n\\item [$\\circ$] If $f\\colon Z\\rightarrow X$ is \\'etale, then $\\nu_Z = f^\\ast \\nu_X$.\n\\item [$\\circ$] If $Z_1$, $Z_2\\subset X$ are disjoint locally closed subschemes, one has\n\\be\\label{ChiSum}\n\\widetilde\\chi(Z_1\\amalg Z_2,X) = \\widetilde\\chi(Z_1,X)+\\widetilde\\chi(Z_2,X).\n\\end{equation}\n\\item [$\\circ$] Given two morphisms $Z_i \\rightarrow X_i$, one has\n\\be\\label{ChiProduct}\n\\widetilde\\chi(Z_1\\times Z_2,X_1\\times X_2)=\\widetilde\\chi(Z_1,X_1)\\cdot \\widetilde\\chi(Z_2,X_2).\n\\end{equation}\n\\item [$\\circ$] If one has a commutative diagram\n\\[\n\\begin{tikzcd} \nZ \\arrow{r}{} \\arrow{d}{}\n&X \\arrow{d}{}\\\\\nW \\arrow{r}{} &Y\n\\end{tikzcd}\n\\]\nwith $X\\rightarrow Y$ smooth and $Z\\rightarrow W$ finite \\'etale of degree $d$, then\n\\be\\label{ChiSquare}\n\\widetilde\\chi(Z,X)=d(-1)^{\\dim X\/Y}\\widetilde\\chi(W,Y).\n\\end{equation}\nAs a special case, if $X\\rightarrow Y$ is \\'etale (for instance, an open immersion) and $Z\\rightarrow X$ is a morphism, then \n\\be\\label{ChiEtale}\n\\widetilde \\chi(Z,X)=\\widetilde\\chi(Z,Y).\n\\end{equation}\n\\end{enumerate}\n\n\n\n\n\n\n\\subsection{Reduction to the deepest stratum}\nConsider the Quot-to-Chow morphism\n\\[\n\\sigma\\colon\\Quot_X(\\F,n) \\rightarrow \\Sym^nX\n\\]\nsending $[\\F\\twoheadrightarrow Q]$ to the zero cycle given by the support of $Q$. This is a morphism of schemes by \\cite[Cor.~7.15]{Rydh1}. For every partition $\\alpha$ of $n$, we have the locally closed subscheme $\\Sym^n_\\alpha X$ of $\\Sym^nX$ parametrising zero-cycles with multiplicities distributed according to $\\alpha$. The restrictions\n\\[\n\\sigma_\\alpha\\colon \\Quot_X(\\F,n)_\\alpha \\rightarrow \\Sym^n_\\alpha X\n\\]\ninduce a locally closed stratification of the Quot scheme.\nThe most interesting stratum is the deepest one, corresponding to the full partition $\\alpha = (n)$.\nThe map \n\\be\\label{mor:phc}\n\\sigma_{(n)}\\colon\\Quot_X(\\F,n)_{(n)}\\rightarrow X\n\\end{equation}\nhas fibre over a point $p\\in X$ the \\emph{punctual} Quot scheme $\\Quot_X(\\F,n)_p$, parametrising quotients whose target is supported entirely at $p$. Note that\n\\[\n\\Quot_X(\\F,n)_p\\isom\\Quot_X(\\mathscr O_X^r,n)_p\\isom \\Quot_{\\mathbb A^3}(\\mathscr O^r,n)_0,\n\\]\nwhere for the first isomorphism we need $\\F$ locally free.\nWe will see in Lemma \\ref{lemma:ZLT} below that \\eqref{mor:phc} is Zariski locally trivial with fibre the punctual Quot scheme. From now on we shorten\n\\[\n\\mathsf P_n = \\Quot_{\\mathbb A^3}(\\mathscr O^r,n)_0\\,,\\quad \\nu_n = \\nu_{\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)}\\big|_{\\mathsf P_n}.\n\\]\nGiven a partition $\\alpha = (\\alpha_1,\\ldots,\\alpha_{s_\\alpha})$ of $n$, we set \n\\[\nQ_\\alpha = \\prod_{i = 1}^{s_\\alpha}\\Quot_X(\\F,\\alpha_i),\n\\]\nand we let \n\\[\nV_\\alpha \\subset Q_\\alpha\n\\]\nbe the open subscheme parametrising quotient sheaves with pairwise disjoint support. Then, by the results of Section \\ref{quotmess} (cf.~Proposition \\ref{prop:etaleV}), $V_\\alpha$ admits an \\'etale map $f_\\alpha$ to the Quot scheme $\\Quot_X(\\F,n)$, and we let $U_\\alpha$ denote its image. \nNote that $U_\\alpha$ contains the stratum $\\Quot_X(\\F,n)_\\alpha$ as a closed subscheme. We can then form the cartesian square\n\\be\\label{diag1313}\n\\begin{tikzcd}[row sep=large,column sep=normal] \nZ_\\alpha\\MySymb{dr}\\arrow[hook]{r}{} \\arrow[swap]{d}{\\textrm{Galois}} & V_\\alpha \\arrow{d}{f_\\alpha}\\arrow[hook]{r}{\\textrm{open}} & Q_\\alpha\\\\\n\\Quot_X(\\F,n)_\\alpha \\arrow[hook]{r}{} & U_\\alpha\\arrow[hook]{r}{\\textrm{open}} & \\Quot_X(\\F,n)\n\\end{tikzcd}\n\\end{equation}\ndefining $Z_\\alpha$. The leftmost vertical map is finite \\'etale with Galois group $G_\\alpha$, the automorphism group of the partition. We observe that $Z_\\alpha = \\prod_i\\Quot_X(\\F,\\alpha_i)_{(\\alpha_i)}\\setminus \\widetilde\\Delta$, where $\\widetilde\\Delta$ parametrises $s_\\alpha$-tuples of sheaves with intersecting supports. Therefore we have a second fibre square\n\\be\\label{diag6565}\n\\begin{tikzcd}[row sep=large,column sep=normal] \nZ_\\alpha\\MySymb{dr} \\arrow[hook]{r}{} \\arrow[swap]{d}{\\pi_\\alpha} & \n\\prod_i\\Quot_X(\\F,\\alpha_i)_{(\\alpha_i)}\\arrow{d} \\\\\nX^{s_\\alpha}\\setminus \\Delta \\arrow[hook]{r} & X^{s_{\\alpha}}\n\\end{tikzcd}\n\\end{equation}\nwhere the vertical map on the right is the product of punctual Quot-to-Chow morphisms \\eqref{mor:phc} and the horizontal inclusions are open immersions. \n\n\n\\subsection{Calculation}\nRecall that the absolute $\\widetilde\\chi$ is not additive on strata, but that the $\\widetilde\\chi$ relative to a morphism is additive.\nExploiting the diagram \\eqref{diag1313}, we compute\n\n\\begin{align*}\n\\widetilde\\chi(\\Quot_X(\\F,n)) &=\n\\sum_{\\alpha}\\widetilde\\chi\\left(\\Quot_X(\\F,n)_\\alpha,\\Quot_X(\\F,n)\\right) \n& \\textrm{by }\\eqref{ChiSum}\\\\\n&= \\sum_{\\alpha}\\widetilde\\chi\\left(\\Quot_X(\\F,n)_\\alpha,U_\\alpha\\right) \n& \\textrm{by }\\eqref{ChiEtale}\\\\\n&= \\sum_{\\alpha}|G_\\alpha|^{-1}\\widetilde\\chi(Z_\\alpha,V_\\alpha)\n& \\textrm{by }\\eqref{ChiSquare}\\\\\n&= \\sum_{\\alpha}|G_\\alpha|^{-1}\\widetilde\\chi\\left(Z_\\alpha,Q_\\alpha\\right).\n& \\textrm{by }\\eqref{ChiEtale}\\\\\n\\end{align*}\n\nBefore giving an expression of $\\widetilde\\chi(Z_\\alpha,Q_\\alpha)$ via the fibre square \\eqref{diag6565}, we make a few technical observations.\n\n\\begin{lemma}\\label{lemma:punctual}\nWe have a canonical isomorphism\n\\[\n\\mathbb A^3\\times \\mathsf P_n\\,\\widetilde{\\rightarrow}\\,\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)_{(n)}\n\\]\nand the restriction of the Behrend function of $\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)$ to its deepest stratum is the pullback of $\\nu_n$ along the projection to $\\mathsf P_n$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows by standard arguments, see \\cite[Lemma 4.6]{BFHilb} or \\cite[Prop. 3.1]{LocalDT}.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:chiPn}\nOne has $\\chi(\\mathsf P_n,\\nu_{n}) = \\widetilde\\chi(\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)) = (-1)^{rn}\\chi(\\mathsf P_n)$.\n\\end{lemma}\n\n\\begin{proof}\nThe torus action of $(\\mathbb C^{\\times})^3$ on $\\mathbb A^3$ lifts to an action on $\\Quot_{\\mathbb A^3}(\\mathscr O^r,n)$.\nIts torus fixed points are isolated and the punctual Quot scheme $\\mathsf P_n$ is a torus invariant subscheme containing all torus fixed points. Then the result follows from \\cite[Cor.~3.5]{BFHilb}.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lemma:ZLT}\nLet $X$ be a smooth $3$-fold, $\\F$ a locally free sheaf of rank $r$, and denote by $\\nu_Q$ the Behrend function of $\\Quot_X(\\F,n)$. Let $\\sigma_{(n)}$ be the morphism defined in \\eqref{mor:phc}. Then there is a Zariski open cover $(U_i)$ of $X$ such that \n\\[\n\\left(\\sigma_{(n)}^{-1}U_i,\\nu_{Q}\\right) \\isom (U_i,\\mathbb{1}_{U_i})\\times \\left(\\mathsf P_n,\\nu_n\\right)\n\\]\nas schemes with constructible functions on them. \n\\end{lemma}\n\n\\begin{proof}\nThis is a standard argument. See for instance \\cite[Cor.~3.2]{LocalDT}.\n\\end{proof}\n\n\n\n\n\\begin{lemma}\\label{lemma:Qalpha}\nOne has the identity\n\\[\n\\widetilde\\chi(Z_\\alpha,Q_\\alpha)\n=\\chi(X^{s_{\\alpha}}\\setminus \\Delta)\\cdot \\prod_{i=1}^{s_\\alpha}\\chi(\\mathsf P_{\\alpha_i},\\nu_{\\alpha_i}).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemma:ZLT}, we can find a Zariski open cover $X^{s_\\alpha}\\setminus \\Delta = \\bigcup_jB_j$ such that \n\\be\\label{zltFibr}\n\\left(\\pi_\\alpha^{-1}B_j,\\nu_{Q_\\alpha}\\right) = \n(B_j,\\mathbb 1_{B_j}) \\times \\left(\\prod_i \\mathsf P_{\\alpha_i},\\prod_i \\nu_{\\alpha_i}\\right),\n\\end{equation}\nwhere $\\pi_\\alpha\\colon Z_\\alpha\\rightarrow X^{s_\\alpha}\\setminus \\Delta$ is the map that appeared for the first time in \\eqref{diag6565}.\nSuch an open covering can be turned into a locally closed stratification $X^{s_\\alpha}\\setminus \\Delta = \\amalg_{\\ell} U_\\ell$ such that each $U_\\ell$ is contained in some $B_j$. Then we have\n\\begin{align*}\n\\widetilde\\chi(Z_\\alpha,Q_\\alpha)\n&= \\sum_\\ell \\widetilde\\chi\\left(\\pi_\\alpha^{-1}U_\\ell,Q_\\alpha\\right) & \\textrm{by }\\eqref{ChiSum}\\\\\n&= \\sum_\\ell\\chi\\bigl(U_\\ell \\times \\prod_i \\mathsf P_{\\alpha_i},\\mathbb 1_{U_\\ell}\\times \\prod_i \\nu_{\\alpha_i}\\bigr) & \\textrm{by } \\eqref{zltFibr}\\\\\n&= \\sum_\\ell\\chi(U_\\ell,\\mathbb 1_{U_\\ell}) \\cdot \\prod_i\\chi\\left(\\mathsf P_{\\alpha_i},\\nu_{\\alpha_i}\\right) & \\textrm{by }\\eqref{ChiProduct}\\\\\n&= \\chi(X^{s_\\alpha}\\setminus \\Delta) \\cdot \\prod_i\\chi\\left(\\mathsf P_{\\alpha_i},\\nu_{\\alpha_i}\\right),\n\\end{align*}\nand the lemma is proved.\n\\end{proof}\n\nWe are now in a position to finish the computation of $\\widetilde{\\chi}(\\Quot_X(\\F,n))$:\n\\begin{align*}\n\\widetilde\\chi(\\Quot_X(\\F,n)) &=\n\\sum_{\\alpha}|G_\\alpha|^{-1}\\chi(X^{s_{\\alpha}}\\setminus \\Delta)\\cdot \\prod_{i}\\chi\\left(\\mathsf P_{\\alpha_i},\\nu_{\\alpha_i}\\right) & \\textrm{by Lemma }\\ref{lemma:Qalpha} \\\\\n&=\\sum_{\\alpha}|G_\\alpha|^{-1}\\chi(X^{s_{\\alpha}}\\setminus \\Delta)\\cdot \\prod_{i}\\,(-1)^{r\\alpha_i}\\chi\\left(\\mathsf P_{\\alpha_i}\\right) & \\textrm{by Lemma }\\ref{lemma:chiPn}\\\\\n&=(-1)^{rn}\\sum_{\\alpha}|G_\\alpha|^{-1}\\chi(X^{s_{\\alpha}}\\setminus \\Delta)\\cdot \\prod_{i}\\chi\\left(\\mathsf P_{\\alpha_i}\\right) \\\\\n&=(-1)^{rn}\\chi(\\Quot_X(\\F,n)).\n\\end{align*}\nBy virtue of formula \\eqref{GKformula} of Gholampour--Kool \\cite[Thm.~1.1]{Gholampour2017}, we conclude that for a locally free sheaf $\\F$ on a smooth quasi-projective $3$-fold $X$ one has\n\\begin{equation}\\label{eq:Thm_Locally_Free_Case}\n\\sum_{n\\geq 0}\\widetilde\\chi(\\Quot_X(\\F,n)) q^n = \\mathsf M((-1)^rq)^{r\\chi(X)}.\n\\end{equation}\nThis completes the proof of Theorem \\ref{Thm:LocallyFree}.\n\n\n\n\n\n\\section{Tools for the proof of Theorem \\ref{thm2}}\nThis section contains the technical preliminaries we will need in Section \\ref{Sec:dtpt} for the proof of Theorem \\ref{thm2}. We mainly follow \\cite{Toda2}.\nThe key objects are the following.\n\\begin{itemize}\n\\item [(i)] Moduli spaces of torsion free sheaves and of PT pairs on a smooth projective $3$-fold $X$. These can be seen as parametrising stable objects in a suitable heart \n\\[\n\\mathscr A_\\mu\\subset D^b(X)\n\\]\nof a bounded t-structure on the derived category of $X$.\n\\item [(ii)] The Hall algebra of the abelian category $\\mathscr A_\\mu$ and the associated (Behrend weighted, unweighted) integration maps.\n\\end{itemize}\nThroughout, let $X$ be a smooth projective $3$-fold, not necessarily Calabi--Yau.\nWe fix an ample class $\\omega$ on $X$, an integer $r\\geq 1$, and a divisor class $D\\in H^2(X,\\mathbb Z)$, satisfying the coprimality condition\n\\be\\label{coprime}\n\\gcd(r,D\\cdot \\omega^2) = 1.\n\\end{equation}\n\n\n\n\\subsection{Moduli of sheaves and PT pairs}\\label{sec:Moduli}\nSlope stability with respect to $\\omega$ is defined in terms of the slope function $\\mu_\\omega$, attaching to a torsion free coherent sheaf $E$ the ratio\n\\[\n\\mu_\\omega(E) = \\frac{c_1(E)\\cdot \\omega^2}{\\rk E}\\in \\mathbb Q\\cup\\set{\\infty}.\n\\]\nThe coherent sheaf $E$ is $\\mu_{\\omega}$-stable or slope-stable if the strict inequality\n\\[\n\\mu_{\\omega}(S) < \\mu_{\\omega}(E)\n\\] \nholds for all proper non-trivial subsheaves $0 \\neq S \\subsetneq E$ with $0 < \\rk(S) < \\rk(E)$.\nThe sheaf is slope-semistable if the same condition holds with $<$ replaced by $\\leq$.\nNote that condition \\eqref{coprime} implies that any slope-semistable sheaf is slope-stable and, hence, that the notions of slope and Gieseker stability coincide; see~\\cite{modulisheaves}.\nIn particular, the coarse moduli space\n\\be\\label{MDT}\nM_{\\DT}(r,D,-\\beta,-m)\n\\end{equation}\nof $\\mu_\\omega$-stable sheaves of Chern character $(r,D,-\\beta,-m)$ is a projective scheme \\cite{modulisheaves}.\n\nIf $X$ is Calabi--Yau, $M_{\\DT}(r,D,-\\beta,-m)$ carries a symmetric perfect obstruction theory and hence a (zero-dimensional) virtual fundamental class by \\cite{ThomasThesis}.\nBy the main result of \\cite{Beh}, the associated DT invariant\n\\be\\label{DThigherrank}\n\\mathsf{DT}(r,D,-\\beta,-m) = \\int_{\\left[M_{\\DT}(r,D,-\\beta,-m)\\right]^{\\vir}}1 \\in \\mathbb Z,\n\\end{equation}\ncoincides with Behrend's virtual Euler characteristic $\\widetilde\\chi(M_{\\DT}(r,D,-\\beta,-m))$.\n\nThe notion of higher rank PT pair originated in the work of Lo \\cite{Lo}, and was revisited by Toda \\cite{Toda2}; also see the example \\cite[Ex.~3.4]{Toda2}.\nThe definition is the following.\n\\begin{definition}[{\\cite[Def.~3.1]{Toda2}}]\\label{ptpair}\n\tA \\emph{PT pair} on $X$ is a two-term complex $J^\\bullet \\in D^b(X)$ such that\n \\begin{enumerate}\n \t\\item $H^i(J^\\bullet) = 0$ if $i \\neq 0,1$,\n \t\\item $H^0(J^\\bullet)$ is $\\mu_{\\omega}$-(semi)stable and $H^1(J^\\bullet)$ is zero-dimensional,\n \\item $\\Hom(Q[-1],J^\\bullet) = 0$ for every zero-dimensional sheaf $Q$.\n \\end{enumerate}\nWe say that a PT pair $J^\\bullet$ is $\\F$-\\emph{local}, or \\emph{based at} $\\F$, if $H^0(J^\\bullet)=\\F$.\n\\end{definition}\nBy \\cite[Thm.~1.2]{Lo}, the coarse moduli space\n\\be\\label{MPT}\nM_{\\PT}(r,D,-\\beta,-m)\n\\end{equation}\nparametrising PT pairs with the indicated Chern character is a proper algebraic space of finite type.\nIf $X$ is in addition Calabi--Yau, the PT moduli space carries a symmetric perfect obstruction theory by the results of Huybrechts--Thomas \\cite{HT}.\nThe PT invariant is defined, just as \\eqref{DThigherrank}, by integration against the associated virtual class.\nSimilarly to the DT case, by \\cite{Beh} it coincides with Behrend's virtual Euler characteristic\n\\[\n\\mathsf{PT}(r,D,-\\beta,-m) = \\widetilde\\chi(M_{\\PT}(r,D,-\\beta,-m)).\n\\]\n\nFor later purposes, we recall the notion of \\emph{family} of PT pairs.\n\\begin{definition}\\label{def:familyPT}\nA \\emph{family of PT pairs} parametrised by a scheme $B$ is a perfect complex $J^\\bullet \\in D^b(X\\times B)$ such that, for every $b\\in B$,\nthe derived restriction of $J^\\bullet$ to $X\\times \\set{b}\\subset X\\times B$ is a PT pair.\n\\end{definition}\n\nNote that no additional flatness requirement is imposed; see Remark~\\ref{flatness} for a brief discussion on \\emph{families of objects} in a heart of a bounded t-structure on $D^b(X)$.\n\n\nWe define the moduli spaces\n\\be\\label{moduli}\nM_{\\DT}(r,D),\\quad M_{\\PT}(r,D)\n\\end{equation}\nof $\\mu_\\omega$-stable sheaves and PT pairs, respectively, as the (disjoint) unions of the moduli spaces \\eqref{MDT} and \\eqref{MPT}, over all $(\\beta,m)\\in H^4(X)\\oplus H^6(X)$.\nElements of these moduli spaces will be called DT and PT objects respectively.\n\n\n\\begin{remark}\\label{rem:stacks_gerbes}\nThe moduli \\emph{stacks} of DT and PT objects \n\\be\\label{DTPT_stacks}\n\\mathcal M_{\\DT}(r,D,-\\beta,-m),\\quad \\mathcal M_{\\PT}(r,D,-\\beta,-m)\n\\end{equation}\nalso enter the picture in Toda's proof of the DT\/PT correspondence in the key Hall algebra identity \\cite[Lemma 3.16]{Toda2} underlying his wall-crossing formula.\nBy the coprimality assumption \\eqref{coprime}, each sheaf or PT pair parametrised by these stacks has automorphism group $\\mathbb G_m$. Thus, the stacks $\\mathcal M_{\\DT}(r,D)$ and $\\mathcal M_{\\PT}(r,D)$, defined as the disjoint unions of the stacks \\eqref{DTPT_stacks} over all $(\\beta,m)$, are $\\mathbb G_m$-gerbes over their coarse moduli spaces \\eqref{moduli}.\n\n\\end{remark}\n\n\n\n\n\n\n\n\\subsubsection{A characterisation of PT pairs}\nLet $C\\subset X$ be a curve, where $X$ is a Calabi--Yau $3$-fold. The formulation of the $C$-local DT\/PT correspondence requires $C$ to be Cohen--Macaulay. This is equivalent to the ideal sheaf $\\mathscr I_C\\subset \\mathscr O_X$ being both a DT and a PT object.\nSimilarly, an $\\mathcal F$-local (higher rank) DT\/PT correspondence requires the sheaf $\\mathcal F$ to be both a DT object and a PT object. This assumption will be crucial, for instance, in the proof of Lemma \\ref{Closed_Points_13}.\n\nWe introduce the following definition, which allows us to recognise a PT pair more easily and to describe the intersection of the spaces \\eqref{moduli} of DT and PT objects.\n\n\\begin{definition}\nA complex $J^\\bullet \\in D^b(X)$ satisfying properties (1) and (2) in Definition \\ref{ptpair} is called a \\emph{pre-PT pair}.\n\\end{definition}\n\nWhether or not a pre-PT pair is a PT pair depends on the vanishing of a particular local Ext sheaf.\nTo see this, recall that the derived dual $(-)^{\\vee} = {\\textbf R}\\lHom(-,\\mathscr O_X)$ is a triangulated anti-equivalence of $D^b(X)$ that restricts to an anti-equivalence of abelian categories\n\\[\n(-)^{\\vee}\\colon \\Coh_0(X)\\,\\widetilde{\\rightarrow}\\,\\Coh_0(X)[-3],\\qquad Q\\mapsto Q^{\\textrm{D}}[-3],\n\\]\nwhere $Q^{\\textrm{D}} = \\lExt^3(Q,\\mathscr O_X)$ is the usual dual of a zero-dimensional sheaf on a $3$-fold.\nIn particular, we also have an anti-equivalence $\\Coh_0(X)[-1]\\,\\widetilde{\\rightarrow}\\,\\Coh_0(X)[-2]$.\n\\begin{lemma}\\label{lem:PrePT_is_PT}\nLet $J^\\bullet$ be a pre-PT pair.\nThen $J^\\bullet$ is a PT pair if and only if $\\lExt^2(J^\\bullet,\\mathscr O_X)=0$.\n\\end{lemma}\n\\begin{proof}\nThere is a natural truncation triangle\n\\be\\label{triangle}\n\\tau_{\\leq 1}(J^{\\bullet \\vee})\\rightarrow J^{\\bullet \\vee}\\rightarrow\n\\lExt^2(J^\\bullet,\\mathscr O_X)[-2],\n\\end{equation}\nwhere $\\tau_{\\leq 1}(J^{\\bullet \\vee})$ and $J^{\\bullet \\vee}$ live in degrees $[0,1]$ and $[0,2]$ respectively.\nNote that the rightmost object is a (shifted) zero-dimensional sheaf because $H^0(J^{\\bullet})$ is torsion free and $H^1(J^{\\bullet})$ is zero-dimensional.\nThe dualising involution yields\n\\[\n\\Hom(Q[-1],J^\\bullet) = \\Hom(J^{\\bullet \\vee},Q^{\\textrm{D}}[-2])\n\\]\nfor every zero-dimensional sheaf $Q \\in \\Coh_0(X)$.\nBut we also have\n\\[\n\\Hom(J^{\\bullet \\vee},Q^{\\textrm{D}}[-2]) = \\Hom(\\lExt^2(J^{\\bullet},\\mathscr O_X),Q^{\\textrm{D}})\n\\]\nby \\eqref{triangle}, since there are no maps from higher to lower degree in $D^b(X)$.\nIn conclusion, it follows that $J^{\\bullet}$ is a PT pair if and only if $\\lExt^2(J^{\\bullet},\\mathscr O_X) = 0$ as claimed.\n\\end{proof}\n\n\\begin{example}\nIn the case of a classical PT pair $I^\\bullet=[\\mathscr O_X\\rightarrow F]$, where $F$ is a pure one-dimensional sheaf, the above vanishing can be deduced directly from the existence of the triangle $I^\\bullet\\rightarrow \\mathscr O_X\\rightarrow F$.\nIndeed, it induces an exact sequence\n\\[\n\\ldots \\to \\lExt^2(\\mathscr O_X,\\mathscr O_X)\\rightarrow \\lExt^2(I^\\bullet,\\mathscr O_X)\\rightarrow \n\\lExt^3(F,\\mathscr O_X) \\to \\ldots\n\\]\nin which the outer two terms vanish, the latter by purity of $F$.\n\\end{example}\n\nThe following corollary explains the assumptions on the sheaf $\\mathcal{F}$ made in \\cite{Gholampour2017}, and the assumptions required to have an $\\F$-local DT\/PT correspondence.\n\\begin{corollary}\\label{cor:DTPT_object}\nLet $\\mathcal F$ be a coherent sheaf on $X$.\nThen $\\mathcal F$ is both a DT and a PT object if and only if $\\mathcal F$ is a $\\mu_{\\omega}$-stable sheaf of homological dimension at most one.\n\\end{corollary}\n\\begin{proof}\nThe coherent sheaf $\\F$ is $\\mu_{\\omega}$-stable if and only if it is a DT object.\nIt is then torsion free, so $\\lExt^{\\geq 3}(\\mathcal F,\\mathscr O_X) = 0$.\nIn addition, $\\F$ has homological dimension at most one if and only if $\\lExt^{\\geq 2}(\\mathcal F,\\mathscr O_X)=0$, but this holds if and only if it is a PT object by Lemma~\\ref{lem:PrePT_is_PT}.\n\\end{proof}\n\n\\begin{remark}\\label{rmk:vanishingext}\nCorollary \\ref{cor:DTPT_object} implies that for every smooth projective $3$-fold $X$, for every $\\mu_{\\omega}$-stable torsion free sheaf $\\mathcal F$ of homological dimension at most one, and for every zero-dimensional sheaf $Q$, the only possibly non-vanishing Ext groups between $\\mathcal F$ and $Q$ are\n\\[\n\\Hom(\\mathcal F,Q)\\isom \\Ext^3(Q,\\mathcal F)^\\ast,\\qquad \\Ext^1(\\mathcal F,Q)\\isom \\Ext^2(Q,\\mathcal F)^\\ast.\n\\]\nThis in fact holds without the stability assumption by \\cite[Lemma 4.1]{Gholampour2017}.\nAs a consequence, one has $\\hom(\\mathcal F,Q)-\\mathrm{ext}^1(\\mathcal F,Q)=r\\cdot \\ell(Q)$.\nNote that the last relation reads $\\mathrm{ext}^1(\\F,Q[-1]) - \\mathrm{ext}^1(Q[-1],\\F) = r \\cdot \\ell(Q)$, which is reminiscent of the DT\/PT wall-crossing.\n\\end{remark}\n\n\n\n\\subsection{Hall algebras and integration maps}\nWe recall from \\cite[Sec.~2.5,~3.4]{Toda2} the Hall algebra of a suitable heart $\\catA_\\mu$ of a bounded t-structure on $D^b(X)$. \nTo introduce it, recall that a \\emph{torsion pair} \\cite{MR1327209} on an abelian category $\\catA$ is a pair of full subcategories $(\\catT,\\catF)$ such that\n\\begin{enumerate}\n \\item for $T \\in \\catT$ and $F \\in \\catF$, we have $\\Hom(T,F) = 0$,\n \\item for every $E \\in \\catA$, there exists a short exact sequence\n \\[\n 0 \\to T_E \\to E \\to F_E \\to 0\n \\]\n in $\\catA$ with $T_E \\in \\catT$ and $F_E \\in \\catF$. This sequence is unique by (1).\n\\end{enumerate}\n\nRecall also that we have fixed an ample class $\\omega$ on the $3$-fold $X$ and a pair $(r,D)$ satisfying the coprimality condition \\eqref{coprime}.\nDefine the number\n\\[\n\\mu = \\frac{D\\cdot \\omega^2}{r}\\in \\mathbb Q.\n\\]\nToda \\cite{Toda2} considers the torsion pair\n\\[\n\\Coh(X) = \\langle \\Coh_{>\\mu}(X),\\Coh_{\\leq\\mu}(X) \\rangle,\n\\]\nwhere for an interval $I\\subset \\mathbb R\\cup\\set{\\infty}$, the category $\\Coh_I(X)$ is the extension-closure of the $\\mu_\\omega$-semistable objects $E\\in \\Coh(X)$ with slope $\\mu_\\omega(E)\\in I$, along with the zero sheaf.\nTilting $\\Coh(X)$ at the above torsion pair produces the heart\n\\[\n\\mathscr A_\\mu = \\langle \\Coh_{\\leq\\mu}(X),\\Coh_{>\\mu}(X)[-1]\\rangle\n\\]\nof a bounded t-structure on $D^b(X)$.\nWe summarise the required properties of $\\catA_{\\mu}$.\n\\begin{lemma}\nThe category $\\catA_{\\mu}$ satisfies the following properties.\n\\begin{enumerate}\n \\item An object $E \\in \\catA_{\\mu}$ is a two-term complex in $D^b(X)$ such that\n \\[\n H^0(E) \\in \\Coh_{\\leq \\mu}(X), \\quad H^1(E) \\in \\Coh_{> \\mu}(X), \\quad \\text{and } H^i(E) = 0 \\text{ otherwise}.\n \\]\n \\item $\\catA_{\\mu}$ is a noetherian and abelian category,\n \\item $\\catA_{\\mu}$ contains the noetherian and abelian full subcategory\n \\[\n \\catB_\\mu = \\langle \\Coh_{\\mu}(X),\\Coh_{\\leq 1}(X)[-1]\\rangle \\subset \\catA_\\mu,\n \\]\n where $\\Coh_{\\leq 1}(X)$ consists of sheaves with support of dimension at most one,\n \\item the category $\\Coh_{\\leq 1}(X) \\subset \\catA_{\\mu}$ is closed under subobjects, extensions, and quotients.\n\\end{enumerate}\n\\end{lemma}\n\n\nLet $\\mathscr M_X$ be Lieblich's moduli stack of semi-Schur objects on $X$; it is an algebraic stack locally of finite type \\cite{Lieblich1}. Recall that a complex $E\\in D^b(X)$ is a semi-Schur object if $\\Ext^i(E,E) = 0$ for all $i < 0$ (see loc.~cit.~for more details).\nBecause $\\mathscr A_\\mu$ is the heart of a bounded t-structure, its objects have no negative self-extensions.\nConsider the substack of $\\mathscr M_X$ parametrising objects in $\\catA_{\\mu}$.\nIt defines an open substack of $\\mathscr M_X$, which we abusively\\footnote{If $\\mathscr C$ is a subcategory of $D^b(X)$ whose objects have no negative self-extensions, we abuse notation and denote the corresponding moduli stack, a substack of $\\mathscr M_X$, by $\\mathscr C$ if it exists.} still denote by $\\catA_{\\mu}$. The category $\\mathscr B_\\mu$ defines, in turn, an open substack of $\\mathscr A_\\mu$. The moduli stacks \\eqref{DTPT_stacks} both carry an open immersion to $\\mathscr B_\\mu$, cf.~\\cite[Remark 3.8]{Toda2}. \n\n\n\\begin{remark}\\label{flatness}\nLet $B$ be a base scheme.\nSince $X$ is smooth and $\\catA_{\\mu} \\subset \\mathscr M_X$ is open, a $B$-valued point of $\\catA_{\\mu}$ is a perfect complex $E$ on $X \\times B$ such that $E_b = \\textbf{L} i_b^*(E) \\in \\catA_{\\mu}$ for every closed point $b \\in B$. (Here $i_b\\colon X\\times \\set{b} \\hookrightarrow X\\times B$ is the natural inclusion.)\nNote that no further flatness assumption is required.\n\nIndeed, Bridgeland shows in \\cite[Lemma~4.3]{Bri2} that a complex $E \\in D^b(X \\times B)$ is a (shifted) sheaf \\emph{flat over $B$} if and only if each \\emph{derived} fibre $E_b \\in \\Coh(X)$ is a sheaf.\nIn other words, flatness of $E$ over $B$ is encoded in the vanishing $H^i(E_b) = 0$ for all $i < 0$.\nThus the right notion of a $B$-family of objects in $\\catA_{\\mu}$ is given by a perfect complex $E$ on $X \\times B$ such that $H^i_{\\catA_{\\mu}}(E_b) = 0$ for all $i \\neq 0$ and for all closed points $b \\in B$. This means precisely that $E_b \\in \\catA_{\\mu}$.\n\\end{remark}\n\n\\subsubsection{Motivic Hall algebra}\nWe recall the definition of the motivic Hall algebra from \\cite{Bri-Hall}.\nLet $S$ be an algebraic stack that is \\emph{locally} of finite type with affine geometric stabilisers.\nA (representable) morphism of stacks is a \\emph{geometric bijection} if it induces an equivalence on $\\mathbb C$-valued points. It is a \\emph{Zariski fibration} if its pullback to any scheme is a Zariski fibration of schemes.\n\\begin{definition}\\label{def:Relative_Grothendieck_Group_of_Stacks}\nThe $S$-relative Grothendieck group of stacks is the $\\mathbb Q$-vector space $K(\\mathsf{St}\/S)$ generated by symbols $[T \\to S]$, where $T$ is a \\emph{finite type} algebraic stack over $\\mathbb C$ with affine geometric stabilisers, modulo the following relations.\n\t\\begin{enumerate}\n\t\t\\item For every pair of $S$-stacks $f_1, f_2 \\in \\mathsf{St}\/S$, we have \n\t\t\t\\begin{equation*}\n\t\t\t\t[T_1 \\sqcup T_2 \\xrightarrow{f_1 \\sqcup f_2} S] =\n\t\t\t\t\t[T_1 \\xrightarrow{f_1} S] + [T_2 \\xrightarrow{f_2} S].\n\t\t\t\\end{equation*}\n\t\t\\item For every geometric bijection $T_1 \\to T_2$ of $S$-stacks, we have\n\t\t\t\\begin{equation*}\n\t\t\t\t[T_1 \\xrightarrow{f_1} S] = [T_2 \\xrightarrow{f_2} S].\n\t\t\t\\end{equation*}\n\t\t\\item For every pair of Zariski locally trivial fibrations $f_i \\colon T_i \\to Y$\n\t\t\twith the same fibres and every morphism $g \\colon Y \\to S$, we have\n\t\t\t\\begin{equation*}\n\t\t\t\t[T_1 \\xrightarrow{g \\circ f_1} S] = [T_2 \\xrightarrow{g \\circ f_2} S].\n\t\t\t\\end{equation*}\n\t\\end{enumerate}\n\\end{definition}\n\\begin{remark}\nThe last relation plays no further role in this paper.\n\\end{remark}\n \nAs a $\\mathbb Q$-vector space, the \\emph{motivic Hall algebra} of $\\catA_{\\mu}$ is \n\\[\nH(\\catA_{\\mu}) = K(\\mathsf{St}\/\\catA_{\\mu}).\n\\]\nWe now define the product $\\star$ on $H(\\mathscr A_\\mu)$.\nLet $\\mathscr A_\\mu^{(2)}$ be the stack of short exact sequences in \n$\\mathscr A_\\mu$, and let \n\\[\np_i\\colon \\mathscr A_\\mu^{(2)}\\rightarrow \\mathscr A_\\mu,\\qquad i=1,2,3,\n\\]\nbe the $1$-morphism sending a short exact sequence $0\\rightarrow E_1\\rightarrow E_3\\rightarrow E_2\\rightarrow 0$ to $E_i$.\n\nThe following is shown in \\cite[App.~B]{BCR} for every heart of a bounded t-structure on $D^b(X)$ whose moduli stack is open in $\\mathscr M_X$, where $X$ is any smooth projective variety.\n\\begin{prop}\nThe stack $\\catA_{\\mu}^{(2)}$ is an algebraic stack that is locally of finite type over $\\mathbb C$.\nThe morphism $(p_1,p_2) \\colon \\catA_{\\mu}^{(2)} \\to \\catA_{\\mu} \\times \\catA_{\\mu}$ is of finite type.\n\\end{prop}\nGiven two $1$-morphisms $f_i\\colon \\mathscr X_i\\rightarrow \\mathscr A_\\mu$, consider the diagram of stacks\n\\be\\label{HAproduct}\n\\begin{tikzcd\n\\mathscr X_1\\star \\mathscr X_2\\MySymb{dr} \\arrow{r}{f}\\arrow{d} &\n\\mathscr A_\\mu^{(2)}\\arrow{r}{p_3}\\arrow{d}{(p_1,p_2)} & \n\\mathscr A_\\mu \\\\\n\\mathscr X_1\\times \\mathscr X_2\\arrow[swap]{r}{f_1\\times f_2} & \n\\mathscr A_\\mu\\times \\mathscr A_\\mu &\n\\end{tikzcd}\n\\end{equation}\nwhere the square is cartesian.\nThen one defines\n\\[\n\\left[f_1\\colon \\mathscr X_1\\rightarrow \\mathscr A_\\mu\\right]\\star \\left[f_2\\colon\\mathscr X_2\\rightarrow \\mathscr A_\\mu\\right] = \n\\left[p_3\\circ f\\colon \\mathscr X_1\\star \\mathscr X_2\\rightarrow \\mathscr A_\\mu\\right].\n\\]\nAs a consequence of the previous proposition, the stack $\\mathscr X_1 \\star \\mathscr X_2$ is algebraic and of finite type.\nIt also has affine geometric stabilisers, and thus defines an element of $H(\\catA_{\\mu})$.\nThe unit is given by the class $1 = [\\Spec\\mathbb C \\to \\catA_{\\mu}]$ corresponding to the zero object $0 \\in \\catA_{\\mu}$.\n\\begin{theorem}\nThe triple $(H(\\catA_{\\mu}),\\star,1)$ defines a unital associative algebra.\n\\end{theorem}\n\\begin{proof}\nThe proof of \\cite[Thm.~4.3]{Bri-Hall} goes through without change.\n\\end{proof}\nLet $\\Gamma\\subset H^\\ast(X,\\mathbb Q)$ be the image of the Chern character map.\nIt is a finitely generated free abelian group.\nThe stack $\\catA_{\\mu}$ decomposes as a disjoint union into open and closed substacks \n\\[\n\\catA_{\\mu} = \\coprod_{v \\in \\Gamma} \\catA_{\\mu}^{(v)}\n\\]\nwhere $\\catA_{\\mu}^{(v)}$ is the substack of objects of Chern character $v$.\nThe Hall algebra is $\\Gamma$-graded\n\\[\nH(\\catA_{\\mu}) = \\bigoplus_{v \\in \\Gamma} H_v(\\catA_{\\mu})\n\\]\nwhere $H_v(\\catA_{\\mu})$ is spanned by classes of maps $[\\mathscr X \\to \\catA_{\\mu}]$ that factor through $\\catA_{\\mu}^{(v)} \\subset \\catA_{\\mu}$.\n\n\\subsubsection{Integration morphism}\nFor simplicity, we denote a symbol $[T \\to \\Spec\\mathbb C]$ in the Grothendieck group $K(\\mathsf{St}\/\\mathbb C)$ by $[T]$.\nThis group has a natural commutative ring structure induced by the fibre product of stacks $[T] \\cdot [U] = [T \\times_{\\mathbb C} U]$.\nIn turn, the Hall algebra has a natural structure of $K(\\mathsf{St}\/\\mathbb C)$-module where the action is given by\n\\[\n[T] \\cdot [U \\xrightarrow{f} \\catA_{\\mu}] = [T \\times U \\xrightarrow{f \\circ \\text{pr}_2} \\catA_{\\mu}].\n\\]\nThe ring $K(\\mathsf{St}\/\\mathbb C)$ is obtained from the classical Grothendieck ring of varieties $K(\\mathsf{Var}\/\\mathbb C)$ by localising at the classes of special algebraic groups \\cite[Lemma 3.8]{Bri-Hall}.\nThere is a subring\n\\[\n\\Lambda = K(\\mathsf{Var}\/\\mathbb C)[\\mathbb{L}^{-1},(1+\\mathbb{L}+\\cdots+\\mathbb{L}^k)^{-1} \\colon k \\geq 1] \\subset K(\\mathsf{St}\/\\mathbb C),\n\\]\nwhere $\\mathbb{L} = [\\mathbb A^1]$ is the Lefschetz motive.\nNote that $\\Lambda$ does not contain $[B\\mathbb C^{\\ast}] = (\\mathbb{L}-1)^{-1}$.\n\nBy \\cite[Thm.~5.1]{Bri-Hall}, the $\\Lambda$-submodule\\footnote{The classes $1+\\mathbb{L}+\\cdots+\\mathbb{L}^k$ of the projective spaces $\\mathbb P^k$ have to be inverted in the definition of the regular Hall subalgebra; see the corrected version \\cite{Bri} available on the arXiv.} \n\\[\nH^{\\textrm{reg}}(\\catA_{\\mu})\\subset H(\\catA_{\\mu})\n\\]\ngenerated by \\emph{regular} elements is closed under the $\\star$ product, where a regular element is an element in the span of the classes $[f \\colon T \\to \\catA_{\\mu}]$ where $T$ is a variety.\nMoreover, the quotient \n\\[\nH^{\\textrm{sc}}(\\mathscr A_\\mu) = H^{\\textrm{reg}}(\\mathscr A_\\mu)\/(\\mathbb{L}-1)\\cdot H^{\\textrm{reg}}(\\mathscr A_\\mu)\n\\]\nof the regular subalgebra $H^{\\textrm{reg}}(\\mathscr A_\\mu)\\subset H(\\mathscr A_\\mu)$ by the ideal generated by $\\mathbb{L}-1$ is a commutative algebra called the \\emph{semi-classical limit}.\nIt is equipped with an induced Poisson bracket\n\\begin{equation}\\label{eq:Poisson_Bracket}\n\\Set{f,g} = \\frac{f\\star g-g\\star f}{\\mathbb{L}-1}\n\\end{equation}\nwith respect to $\\star$, where $f,g \\in H^{\\textrm{sc}}(\\catA_{\\mu})$.\nThe semi-classical limit is the domain of the integration map.\n\nThe codomain of the integration map is the \\emph{quantum torus}.\nThere are two versions depending on the choice of a sign $\\sigma \\in \\{\\pm 1\\}$.\nEither is defined as the $\\mathbb Q$-vector space\n\\[\nC_{\\sigma}(X) = \\bigoplus_{v\\in \\Gamma} \\mathbb Q\\cdot c_v\n\\]\ngenerated by elements $\\Set{c_v \\mid v \\in \\Gamma}$ and equipped with a product given by the rule\n\\[\nc_{v_1}\\star c_{v_2} = \\sigma^{\\chi(v_1,v_2)}\\cdot c_{v_1+v_2}.\n\\]\nThe quantum torus is a Poisson algebra with respect to the bracket\n\\[\n\\Set{c_{v_1},c_{v_2}} = \\sigma^{\\chi(v_1,v_2)}\\chi(v_1,v_2)\\cdot c_{v_1+v_2}.\n\\]\n\n\\begin{remark}\nNote that the product is commutative when $\\sigma = +1$.\nIf $X$ is Calabi--Yau the Euler pairing is anti-symmetric, so the product is also commutative when $\\sigma = -1$.\n\\end{remark}\nThere are two integration morphisms $H^{\\textrm{sc}}(\\mathscr A_\\mu)\\rightarrow C_{\\sigma}(X)$ defined by \n\\be\\label{integrationmaps}\n\\left[f\\colon Y\\rightarrow \\mathscr A_\\mu\\right] \\,\\,\\mapsto\\,\\,\n\\begin{cases}\n\\chi(Y) \\cdot c_v, & \\textrm{if }\\sigma = 1 \\\\\n\\chi(Y,f^\\ast \\nu) \\cdot c_v, & \\textrm{if }\\sigma = -1,\n\\end{cases}\n\\end{equation}\nwhenever $f$ factors through the open and closed substack $\\catA_{\\mu}^{(v)}\\subset \\catA_{\\mu}$ for some $v\\in \\Gamma$, and where $\\nu$ denotes the Behrend weight on $\\mathscr A_\\mu$.\nFor simplicity, we denote these group homomorphisms by $I^E$ (when $\\sigma=1$) and $I^B$ (when $\\sigma=-1$) for Euler and Behrend respectively.\n\n\\begin{remark}\\label{rem:Integration_Morphisms}\nThe result \\cite[Thm.~5.2]{Bri-Hall} gives conditions which guarantee that $I^E$ and $I^B$ are morphisms of commutative or Poisson algebras.\nWe summarise these here.\n\\begin{enumerate}\n\\item If $X$ is any smooth projective $3$-fold, then $I^E$ is a morphism of commutative algebras.\nIt is not a morphism of Poisson algebras in general.\nHowever, if $[f \\colon Y \\to \\catA_{\\mu}] \\in H^{\\textrm{sc}}(\\catA_{\\mu})$ factors through $\\catA_{\\mu}^{(v_0)}$ where $v_0 \\in \\Gamma$ is the Chern character of a zero-dimensional object, then\n\\[\nI^E\\left(\\{f,g\\}\\right) = \\left\\{I^E(f),I^E(g)\\right\\}\n\\]\nholds for all $g \\in H^{\\textrm{sc}}(\\catA_{\\mu})$.\nThis follows from the proof of \\cite[Thm.~5.2]{Bri-Hall}, because Serre duality between a zero-dimensional object $Z$ and any object $E$ in $D^b(X)$ reduces to the Calabi--Yau condition $\\Ext^i(Z,E) \\cong \\Ext^{3-i}(E,Z)$ for all $i \\in \\mathbb Z$.\n\\item If $X$ is Calabi--Yau, then both $I^E$ and $I^B$ are morphism of Poisson algebras.\n\\end{enumerate}\n\\end{remark}\n\nFinally, we need a completed version of both the Hall algebra and the quantum torus.\nToda constructs in \\cite[Sec.~3.4]{Toda2} a completed Hall algebra\n\\[\n\\widehat H_{\\#}(\\mathscr A_{\\mu}) = \\prod_{v \\in \\Gamma_{\\#}} H_v(\\mathscr A_{\\mu}) = \\prod_{v \\in \\Gamma_{\\#}}H_v(\\Coh_{\\leq 1}(X)[-1]),\n\\]\nwhere $\\Gamma_{\\#} = \\set{(0,0,-\\gamma,-m) \\in \\Gamma \\mid \\gamma \\geq 0, m \\geq 0}$ can be seen as the set of Chern characters of objects in $\\mathscr A_\\mu$ belonging to $\\Coh_{\\leq 1}(X)[-1]$, cf.~\\cite[Remark 3.11]{Toda2}.\nMoreover, for each $(r,D)$ satisfying \\eqref{coprime}, he constructs a bimodule over this algebra,\n\\[\n\\widehat H_{r,D}(\\mathscr A_\\mu) = \\prod_{v\\in \\Gamma_{r,D}}H_v(\\mathscr A_\\mu),\n\\]\nwhere $\\Gamma_{r,D} \\subset \\Gamma$ is a suitably bounded subset\\footnote{See \\cite[Lemma~3.9]{Toda2}, which implies that the product and bracket extend to these completions.} of admissible Chern characters of the form $(r,D,-\\beta,-n)$. \nThere is a corresponding $\\Lambda$-submodule of regular elements\n\\[\n\\widehat H_{r,D}^{\\textrm{reg}}(\\mathscr A_\\mu)\\subset \\widehat H_{r,D}(\\mathscr A_\\mu).\n\\]\nThe semi-classical limit $\\widehat H_{r,D}^{\\textrm{sc}}(\\mathscr A_\\mu)$ is defined as the quotient of $\\widehat H_{r,D}^{\\textrm{reg}}(\\mathscr A_\\mu)$ by the submodule generated by $\\mathbb{L}-1$.\nThese are bimodules over the corresponding algebras $\\widehat H_{\\#}^{\\textrm{sc}}(\\mathscr A_\\mu)$ and $\\widehat H_{\\#}^{\\textrm{reg}}(\\mathscr A_\\mu)$ with respect to $\\star$-multiplication on the left and on the right.\n\nThere are (Behrend weighted and unweighted) integration morphisms induced by \\eqref{integrationmaps},\n\\begin{equation}\\label{intmap}\n\\begin{split}\nI_{r,D} &\\colon \\widehat H_{r,D}^{\\textrm{sc}}(\\catA_{\\mu}) \\to \\widehat{C}_{r,D}(\\catA_{\\mu}) = \\prod_{v\\in \\Gamma_{r,D}}C_v(X), \\\\\nI_{\\#} &\\colon \\widehat H_{\\#}^{\\textrm{sc}}(\\catA_{\\mu}) \\to \\widehat{C}_{\\#}(\\catA_{\\mu}) = \\prod_{v \\in \\Gamma_{\\#}} C_v(X),\n\\end{split}\n\\end{equation}\n to the completed quantum tori.\nThese are compatible with the Poisson-bimodule structure of $\\widehat H_{r,D}^{\\textrm{sc}}(\\catA_{\\mu})$ over $\\widehat H_{\\#}^{\\textrm{sc}}(\\catA_{\\mu})$ and $\\widehat{C}_{r,D}(\\catA_{\\mu})$ over $\\widehat{C}_{\\#}(\\catA_{\\mu})$ in the following sense:\n\\begin{equation}\\label{eq:Poisson_bimodule_morphism}\nI_{r,D}\\left(\\{a,x\\}\\right) = \\left\\{ I_{\\#}(a),I_{r,D}(x) \\right\\}\n\\end{equation}\nfor all $a \\in \\widehat H_{\\#}^{\\textrm{sc}}(\\catA_{\\mu})$ and $x \\in \\widehat H_{r,D}^{\\textrm{sc}}(\\catA_{\\mu})$, where $\\{a,-\\} \\colon \\widehat H_{r,D}^{\\textrm{sc}}(\\catA_{\\mu}) \\to \\widehat H_{r,D}^{\\textrm{sc}}(\\catA_{\\mu})$.\n\nIn short, $I_{r,D}$ is a Poisson-bimodule morphism over $\\widehat H_{\\#}^{\\textrm{sc}}(\\catA_{\\mu})$ whereas $I_{\\#}$ is a morphism of Poisson algebras.\nWe use \\eqref{intmap} in Section \\ref{DTPTcorrespondence} to obtain the $\\mathcal F$-local DT\/PT correspondence (Theorem \\ref{thm:DTPT}) and to reprove equation~\\eqref{GKformula} for stable sheaves (Theorem \\ref{thm:gk}).\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The higher rank local DT\/PT correspondence}\\label{Sec:dtpt}\nLet $X$ be a smooth projective $3$-fold and let $\\mathcal{F} \\in M_{\\DT}(r,D)$ be a $\\mu_{\\omega}$-stable sheaf of homological dimension at most one, i.e., $\\mathcal{F}$ satisfies $\\lExt^i(\\F,\\mathscr O_X) = 0$ for $i\\geq 2$.\n\nIn this section, we embed the Quot schemes $\\Quot_X(\\mathcal F)$ and $\\Quot_X(\\lExt^1(\\mathcal F,\\mathscr O_X))$ in suitable moduli spaces of torsion free sheaves and PT pairs, respectively.\nIn the Calabi--Yau case, we use these embeddings to define $\\mathcal F$-local DT and PT invariants of $X$, representing the virtual contributions of $\\mathcal F$ to the global invariants.\\footnote{The definition makes sense for arbitrary $3$-folds, but the enumerative meaning of these numbers is less clear without the Calabi--Yau condition.}\nWe prove Theorem \\ref{thm2} (resp.~Theorem \\ref{thm3}) by applying the integration morphism $I^B$ (resp.~$I^E$) to an identity in the Hall algebra of $\\catA_\\mu$ (Proposition \\ref{HallIdentity2}), which is the $\\F$-local analogue of the global identity \\cite[Thm.~1.2]{Toda2}.\n\n\n\n\n\\subsection{Embedding Quot schemes in DT and PT moduli spaces}\nRecall that \\emph{closed immersions} of schemes are precisely the \\emph{proper monomorphisms} (in the categorical sense), and a monomorphism of schemes is a morphism $Y\\rightarrow Z$ such that the induced natural transformation $\\Hom(-,Y)\\rightarrow \\Hom(-,Z)$ is injective. We will implicitly use this characterisation in Propositions \\ref{DTembedding} and \\ref{PTembedding}.\n\nAs the DT side is more immediate than the PT side, we start there.\n\n\\begin{prop}\\label{DTembedding}\nTaking the kernel of a surjection $\\mathcal F\\twoheadrightarrow Q$ defines a closed immersion \n\\[\n\\phi_{\\F} \\colon \\Quot_X(\\mathcal F) \\hookrightarrow M_{\\DT}(r,D).\n\\]\n\\end{prop}\n\n\\begin{proof}\nLet $B$ be a scheme, $\\pi_X\\colon X\\times B\\rightarrow X$ the projection, and $\\pi_X^\\ast \\F\\twoheadrightarrow \\mathscr Q$ a flat family of zero-dimensional quotients of $\\F$. Then the kernel $\\mathcal K\\subset \\pi_X^\\ast \\F$ is $B$-flat and defines a family of torsion free $\\mu_\\omega$-stable sheaves on $X$. Therefore the association\n\\[\n[\\mathcal F\\twoheadrightarrow Q]\\mapsto \\ker(\\mathcal F\\twoheadrightarrow Q)\n\\]\nis a morphism to the moduli stack ${\\mathcal M}_{\\DT}(r,D)$, and composing with the natural morphism $p \\colon {\\mathcal M}_{\\DT}(r,D) \\to M_{\\DT}(r,D)$ to the coarse moduli space defines $\\phi_{\\F}$.\nSince the Quot scheme is proper and $M_{\\DT}(r,D)$ is separated, the morphism is proper.\nMoreover, $\\phi_{\\mathcal F}$ is injective on all valued points by definition of the Quot functor, identifying two flat quotients precisely when they share the same kernel.\nThis proves that $\\phi_{\\mathcal F}$ is a closed immersion.\n\\end{proof}\n\n\n\\begin{remark}\nAs the proof shows, the initial assumption on the homological dimension of $\\F$ is not needed. It will be needed in Proposition \\ref{PTembedding} (the ``PT side'').\n\\end{remark}\n\n\n\n\nWe now move to the PT side.\nFirst, we construct a map\n\\[\n\\Quot_X(\\lExt^1(\\mathcal F,\\mathscr O_X))\\rightarrow M_{\\PT}(r,D)\n\\]\non the level of $\\mathbb C$-valued points.\nConsider a surjection\n\\[\nt \\colon \\lExt^1(\\mathcal F,\\mathscr O_X) \\twoheadrightarrow Q,\n\\]\nand recall the following identifications, induced by the derived dualising functor,\n\\be\\label{homs617}\n\\Hom(\\mathcal F^{\\vee},Q[-1]) = \\Hom(Q^{\\textrm{D}}[-2],\\mathcal F) = \\Ext^1(Q^{\\textrm{D}}[-1],\\mathcal F).\n\\end{equation}\nWe interpret $t$ as an element of the first Hom-space by precomposing its shift\n\\[\n\\bar{t} \\colon \\mathcal F^{\\vee} \\to \\lExt^1(\\mathcal F,\\mathscr O_X)[-1] \\xrightarrow{t[-1]} Q[-1],\n\\]\nand we associate to $t$ the extension\n\\be\\label{triangle10482}\n\\mathcal F \\hookrightarrow J^\\bullet \\twoheadrightarrow Q^{\\textrm{D}}[-1]\n\\end{equation}\nin $\\mathscr{A}_{\\mu}$ corresponding to $\\bar{t}$ under \\eqref{homs617}. \nNote that $\\rk J^\\bullet = \\rk \\mathcal F = r$ and $c_1(J^\\bullet) = c_1(\\mathcal F) = D$.\nIt is clear that $J^\\bullet$ defines a pre-PT pair based at $\\mathcal F$.\nTo see that $J^\\bullet$ is in fact a PT pair, we dualise again by applying $\\lHom(-,\\mathscr O_X)$ to the defining triangle \\eqref{triangle10482}.\nWe find\n\\[\n\\cdots \\to \\lExt^1(\\mathcal F,\\mathscr O_X) \\xrightarrow{t} Q = \\lExt^3(Q^{\\textrm{D}},\\mathscr O_X) \\to \\lExt^2(J^\\bullet,\\mathscr O_X) \\to 0\n\\]\nwhere the last zero is $\\lExt^2(\\mathcal F,\\mathscr O_X) = 0$.\nHere, we have $t = H^1(\\bar{t})$ since the derived dual is an involution.\nThus the surjectivity of the morphism $t$ is equivalent to the vanishing of $\\lExt^2(J^\\bullet,\\mathscr O_X)$.\nIn turn, by Lemma~\\ref{lem:PrePT_is_PT}, this means that $J^\\bullet$ is a PT pair.\n\n\\begin{remark}\nConversely, any $\\mathcal F$-local PT pair, consisting of an exact triangle of the form \\eqref{triangle10482}, gives rise to a surjection $\\lExt^1(\\mathcal F,\\mathscr O_X) \\twoheadrightarrow Q$ by applying $\\lHom(-,\\mathscr O_X)$.\n\\end{remark}\n\nTo extend this association to families, we make use of the following result.\n\\begin{lemma}\\label{TopExt}\nLet $B$ be a scheme, and let $\\mathscr Q\\in \\Coh(X\\times B)$ be a $B$-flat family of zero-dimensional sheaves on a smooth scheme $X$ of dimension $d$. Then one has\n\\[\n\\lExt^i_{X\\times B}(\\mathscr Q,\\mathscr O_{X\\times B}) = 0, \\qquad i\\neq d.\n\\]\nMoreover, the base change map\n\\[\n\\lExt^d_{X\\times B}(\\mathscr Q,\\mathscr O_{X\\times B})\\otimes_{\\mathscr O_B}k(b) \\rightarrow \\lExt^d_X(\\mathscr Q_b,\\mathscr O_X)\n\\]\nis an isomorphism for all $b\\in B$.\n\\end{lemma}\n\n\\begin{proof}\nThe vanishing follows from \\cite[Theorem 1.10]{PicScheme}.\nThe base change property follows from \\cite[Theorem 1.9]{PicScheme}.\n\\end{proof}\n\n\n\n\n\\begin{prop} \\label{PTembedding}\nThe association $t\\mapsto J^\\bullet$ extends to a closed immersion\n\\[\n\\psi_{\\mathcal F} \\colon \\Quot(\\lExt^1(\\mathcal F,\\mathscr O_X)) \\hookrightarrow M_{\\PT}(r,D).\n\\]\n\\end{prop}\n\n\\begin{proof}\nLet $B$ be a base scheme, and let $\\pi_X\\colon X\\times B\\rightarrow X$ denote the natural projection.\nLet $\\mathscr Q \\in \\Coh(X \\times B)$ be a $B$-flat family of zero-dimensional sheaves on $X$ receiving a surjection\n\\begin{equation*}\nt \\colon \\pi_X^*\\lExt^1(\\mathcal F,\\mathscr O_X) \\twoheadrightarrow \\mathscr Q.\n\\end{equation*}\nIn particular, $\\mathscr Q \\in \\Perf(X \\times B)$ is a perfect object in the derived category of $X \\times B$.\nPulling back the triangle $H^0(\\mathcal F^\\vee)\\rightarrow \\mathcal F^\\vee \\rightarrow \\lExt^1(\\mathcal F,\\mathscr O_X)[-1]$ on $X$ to $X \\times B$ yields the exact triangle\n\\be\\label{triangle7163}\n\\pi_X^*H^0(\\mathcal F^{\\vee}) \\to \\pi_X^*(\\mathcal F^{\\vee}) \\to \\pi_X^*\\lExt^1(\\mathcal F,\\mathscr O_X)[-1],\n\\end{equation}\nand precomposing $t[-1]$ yields the morphism $\\bar{t} \\colon \\pi_X^*(\\mathcal F^{\\vee}) \\to \\mathscr Q[-1]$.\nSince all objects we consider are perfect complexes on $X \\times B$, so are their derived duals.\nWe deduce\n\\begin{equation*}\n\\pi_X^*(\\mathcal F^{\\vee}) = (\\pi_X^*\\mathcal F)^{\\vee}.\n\\end{equation*}\nMore generally, for complexes $E,F \\in \\Perf(X \\times B)$ we have a natural isomorphism\n\\[\n{\\textbf R}\\lHom(E,F) \\cong {\\textbf R}\\lHom(F^{\\vee},E^{\\vee}).\n\\]\nBy Lemma~\\ref{TopExt}, we have a natural isomorphism of perfect complexes on $X \\times B$\n\\begin{equation*}\n{\\textbf R} \\lHom\\left(\\pi_X^*(\\mathcal F^{\\vee}),\\mathscr Q[-1]\\right) \\cong {\\textbf R} \\lHom\\left(\\mathscr Q^{\\textrm{D}}[-2],\\pi_X^*\\mathcal F\\right).\n\\end{equation*}\nTaking derived global sections yields an isomorphism of complexes of vector spaces, and further taking cohomology yields a natural isomorphism of $\\mathbb C$-linear Hom-spaces\n\\be\\label{gnagna311}\n\\Hom_{X \\times B}\\left(\\pi_X^*(\\mathcal F^{\\vee}),\\mathscr Q[-1]\\right) \\cong \\Ext^1_{X \\times B}\\left({\\mathscr Q}^{\\textrm{D}}[-1],\\pi_X^*\\mathcal F\\right).\n\\end{equation}\nWe write the image of $\\bar{t}$ under this identification as an extension\n\\be\\label{Jfamily}\n\\pi_X^*\\mathcal F \\to J^{\\bullet} \\to \\mathscr Q^{\\textrm{D}}[-1].\n\\end{equation}\nWe claim that $J^{\\bullet}$ is a family of PT pairs parametrised by $B$ (cf.~Definition \\ref{def:familyPT}).\n\nFirst, taking the derived fibre of the triangle \\eqref{Jfamily} shows that $J_b^\\bullet = {\\textbf L} i_b^*(J^{\\bullet})$ is a pre-PT pair on $X_b$ \nfor all closed points $b \\in B$, where $X_b = X\\times \\set{b}$ and $i_b\\colon X_b \\hookrightarrow X\\times B$ is the natural closed immersion.\nSecond, $J^{\\bullet}$ is a perfect complex because $\\pi_X^\\ast \\mathcal F$ and $\\mathscr Q^{\\textrm{D}}[-1]$ are.\nThus $J^{\\bullet}$ defines a family of pre-PT pairs based at $\\mathcal F$ in the sense of Definition~\\ref{def:familyPT}.\n\nTo see that each derived fibre $J^{\\bullet}_b$ is a PT pair, recall that for a perfect complex the operations of taking the derived fibre and taking the derived dual commute.\nIn other words, there is a \\emph{canonical} isomorphism\n\\[\n{\\textbf L} i_b^*(E^{\\vee}) \\cong ({\\textbf L} i_b^*E)^{\\vee}\n\\]\nfor all $b \\in B$, where $E \\in \\Perf(X \\times B)$ is a perfect complex.\nAs a consequence, the diagram\n\\begin{equation*}\n\\begin{tikzcd}[row sep=large]\n\\Hom_{X \\times B}\\left(\\pi_X^*\\mathcal F^{\\vee},{\\mathscr Q}[-1]\\right) \\arrow[r,\"(-)^{\\vee}\"]\\arrow[d,\"{\\textbf L} i_b^*\"] & \n\\Hom_{X \\times B}\\left({\\mathscr Q}^{\\textrm{D}}[-2], \\pi_X^*\\mathcal F \\right) \\arrow[d,\"{\\textbf L} i_b^*\"] \\\\\n\\Hom_{X_b}\\left(\\mathcal F^{\\vee},{\\mathscr Q}_b[-1]\\right) \\arrow[r,\"(-)^{\\vee}\"] &\n\\Hom_{X_b}\\left({\\mathscr Q}^{\\textrm{D}}_b[-2],\\mathcal F\\right)\n\\end{tikzcd}\n\\end{equation*}\ncommutes.\\footnote{Here we are implicitly using Lemma \\ref{TopExt}, saying that dualising commutes with base change for a flat family of zero-dimensional sheaves.\nMore generally, dualising commutes with any base change for perfect objects.}\nIn words, given the morphism $\\bar{t} \\colon \\pi_X^*(\\mathcal F^{\\vee}) \\to {\\mathscr Q}[-1]$, we obtain our family of pre-PT pairs $J^{\\bullet}$ on $X \\times B$ by dualising, and taking its derived fibre $J^{\\bullet}_b = {\\textbf L} i_b^*(J^{\\bullet})$ yields a complex that is canonically isomorphic to the pre-PT pair obtained by first restricting $t$ to the derived fibre\n\\[ \n\\bar{t}_b = {\\textbf L} i_b^*(\\bar{t}) \\colon \\mathcal F^{\\vee} \\to \\lExt^1(\\mathcal F,\\mathscr O_X)[-1] \\xrightarrow{t_b[-1]} {\\mathscr Q}_b[-1]\n\\]\nand then dualising.\nBy taking the derived fibre of the triangle \\eqref{Jfamily} and running the argument for a $\\mathbb C$-valued point, it follows that $\\lExt^2(J^{\\bullet}_b,\\mathscr O_{X_b}) = 0$ because each map $t_b$ is surjective.\nThus $J^{\\bullet}$ defines a family of PT pairs by Lemma~\\ref{lem:PrePT_is_PT}.\nWe have obtained a morphism to the moduli stack $\\mathcal{M}_{\\PT}(r,D)$, and composing with $\\mathcal{M}_{\\PT}(r,D) \\to M_{\\PT}(r,D)$, defines $\\psi_{\\mathcal F}$.\n\n\nNote that $\\psi_{\\mathcal F}$ is proper because its domain is proper and the PT moduli space is separated.\nTo prove that it is a closed immersion, it is enough to show it is injective on $B$-valued points.\nLet $t$ and $u$ be two families of surjections with (the same) target ${\\mathscr Q}$, and assume that they give rise to the same family of PT pairs $J^\\bullet$.\nThen, by \\eqref{gnagna311}, we conclude that $\\bar t= t[-1]\\circ g = u[-1]\\circ g=\\bar u$, where\n\\[\ng \\colon \\pi_X^*(\\mathcal F^{\\vee}) \\to \\pi_X^*\\lExt^1(\\mathcal F,\\mathscr O_X)[-1]\n\\]\nis the morphism appearing in \\eqref{triangle7163}.\nBut the natural map\n\\[\n(-)\\circ g\\colon \\Hom\\left(\\pi_X^*\\lExt^1(\\mathcal F,\\mathscr O_X)[-1],\\mathscr Q[-1]\\right)\\rightarrow \\Hom\\left(\\pi_X^*(\\mathcal F^{\\vee}),\\mathscr Q[-1]\\right)\n\\]\nis injective because $\\Hom(\\pi_X^\\ast H^0(\\mathcal F^\\vee)[1],\\mathscr Q[-1]) = \\Ext^{-2}(\\pi_X^*H^0(\\mathcal F^\\vee),{\\mathscr Q}) = 0$, i.e., its kernel vanishes.\nIt follows that $t = u$.\nThis completes the proof.\n\\end{proof}\n\n\n\nSee Section \\ref{UniversalEmbedding}, and in particular Proposition \\ref{prop:UnivPTimmersion}, for a `universal' closed immersion generalising Proposition \\ref{PTembedding}.\n\n\n\n\\subsection{The $\\mathcal F$-local Hall algebra identity}\\label{DTPT_identity}\nIn this section, we prove an $\\F$-local analogue of Toda's Hall algebra identity in \\cite[Lemma~3.16]{Toda2} which gives rise to the higher rank DT\/PT correspondence by applying the integration morphisms $I^B$.\n\nWe introduce the Hall algebra elements.\nLet $\\Coh_0(X)[-1]$ be the shift of the category of zero-dimensional coherent sheaves on $X$, which we denote by $\\mathcal C_{\\infty}$ to be consistent with the notation of \\cite{Toda2}.\nThe moduli stack of objects in ${\\mathcal C}_{\\infty}$ is an open substack $\\mathcal C_{\\infty} \\subset \\mathcal A_{\\mu}$.\nWe obtain an element\n\\[\n\\delta(\\mathcal C_\\infty) = \\left[\\mathcal C_\\infty \\hookrightarrow \\mathcal A_\\mu\\right]\\in \\widehat H_{\\#}(\\mathscr A_\\mu).\n\\]\n\n\nLet $p\\colon \\mathcal M_{\\DT}(r,D)\\rightarrow M_{\\DT}(r,D)$ denote the natural morphism from the moduli stack of DT objects to its coarse moduli space.\nConsider the cartesian diagram\n\\[\n\\begin{tikzcd}\n\\mathcal Q_X(\\F)\\MySymb{dr} \\ar[hook]{r}{\\iota_{\\F}}\\ar[swap]{d}{p'} & \\mathcal M_{\\DT}(r,D)\\ar{d}{p}\\arrow[hook]{r}{\\textrm{open}} & \\catA_\\mu\\\\\n\\Quot_X(\\F) \\ar[hook,swap]{r}{\\phi_{\\F}} & M_{\\DT}(r,D) &\n\\end{tikzcd}\n\\]\ndefining $\\mathcal Q_X(\\F)$. The map $\\iota_{\\F}$ is a closed immersion by Proposition \\ref{DTembedding} and base change.\nHence $\\mathcal Q_X(\\F)$ defines a locally closed substack of $\\catA_\\mu$. A similar picture holds on the PT side by replacing $\\F$ by $\\lExt^1(\\F,\\mathscr O_X)$.\n\nFor the sake of brevity, we rename these objects \n\\[\n\\mathcal Q_{\\DT}^{\\F} = \\mathcal Q_X(\\F),\\quad \\mathcal Q_{\\PT}^{\\F} = \\mathcal Q_X(\\lExt^1(\\F,\\mathscr O_X)).\n\\]\nWe now obtain Hall algebra elements\n\\[\n \\delta_{\\DT}^{\\F} = \\left[\\mathcal Q_{\\DT}^{\\F}\\rightarrow \\catA_\\mu\\right],\\quad \n \\delta_{\\PT}^{\\F} = \\left[\\mathcal Q_{\\PT}^{\\F}\\rightarrow \\catA_\\mu\\right]\n\\]\nin $\\widehat H_{r,D}(\\catA_\\mu)$.\nBy base change, the morphism $p'$ and its analogue for $\\lExt^1(\\F,\\mathscr O_X)$ are (trivial) $\\mathbb G_m$-gerbes (cf.~Remark~\\ref{rem:stacks_gerbes}).\nThus it follows that the elements\n\\[\n\\overline \\delta_{\\DT}^{\\F} = (\\mathbb{L}-1)\\cdot \\delta_{\\DT}^{\\F},\\quad \\overline \\delta_{\\PT}^{\\F} = (\\mathbb{L}-1)\\cdot \\delta_{\\PT}^{\\F}\n\\]\nlie in the regular submodule $\\widehat H_{r,D}^{\\textrm{reg}}(\\mathscr A_\\mu)$. \nProjecting to the semiclassical limit yields elements\n\\be\\label{Main_HA_elements}\n\\overline \\delta_{\\DT}^{\\F},\\,\\,\\,\\overline \\delta_{\\PT}^{\\F} \\,\\in\\,\n\\widehat H_{r,D}^{\\textrm{sc}}(\\mathscr A_\\mu).\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\nLet us form the $\\mathscr A_\\mu$-stacks \n\\[\n\\mathcal Q_{\\DT}^{\\mathcal F}\\star \\mathcal C_\\infty,\\quad \\mathcal C_\\infty\\star \\mathcal Q_{\\PT}^{\\mathcal F}\n\\]\nvia the pullback construction described in \\eqref{HAproduct}. For a scheme $B$, a $B$-valued point of $\\mathcal Q_{\\DT}^{\\mathcal F}\\star \\mathcal C_\\infty$ is an exact triangle $E_1 \\to E \\to E_3$ in $\\Perf(X \\times B)$ such that $E_1$, $E_3 \\in \\catA_{\\mu}(B)$, $E_1$ is a $B$-valued point of $\\mathcal Q_{\\DT}^{\\mathcal F}$, and $E_3$ is a $B$-valued point of $\\mathcal C_{\\infty}$.\nSimilarly, a $B$-valued point of $\\mathcal C_\\infty\\star \\mathcal Q_{\\PT}^{\\mathcal F}$ is an exact triangle $E_1 \\to E \\to E_3$ in $\\Perf(X \\times B)$ where $E_1$ is a $B$-valued point of $\\mathcal C_\\infty$ and $E_3$ is a $B$-valued point of $\\mathcal Q_{\\PT}^{\\mathcal F}$.\n\n\n\\begin{lemma}\\label{Closed_Points_13}\nThere is an equivalence at the level of $\\mathbb C$-valued points\n\\[\n\\left(\\mathcal Q_{\\DT}^{\\mathcal F}\\star \\mathcal C_\\infty\\right)(\\mathbb C) = \\left(\\mathcal C_\\infty\\star \\mathcal Q_{\\PT}^{\\mathcal F}\\right)(\\mathbb C).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nLet $E$ be a $\\mathbb C$-valued point of $\\mathcal C_\\infty\\star \\mathcal Q_{\\PT}^{\\mathcal F}$. Then we can decompose $E$ as an extension of an object $J^\\bullet \\in \\mathcal Q_{\\PT}^{\\mathcal F}$ by a (shifted) zero-dimensional object $Q[-1]\\in\\mathcal C_\\infty$.\nWe obtain the following diagram\n\\[\n\\begin{tikzcd}\n\tH^0(E) \\arrow[r,hook] \\arrow[d,\"f\"] & E \\arrow[r,two heads] \\arrow[d,two heads] & H^1(E)[-1] \\arrow[d,\"g\"]\\\\\n \\mathcal F \\arrow[r,hook] & J^\\bullet \\arrow[r,two heads] & P[-1]\n\\end{tikzcd}\n\\]\nin $\\mathscr{A}_{\\mu}$, where $P\\in \\Coh_0(X)$.\nThe snake lemma in $\\catA_{\\mu}$ induces a four-term exact sequence\n\\[\n0 \\to \\ker(f) \\to Q[-1] \\to \\ker(g) \\to \\coker(f) \\to 0\n\\]\nand implies that $g$ is surjective.\nSince $\\Coh_{\\leq 1}(X)[-1] \\subset \\mathscr{A}_{\\mu}$ is closed under subobjects, extensions, and quotients, and $Q$ is zero-dimensional, we deduce that the above exact sequence lies in $\\mathcal C_\\infty$ entirely.\nBut there are no morphisms in negative degree, so $\\coker(f) = 0$ since $\\mathcal F \\in \\Coh_{\\mu}(X)$ is a sheaf.\nWe obtain the exact sequence\n\\[\n\t0 \\to H^0(E) \\to \\F \\to \\ker(f)[1] \\to 0\n\\]\nin $\\Coh(X)$, proving that $H^0(E) \\in \\mathcal Q_{\\DT}^{\\mathcal F}$ as claimed.\n\nConversely, let $E$ be an extension of a (shifted) zero-dimensional object $P[-1]$ and an object $K \\in \\mathcal Q_{\\DT}^{\\mathcal F}$.\nWe obtain the diagram\n\\[\n\\begin{tikzcd}\n\tQ[-1] \\arrow[r,hook] \\arrow[d,\"f'\"] & K \\arrow[r,two heads] \\arrow[d,hook] & \\mathcal F \\arrow[d,\"g'\"]\\\\\n S[-1] \\arrow[r,hook] & E \\arrow[r,two heads] & J^\\bullet\n\\end{tikzcd}\n\\]\nin $\\mathscr{A}_{\\mu}$, where $S[-1] \\subset E$ is the largest subobject of $E$ in $\\Coh_0(X)[-1]$; this object exists since $\\catA_{\\mu}$ is noetherian and $\\mathcal C_\\infty \\subset \\catA_{\\mu}$ is closed under extensions and quotients.\nThe snake lemma in $\\mathscr{A}_{\\mu}$ induces a four-term exact sequence\n\\[\n0 \\to \\ker(g') \\to \\coker(f') \\to P[-1] \\to \\coker(g') \\to 0\n\\]\nand implies that $f'$ is injective.\nAs before, we deduce that the above exact sequence lies in $\\mathcal C_\\infty$ entirely.\nBy assumption $\\mathcal F$ is both a DT and PT object, hence $\\Hom(\\mathcal C_\\infty,\\mathcal F) = 0$ and so $\\ker(g') = 0$.\nWe obtain the exact sequence\n\\[\n\t0 \\to \\mathcal F \\to J^{\\bullet} \\to \\coker(g') \\to 0\n\\]\nin $\\mathscr{A}_{\\mu}$, proving that $J^\\bullet \\in \\mathcal Q_{\\PT}^{\\mathcal F}$.\nThe constructions are clearly inverse to each other.\n\\end{proof}\n\nWe now refine the above identification to a Hall algebra identity. \n\n\\begin{prop}\\label{HallIdentity2}\nIn $\\widehat H_{r,D}(\\mathscr A_\\mu)$, one has the identity\n\\begin{equation}\\label{eq:Hall_Identity_Flocal}\n\\delta^{\\F}_{\\DT}\\star \\delta(\\mathcal C_\\infty) = \\delta(\\mathcal C_\\infty) \\star \\delta^{\\F}_{\\PT}.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nRecall the open immersion of stacks $\\mathscr B_{\\mu} \\subset \\mathscr A_{\\mu}$.\nThe key Hall algebra identity proven by Toda relies on the existence of geometric bijections\n\\be\\label{amustacks}\n\\begin{tikzcd}\n\\mathcal M_{\\DT}(r,D)\\star \\mathcal C_\\infty \\arrow{dr} & & \\mathcal C_\\infty \\star \\mathcal M_{\\PT}(r,D)\\arrow{dl} \\\\\n& \\widetilde{\\mathscr B}_{\\mu} & \n\\end{tikzcd}\n\\end{equation}\nwhere $\\widetilde{\\mathscr B}_{\\mu}$ is the open and closed substack of $\\mathscr B_\\mu$ parametrising objects $E\\in\\mathscr B_\\mu$ such that $H^1(E)\\in \\Coh_0(X)$. \nPulling back the closed immersions $\\phi_{\\mathcal F}$ and $\\psi_{\\mathcal F}$, the stacks $\\mathcal Q_{\\DT}^{\\mathcal F}\\star \\mathcal C_\\infty$ and $\\mathcal C_\\infty\\star \\mathcal Q_{\\PT}^{\\mathcal F}$ embed as closed substacks of the corresponding $\\widetilde{\\mathscr B}_{\\mu}$-stacks \\eqref{amustacks}.\nRecall that the class of an $\\catA_{\\mu}$-stack $S \\to \\catA_{\\mu}$ in the Hall algebra is equal to the class of its reduction,\n\\[\n\\left[S_{\\text{red}} \\to \\catA_{\\mu} \\right] = \\left[S \\to \\catA_{\\mu} \\right],\n\\]\nvia the geometric bijection $S_{\\text{red}} \\to S$ of $\\catA_{\\mu}$-stacks.\nThus we may assume that $\\mathcal Q_{\\DT}^{\\mathcal F}\\star \\mathcal C_\\infty$ and $\\mathcal C_\\infty\\star \\mathcal Q_{\\PT}^{\\mathcal F}$ are reduced.\nLet $\\mathscr Z_{\\DT}$ and $\\mathscr Z_{\\PT}$ denote their stack-theoretic images in $\\widetilde{\\catB}_{\\mu}$, which are reduced closed substacks.\n\\[\n\\begin{tikzcd}\n{\\mathscr Z}_{\\DT} \\arrow[swap,hook]{dr} & & {\\mathscr Z}_{\\PT} \\arrow[left hook->]{dl} \\\\\n& \\widetilde{\\mathscr B}_{\\mu} & \n\\end{tikzcd}\n\\]\nWe claim that ${\\mathscr Z}_{\\DT} = {\\mathscr Z}_{\\PT}$ as substacks.\nThis establishes the identity~\\eqref{eq:Hall_Identity_Flocal} because, crucially, we have $\\delta_{\\DT}^{\\mathcal{F}} \\star \\delta(\\mathcal{C}_{\\infty}) = [\\mathscr{Z}_{\\DT} \\to \\catA_{\\mu}]$ in $\\widehat{H}_{r,D}(\\catA_{\\mu})$ by the geometric bijection $\\mathcal{Q}_{\\DT}^{\\F} \\star \\mathcal{C}_{\\infty} \\to \\mathscr{Z}_{\\DT}$ over $\\widetilde{\\mathscr B}_{\\mu} \\subset \\catA_{\\mu}$; similarly, we have the identity $\\delta(\\mathcal{C}_{\\infty}) \\star \\delta_{\\PT}^{\\mathcal{F}} = [\\mathscr{Z}_{\\PT} \\to \\catA_{\\mu}]$.\n\nTo prove this claim, recall that the stack $\\catA_{\\mu}$ is locally of finite type.\nSince $\\catA_{\\mu} \\subset D^b(X)$ is the heart of a bounded t-structure, it follows that the stack $\\catA_{\\mu}$ has affine geometric stabilisers \\cite[Lemma~2.3.9]{SjoerdThesis}.\nBy a result of Kresch \\cite[\\textsection~4.5]{kreschcycle}, it is locally a global quotient stack $[V\/G]$ where $V$ is a variety and $G$ is a linear algebraic group.\nThus locally ${\\mathscr Z}_{\\DT}$ and ${\\mathscr Z}_{\\PT}$ correspond to $G$-invariant closed subvarieties of $V$.\nBut the $\\mathbb C$-valued points of these (reduced) closed subvarieties coincide by Lemma~\\ref{Closed_Points_13}.\nThus ${\\mathscr Z}_{\\DT} = {\\mathscr Z}_{\\PT}$ as claimed. \n\\end{proof}\n\n\n\n\n\\subsection{The $\\mathcal F$-local DT\/PT correspondence}\\label{DTPTcorrespondence}\nIn this section, we prove Theorem \\ref{thm2} and Theorem \\ref{thm3}.\nFor $n\\geq 0$, we define\n\\begin{align*}\n\\mathsf{DT}_{\\mathcal F,n} &= \\chi(\\Quot_X(\\mathcal F,n),\\nu_{\\DT}) \\\\\n\\mathsf{PT}_{\\mathcal F,n} &= \\chi(\\Quot_X(\\lExt^1(\\mathcal F,\\mathscr O_X),n),\\nu_{\\PT})\n\\end{align*}\nwhere the Behrend weights come from the full DT and PT moduli spaces and are restricted via the closed immersions of Propositions \\ref{DTembedding} and \\ref{PTembedding}. \nWe form the generating functions\n\\[\n\\mathsf{DT}_{\\mathcal F}(q) = \\sum_{n\\geq 0}\\mathsf{DT}_{\\mathcal F,n}q^n,\\quad \\mathsf{PT}_{\\mathcal F}(q) = \\sum_{n\\geq 0} \\mathsf{PT}_{\\mathcal F,n}q^n\n\\]\nin the completed quantum torus $\\widehat C_{r,D}^{\\sigma}(X)= \\prod_{v\\in \\Gamma_{r,D}}C_v(X)$, where we use the shorthand $q = c_{(0,0,0,1)}$.\nIn the Calabi--Yau case, these series can be interpreted as the ``contribution'' of $\\F$ to the global DT and PT invariants. \n\nRecall the Hall algebra elements \\eqref{Main_HA_elements}.\nNote that \n\\be\\label{Integration918}\nI^B\\left(\\overline\\delta_{\\DT}^{\\F}\\right) = -\\mathsf{DT}_{\\F}(q^{-1})c_{\\nu},\\quad I^B\\left(\\overline\\delta_{\\PT}^{\\F}\\right) = -\\mathsf{PT}_{\\F}(q^{-1}) c_{\\nu},\n\\end{equation}\nwhere $I^B$ is the Behrend weighted version of the map $I_{r,D}$ from \\eqref{intmap}, and $\\nu = \\ch(\\mathcal{F})$.\nThe minus sign is a consequence of property \\eqref{ChiSquare} of the Behrend function, taking into account that the moduli stacks are $\\mathbb G_m$-gerbes over the coarse moduli spaces of DT and PT objects.\n\n\n\\begin{remark}\nThe Behrend weights $\\nu_{\\DT}$ and $\\nu_{\\PT}$ are not unrelated. Indeed, up to the $\\mathbb G_m$-gerbes\n\\[\n\\mathcal M_{\\DT}(r,D) \\rightarrow M_{\\DT}(r,D),\\quad \\mathcal M_{\\PT}(r,D) \\rightarrow M_{\\PT}(r,D),\n\\]\nthey are both restrictions of the Behrend function of Lieblich's moduli stack $\\mathscr M_X$, studied in \\cite{Lieblich1}, along their respective open immersions $\\mathcal M_{\\DT}(r,D) \\subset \\mathscr{M}_X$ and $\\mathcal M_{\\PT}(r,D) \\subset \\mathscr{M}_X$.\n\\end{remark}\n\n\n\n\\begin{theorem}\\label{thm:DTPT}\nLet $X$ be a Calabi--Yau 3-fold, and let $\\F$ be a $\\mu_{\\omega}$-stable sheaf of rank $r$ and homological dimension at most one.\nThere is an equality of generating series\n\\begin{equation}\\label{FlocalDTPT}\n\\mathsf{DT}_{\\mathcal F}(q) = \\mathsf M((-1)^rq)^{r\\chi(X)}\\cdot \\mathsf{PT}_{\\mathcal F}(q).\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nThe equality follows by applying the integration morphism $I_{r,D}$ from \\eqref{intmap} taken with $\\sigma = -1$, precisely as in the proof of \\cite[Thm.~3.17]{Toda2}, after replacing the Hall algebra identity of \\cite[Lemma~3.16]{Toda2} by the identity of equation~\\eqref{eq:Hall_Identity_Flocal}.\nWe give the argument in full.\n\nBy Proposition \\ref{HallIdentity2}, we obtain the identity\n\\[\n\\delta^{\\mathcal F}_{\\DT} =\n\\delta(\\catC_{\\infty}) \\star \\delta^{\\mathcal F}_{\\PT} \\star \\delta(\\catC_{\\infty})^{-1}\\,\\in \\,\\widehat{H}_{r,D}(\\catA_{\\mu}).\n\\]\nSince $\\Coh_0(X)$ is artinian, we have a well-defined logarithm $\\epsilon(\\catC_{\\infty}) = \\log \\delta(\\catC_{\\infty}) \\in \\widehat{H}_{\\#}(\\catA_{\\mu})$.\nWe obtain in $\\widehat{H}_{r,D}(\\catA_{\\mu})$ the identity\n\\[\n\\delta^{\\mathcal F}_{\\DT} =\n\\exp\\left(\\epsilon(\\catC_{\\infty})\\right) \\star \\delta^{\\mathcal F}_{\\PT} \\star \\exp\\left(-\\epsilon(\\catC_{\\infty})\\right).\n\\]\nThe $\\widehat{H}_{\\#}(\\catA_{\\mu})$-bimodule structure of $\\widehat{H}_{r,D}(\\catA_{\\mu})$ induces an adjoint action of $a \\in \\widehat{H}_{\\#}(\\catA_{\\mu})$ on $\\widehat{H}_{r,D}(\\catA_{\\mu})$ via the equation\n\\[\n\\textrm{Ad}(a) \\circ x = a \\star x - x \\star a \\colon \\widehat{H}_{r,D}(\\catA_{\\mu}) \\to \\widehat{H}_{r,D}(\\catA_{\\mu}).\n\\]\nBy the Baker--Campbell--Hausdorff formula the above equation becomes\n\\[\n\\delta^{\\mathcal F}_{\\DT} =\n\\exp\\bigl(\\textrm{Ad}(\\epsilon(\\catC_{\\infty}))\\bigr) \\circ \\delta^{\\mathcal F}_{\\PT}\n\\]\nin $\\widehat{H}_{r,D}(\\catA_{\\mu})$.\nMultiplying both sides of the equation by $\\mathbb{L}-1$ and projecting the resulting equation in $\\widehat{H}^{\\textrm{reg}}_{r,D}(\\catA_{\\mu})$ to the semi-classical quotient $\\widehat{H}^{\\textrm{sc}}_{r,D}(\\catA_{\\mu})$, we obtain the identity\n\\[\n\\overline{\\delta}^{\\mathcal F}_{\\DT} =\n\\exp\\bigl(\\textrm{Ad}^{\\textrm{sc}}(\\overline{\\epsilon}(\\catC_{\\infty}))\\bigr) \\circ \\overline{\\delta}^{\\mathcal F}_{\\PT} ~\\in~ \\widehat{H}^{\\textrm{sc}}_{r,D}(\\catA_{\\mu}),\n\\]\nwhere we have written the adjoint action of $a \\in \\widehat{H}^{\\textrm{sc}}_{\\#}(\\catA_{\\mu})$ as\n\\[\n\\textrm{Ad}^{\\textrm{sc}}(a) \\circ x \\defeq \\{a,x\\} \\colon \\widehat{H}^{\\textrm{sc}}_{r,D}(\\catA_{\\mu}) \\to \\widehat{H}^{\\textrm{sc}}_{r,D}(\\catA_{\\mu}),\n\\]\nin terms of the Poisson bracket of equation~\\eqref{eq:Poisson_Bracket}, and we have applied Joyce's No-Poles Theorem which states that $\\overline{\\epsilon}(\\catC_{\\infty}) = (\\mathbb{L} - 1)\\epsilon(\\catC_{\\infty}) \\in \\widehat{H}^{\\textrm{reg}}_{\\#}(\\catA_{\\mu})$; cf.~\\cite[Thm.~3.12]{Toda2}.\n\nApplying the integration morphism $I_{r,D}$, and using the Euler pairing computation\n\\begin{align*}\n\\chi\\bigl((0,0,-\\gamma,-m),(r,D,-\\beta,-n) \\bigr) = rm-D\\gamma,\n\\end{align*}\nalong with the identities \\eqref{Integration918}, we obtain the formula\n\\begin{equation}\\label{almostthere}\n \\mathsf{DT}_{\\mathcal F}(q) = \\exp\\left( \\sum_{m > 0} (-1)^{rm-1} rm \\cdot \\mathsf N_{m,0} q^m\\right)\\cdot \\mathsf{PT}_{\\mathcal F}(q),\n\\end{equation}\nafter formally sending $q^{-1} \\mapsto q$.\nHere the ``$\\mathsf N$-invariants'' $\\mathsf N_{m,0} \\in \\mathbb Q$ count semistable zero-dimensional sheaves $E$ with $\\chi(E)=m$, and are defined by the relation\n\\[\nI_{\\#}\\left(\\overline{\\epsilon}(\\catC_{\\infty})\\right) = - \\sum_{m \\geq 0} \\mathsf N_{m,0} (q^{-1})^m.\n\\]\nSee \\cite[Sec.~3.6]{Toda2} and the references therein for more details.\nUsing the rank one identity\n\\[\n\\exp\\left( \\sum_{m > 0} (-1)^{m-1} m \\cdot \\mathsf N_{m,0} q^m\\right) = \\mathsf M(-q)^{\\chi(X)}\n\\]\nestablished in \\cite{BFHilb,LEPA,JLI}, the relation \\eqref{almostthere} becomes precisely\n\\[\n\\mathsf{DT}_{\\mathcal F}(q) = \\mathsf M((-1)^rq)^{r\\chi(X)} \\cdot \\mathsf{PT}_{\\mathcal F}(q).\n\\]\nThis completes the proof.\n\\end{proof}\n\n\n\nWe now prove Theorem \\ref{thm3}, namely the main result of \\cite{Gholampour2017} in the special case of a stable sheaf $\\F$. We do not require $X$ to be Calabi--Yau.\n\\begin{theorem}\\label{thm:gk}\nLet $X$ be a smooth projective $3$-fold, and let $\\mathcal F$ be as in Theorem \\ref{thm:DTPT}. \nThen\n\\be \\label{GK937}\n\\sum_{n\\geq 0}\\chi(\\Quot_X(\\mathcal F,n))q^n = \\mathsf M(q)^{r\\chi(X)}\\cdot \\sum_{n\\geq 0}\\chi(\\Quot_X(\\lExt^1(\\mathcal F,\\mathscr O_X),n))q^n.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe equality follows from the proof of \\cite[Thm.~3.17]{Toda2} by replacing the Hall algebra identity of \\cite[Lemma~3.16]{Toda2} by the identity of equation~\\eqref{eq:Hall_Identity_Flocal} \\emph{and} by replacing the Behrend weighted integration morphism $I^B$ by the integration morphism $I^E$ taking Euler characteristics.\nThe compatibility of $I^E$ with the Poisson brackets is explained in Remark~\\ref{rem:Integration_Morphisms}.\n\\end{proof}\n\\begin{remark}\nIn \\cite{Gholampour2017}, formula \\eqref{GK937} is obtained by reducing to the affine case, and carrying out an inductive procedure on the rank.\nThe base case of rank two, established on an affine $3$-fold, relies on the existence of an auxiliary \\emph{cosection} $\\mathcal F\\rightarrow \\mathscr O_X$.\n\nThe above result yields an interpretation of equation~\\eqref{GK937} as the Euler characteristic shadow of the $\\F$-local higher rank DT\/PT correspondence \\eqref{FlocalDTPT}. This question was raised by Gholampour and Kool in \\cite[Sec.~1]{Gholampour2017}. \n\n\nNote that our proof only produces the formula for $\\mu_{\\omega}$-stable sheaves $\\F$ (of homological dimension at most one).\nThis is a consequence of producing the identity \\eqref{eq:Hall_Identity_Flocal} in the Hall algebra of $\\catA_{\\mu}$.\nIn contrast, \\cite{Gholampour2017} proves formula \\eqref{GK937} for \\emph{all} torsion free sheaves (of homological dimension at most one).\n\\end{remark}\n\n\\begin{remark}\\label{rmk:cyclelocal}\nLet $X$ be a Calabi--Yau $3$-fold, $C\\subset X$ a Cohen--Macaulay curve, and $\\F = \\mathscr I_C$ of $\\ch(\\mathscr I_C) = (1,0,-\\beta,p_a(C)-1)$.\nThe \\emph{cycle-local} invariants of \\cite[Sec.~4]{Ob1} may in general differ from those of Theorem \\ref{thm:DTPT}.\nIndeed, the former express the contribution of all ideal sheaves $\\mathscr I_{Z}$ such that $[Z] = [C]$ in $\\Chow_1(X,\\beta)$, and this condition is in general weaker than having an inclusion $\\mathscr I_Z\\hookrightarrow \\mathscr I_C$.\nThe two types of invariants do agree when $C$ is smooth: in this case, \\cite[Thm. 2.1]{Ricolfi2018} proves that $\\Quot_X(\\mathscr I_C,n)$ is precisely the fibre of the Hilbert--Chow morphism over the cycle of $C$. They also agree when $\\beta = \\ch_2(\\mathscr O_C)$ is irreducible, see Proposition \\ref{prop:cycleVSframed}.\n\nAlso the method of proof differs: in \\cite{Ob1}, the author restricts Bridgeland's global DT\/PT identity to the Hall subalgebra of the abelian category of sheaves supported on $C$ in dimension one, whereas in this paper an $\\mathscr I_C$-local identity is established directly in the Hall algebra of $\\mathscr A_\\mu$.\nWe emphasise, however, that both local proofs make use of the established proof of the global correspondence in some way. \n\\end{remark}\n\n\n\n\n\n\n\n\\section{Applications}\nIn this section we discuss a few results and special cases, directly linked to Theorem~\\ref{thm:DTPT}, concerning the PT series $\\mathsf{PT}_{\\F}$ and its properties.\nThroughout, as before, $X$ is a smooth projective 3-fold and $\\mathcal{F}$ is a $\\mu_{\\omega}$-stable sheaf of homological dimension at most one.\n\n\\subsection{Tensoring by a line bundle}\nWe establish a general relation between the generating functions of $\\mathcal F$-local and of $(\\mathcal F \\otimes L)$-local PT invariants, where $L$ is a line bundle on $X$.\n\\begin{prop}\\label{prop:PT_tensor_by_line_bundle}\n We have $\\mathsf{PT}_{\\mathcal F \\otimes L}(q) = \\mathsf{PT}_{\\mathcal F}(q)$ for every line bundle $L$ on $X$.\n\\end{prop}\n\\begin{proof}\nSince $\\mathcal F$ is a $\\mu_{\\omega}$-stable sheaf of homological dimension at most 1, the same holds for $\\mathcal F \\otimes L$. \nMoreover, $(r,D \\cdot \\omega^2)$ are coprime so $(r,(rc_1(L) + D)\\cdot \\omega^2)$ are coprime as well.\n\nLet $t \\colon \\lExt^1(\\mathcal F,\\mathscr O_X) \\twoheadrightarrow Q$ be a zero-dimensional quotient.\nTensoring by $L^{-1}$ induces the identification\n\\[\n\\begin{tikzcd}\n\\lExt^1(\\mathcal F,\\mathscr O_X) \\otimes L^{-1} \\arrow[two heads]{r}\\isoarrow{d} & Q \\otimes L^{-1} \\arrow[equal]{d} \\\\\n\\lExt^1(\\mathcal F \\otimes L,\\mathscr O_X) \\arrow[two heads]{r} & Q \\otimes L^{-1} \n\\end{tikzcd}\n\\]\nusing that Ext-sheaves are local; note that $Q$ is zero-dimensional so $Q \\otimes L^{-1} \\cong Q$, but for reasons of naturality we do not choose an isomorphism here.\nTensoring by line bundles behaves well in flat families, so we obtain an isomorphism\n\\[\n- \\otimes L^{-1} \\colon \\Quot_X(\\lExt^1(\\F,\\mathscr O_X),n) \\,\\widetilde{\\rightarrow}\\, \\Quot_X(\\lExt^1(\\mathcal F \\otimes L,\\mathscr O_X),n).\n\\]\n\nWe claim that the following diagram\n\\begin{equation}\\label{eq:Tensoring_PTmoduli}\n\\begin{tikzcd}[row sep=large]\n\\Quot_X(\\lExt^1(\\F,\\mathscr O_X),n) \\ar[hook,swap]{d}{\\psi_{\\mathcal F}} \\ar{r}{- \\otimes L^{-1}} & \\Quot_X(\\lExt^1(\\mathcal F \\otimes L,\\mathscr O_X),n) \\ar[hook]{d}{\\psi_{\\mathcal{F} \\otimes L}} \\\\\nM_{\\PT}(r,D) \\ar{r}{- \\otimes L} & M_{\\PT}(r,D+rc_1(L))\n\\end{tikzcd}\n\\end{equation}\ncommutes. Our closed immersion into the PT moduli space proceeds by dualising\n\\[\n\\bar{t} \\colon \\mathcal{F}^{\\vee} \\to \\lExt^1(\\mathcal F,\\mathscr O_X)[-1] \\to Q[-1].\n\\]\nTo obtain the corresponding $\\F$-local PT pair, we dualise \\emph{again} to obtain the extension\n\\[\n\\mathcal{F} \\hookrightarrow J^{\\bullet} \\twoheadrightarrow Q^{\\textrm{D}}[-1],\n\\]\nwhich corresponds to $\\bar{t}$ under $\\Hom(\\mathcal F^{\\vee},Q[-1]) \\cong \\Ext^1(Q^{\\textrm{D}}[-1],\\mathcal F )$.\nTensoring this exact sequence by the line bundle $L$ yields an $(\\mathcal{F} \\otimes L)$-local PT pair $J^{\\bullet} \\otimes L$, namely\n\\[\n\\mathcal{F} \\otimes L \\hookrightarrow J^{\\bullet} \\otimes L \\twoheadrightarrow Q^{\\textrm{D}} \\otimes L[-1].\n\\]\n\nWe also obtain a PT pair by performing the operations the other way around.\nIndeed, first tensoring $\\bar t$ by the line bundle $L^{-1}$ yields the morphism\n\\[\n\\bar{t}_L \\colon \\mathcal{F}^{\\vee} \\otimes L^{-1} \\cong (\\mathcal{F} \\otimes L)^{\\vee} \\to \\lExt^1(\\mathcal F \\otimes L,\\mathscr O_X)[-1] \\to (Q \\otimes L^{-1})[-1].\n\\]\nTo obtain the corresponding local PT pair, we dualise \\emph{again} to obtain the extension\n\\[\n\\mathcal{F} \\otimes L \\hookrightarrow J^{\\bullet}_L \\twoheadrightarrow Q^{\\textrm{D}} \\otimes L[-1],\n\\]\ncorresponding to $\\bar{t}_L$ under $\\Hom((\\mathcal F \\otimes L)^{\\vee},(Q \\otimes L^{-1})[-1]) \\cong \\Ext^1(Q^{\\textrm{D}} \\otimes L[-1],\\mathcal F \\otimes L)$.\nWe claim that $J^{\\bullet}_L$ and $J^{\\bullet} \\otimes L$ are canonically isomorphic $(\\mathcal{F} \\otimes L)$-PT pairs.\n\nTo see this, consider the following diagram of canonical and commuting isomorphisms:\n\\[\n\\begin{tikzcd}\n\\Hom(\\lExt^1(\\mathcal F,\\mathscr O_X),Q) \\ar[hook]{d} \\ar[equal]{r}{-\\otimes L^{-1}} & \\Hom(\\lExt^1(\\mathcal F \\otimes L,\\mathscr O_X), Q \\otimes L^{-1}) \\ar[hook]{d} \\\\\n\\Hom(\\mathcal F^{\\vee},Q[-1]) \\ar[equal,swap]{d}{(-)^{\\vee}} \\ar[equal]{r}{- \\otimes L^{-1}} & \\Hom((\\mathcal F \\otimes L)^{\\vee}, Q \\otimes L^{-1}[-1]) \\ar[equal]{d}{(-)^{\\vee}} \\\\\n\\Ext^1(Q^{\\textrm{D}}[-1],\\mathcal F) \\ar[equal]{r}{- \\otimes L} & \\Ext^1(Q^{\\textrm{D}} \\otimes L[-1], \\mathcal F \\otimes L)\n\\end{tikzcd}\n\\]\nwhich proves that the operations indeed commute, so $J^{\\bullet}_L \\cong J^{\\bullet} \\otimes L$ canonically, because the extensions are equal.\nThus the diagram displayed in \\eqref{eq:Tensoring_PTmoduli} commutes.\n\nAs a consequence, pulling back the Behrend functions of $M_{\\PT}(r,D)$ and $M_{\\PT}(r,D+rc_1(L))$ to either moduli space induces the \\emph{same} constructible function on the isomorphic Quot schemes.\nIn particular, their Behrend weighted Euler characteristics are equal, which means that $\\mathsf{PT}_{\\mathcal F,n} = \\mathsf{PT}_{\\mathcal F \\otimes L,n}$ for all $n \\in \\mathbb Z$.\nWe infer $\\mathsf{PT}_{\\mathcal F}(q) = \\mathsf{PT}_{\\mathcal F \\otimes L}(q)$ as claimed.\n\\end{proof}\n\nConsider the generating function of topological Euler characteristics\n\\[\n\\widehat{\\mathsf{PT}}_{\\mathcal{F}}(q) = \\sum_{n\\geq 0}\\chi(\\Quot_X(\\lExt^1(\\mathcal F,\\mathscr O_X),n))q^n.\n\\]\nThe above proof directly carries over to the Euler characteristic setting.\n\\begin{corollary}\nWe have $\\widehat{\\mathsf{PT}}_{\\mathcal{F} \\otimes L}(q) = \\widehat{\\mathsf{PT}}_{\\mathcal{F}}(q)$ for every line bundle $L$ on $X$.\n\\end{corollary}\n\n\n\n\\subsection{Special cases}\nLet $X$ be a smooth projective Calabi--Yau 3-fold.\nWe collect some special cases of Theorem~\\ref{thm:DTPT} by imposing further restrictions on the sheaf $\\mathcal{F}$.\n\\begin{corollary}\\label{cor101}\nWith the assumptions of Theorem~\\ref{thm:DTPT}, if $\\F$ is locally free then the generating series of $\\mathcal F$-local PT invariants is trivial: $\\mathsf{PT}_{\\mathcal F}(q) = 1$.\nIn particular, \n\\[\n\\mathsf{DT}_{\\F}(q) = \\mathsf M((-1)^rq)^{r\\chi(X)}.\n\\]\n\\end{corollary}\n\\begin{proof}\nSince $\\lExt^1(\\F,\\mathscr O_X) = 0$, one deduces from the definitions that $\\mathsf{PT}_{\\F,n} = 0$ for all $n \\neq 0$ and that $\\mathsf{PT}_{\\F,0} = 1$.\nThe formula for $\\mathsf{DT}_{\\F}$ then follows from Theorem~\\ref{thm:DTPT}.\n\\end{proof}\n\\begin{remark}\nRecall that any rank one PT pair is of the form $I^\\bullet = [s \\colon \\mathscr O_X \\to F]$ where $F$ is a pure one-dimensional sheaf and $\\coker(s)$ is zero-dimensional; in particular, $\\ker(s) = H^0(I^\\bullet)$ is the ideal sheaf of a Cohen--Macaulay curve $C \\subset X$.\nThe above corollary generalises the fact that the only rank one PT pair $I^\\bullet$ with $H^0(I^\\bullet)$ a line bundle is the trivial one with $F = 0$. \n\\end{remark}\n\n\\begin{remark}\\label{rmk:comparison}\nCombining Corollary \\ref{cor101} with Theorem \\ref{Thm:LocallyFree}, we observe that for a stable vector bundle $\\F$ of rank $r$ on a Calabi--Yau $3$-fold $X$, one has the identity\n\\[\n\\mathsf{DT}_{\\F,n} = \\widetilde\\chi(\\Quot_X(\\F,n)).\n\\]\nNote that the left hand side depends, a priori, on the embedding\n\\[\n\\phi_{\\F,n}\\colon\\Quot_X(\\F,n) \\hookrightarrow M_{\\DT}(r,D,\\ch_2(\\F),\\ch_3(\\F)-n)\n\\]\nwhereas the right hand side is completely intrinsic to the Quot scheme. The above identity is trivial in the case $(r,D)=(1,0)$, for then $\\Quot_X(\\F,n)=\\Hilb^n X=M_{\\DT}(1,0,0,-n)$.\n\\end{remark}\n\n\n\n\n\n\nNext, we assume that $\\mathcal F$ is a \\emph{reflexive} $\\mu_{\\omega}$-stable sheaf.\nLet $(-)^*$ denote the usual $\\mathscr O_X$-linear dual of a sheaf, and note that $\\mathcal F^*$ is again $\\mu_{\\omega}$-stable and reflexive; see for example \\cite[Lemma~2.1(2)]{Gholampour2017}. In particular, it is both a DT and PT object by Corollary~\\ref{cor:DTPT_object} and the series $\\mathsf{PT}_{\\mathcal F^*}(q)$ is well-defined.\nWe denote the \\emph{reciprocal} of a polynomial $P(q)$ of degree $d$ by\n\\[\nP^*(q) = q^dP(q^{-1}),\n\\]\nand we let $\\ell(T)$ denote the length of a zero-dimensional sheaf $T$.\n\nFollowing its proof, we obtain the virtual analogue of \\cite[Thm.~1.2]{Gholampour2017}.\n\n\\begin{corollary}\\label{cor292}\nWith the assumptions of Theorem~\\ref{thm:DTPT}, if $\\mathcal F$ is reflexive then the series\n\\[\n\\dfrac{\\mathsf{DT}_{\\mathcal F}(q)}{\\mathsf M((-1)^rq)^{r\\chi(X)}} = \\mathsf{PT}_{\\mathcal F}(q)\n\\]\nof $\\mathcal F$-local PT invariants is a \\emph{polynomial} of degree $\\ell(\\lExt^1(\\mathcal F,\\mathscr O_X))$.\nMoreover, this polynomial has a symmetry induced by the derived dualising functor\n\\begin{equation}\\label{eq:PTF_symmetry}\n\\mathsf{PT}^*_{\\mathcal F}(q) = \\mathsf{PT}_{\\mathcal F^*}(q).\n\\end{equation}\nAnd finally, if $\\rk(\\mathcal F) = 2$ then $\\mathsf{PT}^*_{\\mathcal F}(q) = \\mathsf{PT}_{\\mathcal F}(q)$ is palindromic.\n\\end{corollary}\n\\begin{proof}\nThe final claim requires the Behrend weighted identity $\\mathsf{PT}_{\\mathcal F \\otimes L}(q) = \\mathsf{PT}_{\\mathcal F}(q)$ for any line bundle $L$, which is provided by Proposition~\\ref{prop:PT_tensor_by_line_bundle}, and $\\F^* \\cong \\F \\otimes \\det(\\F)^{-1}$ if $\\rk(\\F) = 2$.\n\\end{proof}\n\n\\subsection{Rationality: open questions}\nLet $\\beta\\in H_2(X,\\mathbb Z)$ be a curve class on a Calabi--Yau $3$-fold $X$, and let $\\mathsf{PT}_{n,\\beta}$ be the (rank one) PT invariant, defined as the degree of the virtual fundamental class of $P_n(X,\\beta)=M_{\\PT}(1,0,-\\beta,-n)$. In \\cite[Conj.~3.2]{PT}, Pandharipande and Thomas conjectured that the Laurent series\n\\[\n\\mathsf{PT}_\\beta (q) = \\sum_{n\\in \\mathbb Z}\\mathsf{PT}_{n,\\beta} q^n\n\\]\nis the expansion of a \\emph{rational function} in $q$ (invariant under $q\\leftrightarrow q^{-1}$).\nThis was proved by Bridgeland \\cite{Bri}. More generally, Toda \\cite[Thm.~1.3]{Toda2} proved the rationality of the series\n\\[\n\\mathsf{PT}_{r,D,\\beta}(q)=\\sum_{6n\\in \\mathbb Z}\\mathsf{PT}(r,D,-\\beta,-n) q^n\n\\]\nfor arbitrary $(r,D,\\beta)$. Moreover, Toda previously proved \\cite{Toda0} the rationality of the unweighted generating function\n\\[\n\\widehat{\\mathsf{PT}}_{\\beta}(q) = \\sum_{n\\in \\mathbb Z}\\chi(P_n(X,\\beta)) q^n.\n\\]\nOne may ask similar questions about the local invariants studied in the present paper.\n\nLet $\\F$ be a $\\mu_{\\omega}$-stable sheaf of homological dimension at most one on the Calabi--Yau $3$-fold $X$. It makes sense to ask the following:\n\n\\begin{question}\nOn a Calabi--Yau $3$-fold $X$, is $\\mathsf{PT}_{\\F}(q)$ the expansion of a rational function?\n\\end{question}\n\nOne can of course ask the same question for the unweighted invariants\n\\[\n\\widehat{\\mathsf{PT}}_{\\F}(q)=\\sum_{n\\geq 0}\\chi(\\Quot_X(\\lExt^1(\\F,\\mathscr O_X),n)) q^n,\n\\]\nwhere $X$ is now an arbitrary smooth projective $3$-fold and $\\F$ is a torsion free sheaf of homological dimension at most one (not necessarily stable).\n\n\\begin{question}\nOn a $3$-fold $X$, is $\\widehat{\\mathsf{PT}}_{\\F}(q)$ the expansion of a rational function?\n\\end{question}\nRationality of $\\widehat{\\mathsf{PT}}_{\\F}$ has been announced for toric $3$-folds in \\cite{Gholampour2017}.\nAs for the weighted version, there is a partial answer in the rank one case, building upon work of Pandharipande and Thomas \\cite{BPS}.\nWe will investigate such rationality questions in future work.\n\n\\begin{prop}\\label{prop:cycleVSframed}\nLet $X$ be a Calabi--Yau $3$-fold, $C\\subset X$ a Cohen--Macaulay curve in class $\\beta\\in H_2(X,\\mathbb Z)$. If $\\beta$ is irreducible, then $\\mathsf{PT}_{\\mathscr I_C}(q)$ is the expansion of a rational function in $q$.\n\\end{prop}\n\n\\begin{proof}\nLet $g=1-\\chi(\\mathscr O_C)$ be the arithmetic genus of $C$, and denote by $P_n(X,C)\\subset P_{1-g+n}(X,\\beta)$ the closed subset parametrising stable pairs $[\\mathscr O_X\\rightarrow F]$ such that the fundamental one-cycle of $F$ equals $[C]\\in \\Chow_1(X,\\beta)$. Let $\\mathsf P_{n,C}$ denote the virtual contribution of $P_n(X,C)$. Under the irreducibility assumption on $\\beta$, Pandharipande and Thomas showed in \\cite[Sec.~3.1]{BPS} that the generating function of cycle-local invariants\n\\[\n\\mathsf Z_C(q) = \\sum_{n\\geq 0} \\mathsf P_{n,C}q^{1-g+n}\n\\]\nadmits the unique expression\n\\[\n\\mathsf Z_C(q) = \\sum_{r=0}^g\\mathsf n_{r,C}q^{1-r}(1+q)^{2r-2}\n\\]\nas a rational function of $q$, where $\\mathsf n_{r,C}$ are integers, called the ``BPS numbers'' of $C$. Since $\\beta$ is irreducible, however, we have\n\\[\n\\mathsf P_{n,C} = \\mathsf{PT}_{\\mathscr I_C,n}.\n\\]\nIndeed, the Chow variety parametrises Cohen--Macaulay curves on $X$ in class $\\beta$, therefore given a stable pair $[s \\colon \\mathscr O_X\\rightarrow F]\\in P_n(X,C)$ along with its induced short exact sequence\n\\[\n0\\rightarrow \\mathscr O_X\/\\ker s\\rightarrow F \\rightarrow Q \\rightarrow 0,\n\\]\nthe condition $[F] = [C] \\in \\Chow_1(X,\\beta)$ implies the identity $\\ker s = \\mathscr I_C$. It follows that $\\mathsf{PT}_{\\mathscr I_C} = \\mathsf Z_C$, whence the result.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe question of existence of initial data for isolated multiple black hole systems can be best analyzed according to whether any of the black holes is extremal or not. This is related\nto the different topologies in the initial manifold. For non-extremal black holes, the initial manifold contains one asymptotically flat (AF) end for each black hole, but for \n extremal black holes, the associated ends are asymptotically cilindrical (AC) \\cite{dain10d}. This change in topology translates into different behaviours of the initial data \n near the ends, and results in that \n for extremal black holes, the familiar theorems for elliptic \nequations that use standard weighted Sobolev spaces on AF manifolds, need to be adapted. \n\n\nThe treatment of the Lichnerowicz equation (arising from the conformal method) in the non-extremal case, \\textit{i.e.} initial manifold only\nwith AF ends, is similar to the case of \ncompact manifolds, see for instance \\cite{Choquet99}, \\cite{Chrusciel02a}, \\cite{Maxwell:2004yb}, \\cite{Maxwell:2005bd} (cf. \\cite{Chrusciel:2010my}). \nWe focus here on the extremal case, that is, when there is one AF end representing the region at \nspatial infinity, and at least two AC ends, representing the extremal black holes.\n\n\nIn the literature, the many ends are usually treated individually, with cutoff functions that single out one asymptotic end at a time.\nIn this line, Chrusciel \\textit{et al}, \\cite{Chrusciel:2012np}, prove that if the scalar curvature of the seed metric is \npositive, a solution to the Lichnerowicz equation on a manifold with a finite number of AF and AC ends, exists. \nThis is particularly relevant after the result of Leach \\cite{Leach:2016jra} on manifolds with one AF and one AC end that states that if the \nRiemannian manifold is Yamabe positive, then there exists a conformal transformation to a metric with positive scalar curvature \n(see Proposition 3.5 in \\cite{Leach:2016jra}). Also in the same article, an existence result for the existence of far from Constant Mean Curvature (CMC) initial data \ndescribing an isolated black hole is given.\n\n\n\nSobolev spaces arise naturally when studying the elliptic equations coming from the Einstein constraints. Concerning manifolds with one AF and one AC end, Bartnik \\cite{Bartnik86} introduced weighted Sobolev spaces $W'^{k,p}_\\delta$, that not only \ndescribe asymptotically flat solutions, but that also single out a point in the initial \nmanifold in such a way that by an appropriate choice of $\\delta$, the origin in $\\mathbb R^3$ may be made to describe another asymptoric end. These weighted spaces \nwere crucially used in \\cite{Dain:2010uh,gabach09} for studying CMC, single, extremal black hole initial data, and further results concerning \nthese spaces were proven.\n\n\n \nIn this article we present weighted Sobolev spaces with weights that reflect the existence of more than one black hole. We also extend the \nresults in \\cite{Bartnik86, Dain:2010uh,gabach09}. The aim of these results is to provide an adequate framework to study initial \ndata for many black holes. In particular we use this framework to study existence of initial data for Einstein equations that \ndescribe many extremal black holes. \n\n\nWe will consider initial data with one AF end and several AC ends. The initial data is the set $(M,g_{ij},K_{ij}, E^i,B^i)$ subjected to the constraint equations\n\\begin{equation}\nR+K^2-K_{ij}K^{ij}=2(E_iE^i+B_iB^i),\n\\end{equation}\n\\begin{equation}\\label{mom}\nD_jK^j\\,_i-D_iK=0,\n\\end{equation}\n\\begin{equation}\\label{maxcon}\nD_iE^i=0,\\qquad D_iB^i=0,\n\\end{equation}\nwhere $K=K_{ij}g^{ij}$, $D_i$ and $R$ are respectively the covariant derivative and \nthe curvature scalar associated to the metric $g_{ij}$.\n\nIn the Conformal Method with constant mean curvature, we consider the rescaling\n\\begin{equation}\\label{rescaling}\ng_{ij}=\\Phi^4\\tilde g_{ij},\\quad K_{ij}=\\Phi^{-2}\\tilde K_{ij},\\quad E^i=\\Phi^{-6}\\tilde E^i,\\quad B^i=\\Phi^{-6}\\tilde B^i.\n\\end{equation}\nThe constraint equations in terms of the conformal quantities are\n\\begin{eqnarray}\n\\label{constEq} \\t{\\Delta}\\Phi = \\frac{1}{8}\\t{R}\\Phi+\\frac{\\t{K}^2-\\t{K}_{ij}\\t{K}^{ij}}{8\\Phi^7}-\\frac{\\t{E}_i\\t{E}^i+\\t{B}_i\\t{B}^i}{4\\Phi^3}, \\\\\n \\t{D}_j\\t{K}^j\\,_i-\\t{D}_i\\t{K}+4\\Phi^{-1}\\t{K}\\t{D}_i\\Phi = 0, \\\\\n \\t{D}_i\\t{E}^i = 0, \\\\\n \\t{D}_i\\t{B}^i = 0.\n\\end{eqnarray}\n\n\nAs a concrete example, one of the most relevant multi-black hole initial data was found by Majumdar and Papapetrou \\cite{Papapetrou45,Majumdar47} \nand consists of $N$ charged black holes with equal mass and charge parameters for each black hole. It is interpreted as a set of extremal Reissner-Nordstr\\\"om punctures, \nheld in equilibrium\nby the balance between the gravitational atraction and the electrostatic repulsion.\nThe space metric and electromagnetic potential are given by\n\\begin{equation}\nds^2=\\Phi^{4}ds^2_{flat},\n\\end{equation}\n\n\\begin{equation}\nA=\\pm \\Phi^{-2}dt,\n\\end{equation}\nwhere $ds^2_{flat}$ is the 3-metric of Euclidean space, and \n\\begin{equation}\n\\Phi=\\sqrt{1+\\sum_i^N\\frac{m_i}{|x-x_i|}},\n\\end{equation}\nwhere $m_i, \\; i=1,\\ldots,N$, is the positive electric charge of the $i-$th black hole located at $x=x_i$.\nThis solution contains $N+1$ ends, the end $|x|\\to\\infty$ is AF and the $N$ ends \n$x\\to x_i$ are AC. The latter can be seen by taking the limit of the metric and ckecking that it goes to the standard metric on the cylinder\n(see \n\\cite{Acena:2015rxa} for details).\n\nThe article is organized as follows. In section \\ref{Sobolev} we define the functional spaces needed for dealing with many AC ends. We prove estimates, study the behaviour of functions and properties of operators on these spaces. \nIn section \\ref{initial} we apply these results to study perturbations \nof axially symmetric initial data for many extremal black holes. \n\n\n\n\\section{Weighted Sobolev spaces}\\label{Sobolev}\n\nThe functional spaces that we need in order to deal with many asymptotic ends are extensions of the standard weighted Lebesgue and Sobolev spaces. For comparison we state the norms of those spaces as\npresented by Bartnik \\cite{Bartnik86}. Given locally $p$-measurable functions in $\\mathbb R^n$ and $\\mathbb R^n\\setminus\\{0\\}$ respectively, define\n\\begin{equation}\\label{LLnorm00}\n L^p_\\delta:\\qquad ||u||_{p,\\delta} := \\left\\{\\begin{array}{cc}\n \\left(\\int_{\\mathbb{R}^n}|u|^p\\, \\sigma^{-\\delta p -n} dx\\right)^\\frac{1}{p},&p<\\infty\\\\\n \\mbox{ess sup}_{\\mathbb R^n}(\\sigma^{-\\delta}|u|),& p=\\infty \n \\end{array}\\right.\n\\end{equation}\n\\begin{equation}\\label{Lnorm00}\n W^{k,p}_\\delta:\\qquad ||u||_{k,p,\\delta} := \\sum_{j=0}^k ||D^ju||_{p,\\delta-j},\n\\end{equation}\n \n\\begin{equation}\\label{LLnorm0}\n L'^p_\\delta:\\qquad ||u||'_{p,\\delta} := \\left\\{\\begin{array}{cc}\n \\left(\\int_{\\mathbb{R}^n\\backslash\\{0\\}}|u|^p\\, r^{-\\delta p -n} dx\\right)^\\frac{1}{p},&p<\\infty\\\\\n \\mbox{ess sup}_{\\mathbb R^n\\setminus\\{0\\}}(r^{-\\delta}|u|),& p=\\infty \n \\end{array}\\right.\n\\end{equation}\n\\begin{equation}\\label{Lnorm0}\n W'^{k,p}_\\delta:\\qquad ||u||'_{k,p,\\delta} := \\sum_{j=0}^k ||D^ju||'_{p,\\delta-j},\n\\end{equation}\nwhere $\\sigma:=(1+r^2)^{1\/2}$. Both $\\sigma$ and $r$ are in $L^1_{loc}(\\mathbb R^3)$ and $L^1_{loc}(\\mathbb R^3\\setminus\\{0\\})$ respectively. The unprimed set \\eqref{LLnorm00}, \\eqref{Lnorm00} \nis best suited to problems involving AF initial data, with only one asymptotic end. For single black hole initial data, with two asymptotic ends, \none AF and one at the origin, the spaces \\eqref{LLnorm0}, \\eqref{Lnorm0} are more appropriate, as one locates the black hole puncture precisely at $r=0$. \nIn order to extend these spaces to account for many punctures we introduce the following definitions.\n\n\\begin{definition}\\label{def1}\nLet $\\Pi:=\\{ x_i\\in \\mathbb{R}^n,\\;i=1,\\ldots,N\\}$, and the function $w\\in L^1_{loc}(\\mathbb{R}^n\\setminus \\Pi)$ be\n\\begin{equation}\n w:= \\left(\\sum_{i=1}^N\\frac{1}{r_i}\\right)^{-1},\n\\end{equation}\nwhere $r_i:=|x-x_i|$ is the Euclidean distance from the point $ x_i$.\nThen we define the norms and associated functional spaces \n\\begin{equation}\\label{Lnorm}\nL'^p_{w,\\delta}:\\qquad ||u||'_{w,p,\\delta} := \\left\\{\\begin{array}{cc}\n \\left(\\int_{\\mathbb{R}^n\\setminus\\Pi}|u|^p\\, w^{-\\delta p -n} dx\\right)^\\frac{1}{p},&p<\\infty\\\\\n \\textnormal{ess sup}_{\\mathbb R^n\\setminus\\Pi}(w^{-\\delta}|u|),& p=\\infty \n \\end{array}\\right.\n\\end{equation}\n\n\\begin{equation}\\label{Wnorm}\n W'^{k,p}_{w,\\delta}:\\qquad ||u||'_{w,k,p,\\delta} := \\sum_{j=0}^k ||D^ju||'_{w,p,\\delta-j}.\n\\end{equation}\n\\end{definition}\nThe particular choice of $w$ responds to its behaviour near the $N$ punctures and the extra asymptotic end.\nIf we approach one of the punctures, say $l$, we have $r_l \\rightarrow 0$, and then\n\\begin{equation}\n w \\rightarrow r_l.\n\\end{equation}\nAlso, if $r_l \\rightarrow \\infty$, then all $r_i \\rightarrow\\infty$, and\n\\begin{equation}\n w \\rightarrow \\frac{r}{N}.\n\\end{equation}\nIt is clear that $w$ reduces to $r$ and the norms \\eqref{Lnorm}-\\eqref{Wnorm} reduce to \\eqref{LLnorm0}-\\eqref{Lnorm0} when there is only one puncture at the origin ($N =1,\\,r_1=r$). \nMoreover, Definition \\ref{def1} treats each puncture in the same manner, \\textit{i.e}. there is one $\\delta$ for all the punctures. This is particularly relevant for the case where\nall the black holes are extremal. If some of the ends are AC and some are AF (apart from the region at infinity), then a different weight should be used, with specific behaviour near \nthe two types of punctures.\n\nThe importance of the behaviour of the function $w$ lies in making the standard norms and the new norms equivalent when we restrict the domain to a neighborhood of only one end. For this, we need to recenter the standard norms with the origin in the corresponding puncture. To show that they are indeed equivalent we need to isolate each end, including the $r\\rightarrow\\infty$ end, and therefore we define the following quantities\n\\begin{equation}\nd_{ij} := | x_i- x_j|,\n\\end{equation}\n\\begin{equation}\nR_i := \\frac{1}{3}\\min_{j\\neq i}\\{d_{ij}\\},\n\\end{equation}\n\\begin{equation}\nR := 3\\max_{i}\\{| x_i|\\}.\n\\end{equation}\nWith these quantities we define the sets\n\\begin{equation}\nB_i := \\{r_i \\leq R_i\\},\n\\end{equation}\n\\begin{equation}\nB := \\{r \\geq R\\}. \n\\end{equation}\nIt is straightforward to show that\n\\begin{equation}\nB_i \\cap B = \\emptyset, \\qquad B_i \\cap B_j = \\emptyset,\\qquad i\\neq j.\n\\end{equation}\nTherefore each set $B_i$, $B$ contains only one end, and we show that the norms restricted to each of these sets are equivalent, which is a direct corollary to the following proposition.\n\\begin{proposition}\\label{prop1}\nIf $r_i\\leq R_i$, then there is a constant $C_i$, $0 \\frac{1}{r_i},\\qquad \\forall\\, i=1,\\ldots,N.\n\\end{equation} \nFor the left hand side inequality, as $r_i \\leq R_i$, then for $j\\neq i$\n\\begin{equation}\nr_j \\geq d_{ij} - R_i > 0\n\\end{equation}\nand\n\\begin{eqnarray}\n\\frac{1}{w} & = & \\frac{1}{r_i} + \\sum_{j\\neq i} \\frac{1}{r_j} \\leq \\frac{1}{r_i} + \\sum_{j\\neq i} \\frac{1}{d_{ij} - R_i} \\\\\n& \\leq & \\frac{1}{r_i} + \\left(\\sum_{j\\neq i} \\frac{1}{d_{ij} - R_i}\\right)\\frac{R_i}{r_i} \n = \\frac{1}{r_i}\\left(1 + R_i \\sum_{j\\neq i} \\frac{1}{d_{ij} - R_i}\\right).\n\\end{eqnarray}\nIf we define\n\\begin{equation}\nC_i := \\left(1 + R_i \\sum_{j\\neq i} \\frac{1}{d_{ij} - R_i}\\right)^{-1}\n\\end{equation}\nthe proof of \\eqref{ineqw1} is complete.\n\nNow we prove \\eqref{ineqw2}. If $r\\geq R$,\n\\begin{equation}\n\\frac{r}{2}\\leq r-\\frac{R}{2} < r_i < r+\\frac{R}{2} <2r,\n\\end{equation}\ntherefore\n\\begin{equation}\n\\frac{1}{2r} < \\frac{1}{r_i} < \\frac{2}{r},\n\\end{equation}\nand\n\\begin{equation}\n\\frac{N}{2r} < \\frac{1}{w} < \\frac{2N}{r},\n\\end{equation}\nwhich gives \\eqref{ineqw2}.\n\\end{proof}\nFor the corollary we denote by $||\\cdot||'_{w,k,p,\\delta;B_i}$ the norm with the domain of integration restricted to $B_i$ (resp. for the domain $B$). In the case of the standard norms, $||\\cdot||'_{k,p,\\delta;B_i}$ also means that in the integration the replacement $r\\rightarrow r_i$ has been made, centering the norm at $x_i$. \n\\begin{corollary}\\label{equivNorms}\n The norms $||\\cdot||'_{k,p,\\delta;B_i}$ and $||\\cdot||'_{w,k,p,\\delta;B_i}$ (resp. $||\\cdot||'_{k,p,\\delta;B}$ and $||\\cdot||'_{w,k,p,\\delta;B}$) are equivalent.\n\\end{corollary}\n\nNext we prove important estimates on functions $u\\in W'^ {k,p}_{w,\\delta}$ near the asymptotic ends. This is analogous to and follows Lemma A.1 in \\cite{Dain:2010uh}.\n\\begin{lemma}\\label{decay}\nAssume $u\\in W'^{k,p}_{w,\\delta}$ with $n-kp<0$, then we have the following estimate\n \\begin{equation}\n \\label{eq:41}\n w^{-\\delta}|u|\\leq C \\left\\lVert u \\right\\rVert'_{w,k,p,\\delta}. \n \\end{equation}\nMoreover, we have\n \\begin{equation}\n\\label{eq:44}\n \\lim_{r_i\\to 0}w^{-\\delta} |u|= \\lim_{r\\to \\infty}w^{-\\delta} |u|= 0. \n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nRestricting the domain of $u$ to $B_i$ there are constants $C_k$ such that\n\\begin{equation}\nw^{-\\delta}|u| \\leq C_1 r_i^{-\\delta}|u|\\leq C_2 ||u||'_{k,p,\\delta;B_i} \\leq C_3 ||u||'_{w,k,p,\\delta;B_i} \\leq C_4 ||u||'_{w,k,p,\\delta},\n\\end{equation}\nwhere we have used first Proposition \\ref{prop1}, then Lemma A.1 in \\cite{Dain:2010uh}, noticing that in that lemma the inequality and the \nlimits can also be proven separately for a ball around the origin and for the asymptotic region $r>R$, therefore we can use separately the\nbounds in each of $B_i$ and in $B$. For the third inequality Corollary \\ref{equivNorms} was used and the last inequality comes \nfrom the fact that the norm of $u$ on the whole domain is bigger than its norm restricted to $B_i$. The same sequence of inequalities \nhold for the domain $B$. Also, the set\n\\begin{equation}\nA = \\{r_i \\geq R_i, r\\leq R\\},\n\\end{equation}\nis compact and hence the function\n\\begin{equation}\nw^{-\\delta}|u|\n\\end{equation}\nhas a maximum there, this together with the previous inequalities means that there exists a constant $C$ such that \\eqref{eq:41} \nis satisfied in $\\mathbb{R}^n\\backslash \\Pi$. \nAlso, using the same argument as in Lemma A.1 in \\cite{Dain:2010uh}, we have \\eqref{eq:44}.\n\\end{proof}\n\nAnother important result that will be useful later is the following. \n\n\n\\begin{lemma}\\label{lemab1}\n If $u\\in W'^{1,p}_{w,-1\/2}$, then $w^{\\frac{1}{2}-\\frac{n}{p}}u\\in L^p$ and $w^{\\frac{3}{2}-\\frac{n}{p}}\\partial u \\in L^p$.\n\\end{lemma}\n\\begin{proof}\n\\begin{equation}\n \\|u\\|_{W'^{1,p}_{w,-1\/2}} = \\|u\\|_{L'^p_{w,-1\/2}} + \\|\\partial u\\|_{L'^p_{w,-3\/2}} = \\|w^{\\frac{1}{2}-\\frac{n}{p}}u\\|_{L^p} + \\|w^{\\frac{3}{2}-\\frac{n}{p}}\\partial u\\|_{L^p},\n \\end{equation}\n and therefore if $\\|u\\|_{W'^{1,p}_{w,-1\/2}}$ is bounded, then $\\|w^{\\frac{1}{2}-\\frac{n}{p}}u\\|_{L^p}$ and $\\|w^{\\frac{3}{2}-\\frac{n}{p}}\\partial u\\|_{L^p}$ are also bounded.\n\\end{proof}\n\n\nThe main result we prove in this section is the fact that the Laplace operator is an isomorphism $\\Delta: W'^{k+2,p}_{w,\\delta}\\rightarrow W'^{k,p}_{w,\\delta-2}$. We follow \nBartnik's arguments in the proof of Theorem 1.7 in \\cite{Bartnik86}. We need to exclude some exceptional values of $\\delta$. The exceptional values are $\\{ m'\\in\\mathbb{Z}, m'\\neq -1,-2,\\ldots,3-n\\}$, $\\delta$ is said to be nonexceptional if it is not one of those values, and it is convenient to define\n\\begin{equation}\n m = \\max \\,\\{ m' \\mbox{ exceptional}, m'<\\delta \\}.\n\\end{equation}\n\n\\begin{theorem}\\label{lemaIsom}\n Let $\\delta$ be nonexceptional, $10$ such that for all $\\lambda\\in (-\\lambda_0, \\lambda_0)$ there\nexists a solution $u(\\lambda)\\in H'^{2}_{w,-1\/2}$ of equation\n\\eqref{eqG}. The solution $u(\\lambda)$ is continuously differentiable in\n$\\lambda$ and satisfies $\\Phi_0+u(\\lambda)>0$. Moreover, for small $\\lambda$\nand small $u$ (in the norm $ H^{'2}_{w,-1\/2}$) the solution $u(\\lambda)$ is the\nunique solution of equation \\eqref{eqG}. \n\\end{theorem}\n\n\n\\begin{proof}\nThe proof of this theorem follows the lines of \\cite{Acena:2015rxa} and makes strong use of the Inverse Function theorem. We will ommit some arguments that are carried over from one cylindrical end to many such ends. \nThe proof can be thought of as showing that $u$ gives rise to a good conformal factor and collecting the necessary conditions for the Inverse Function theorem to hold.\n\n\\begin{itemize} \n\\item \\textit{$\\Phi_0+u$ is positive.}\n \nWe are considering the map $G:\\mathbb{R}\\times H'^2_{w,-1\/2}\\rightarrow L'^2_{w,-5\/2}$, but for a general $u\\in H'^2_{w,-1\/2}$ \nthe function $\\Phi=\\Phi_0+u$ does not have a definite sign. In order for $\\Phi$ to be a \nconformal factor we need it to be positive. As we assume $\\Phi_0$ to be a conformal factor, then we can \nconjecture that if $u$ is small enough, then $\\Phi$ is also going to be a conformal factor. This can be achieved by restricting $u$ to be in a ball around the origin in $H'^2_{w,-1\/2}$. We therefore define the subset $V$ of $H'^2_{w,-1\/2}$ as\n\\begin{equation}\n V = \\{v\\in H'^2_{w,-1\/2} : ||v||_{H'^2_{w,-1\/2}}<\\xi\\},\n\\end{equation}\nwhere $\\xi>0$ is a constant that can always be chosen and whose particular value depends on $\\Phi_0$. We restrict the map, \n$G:\\mathbb{R}\\times V\\rightarrow L'^2_{w,-5\/2}$, having that if $u\\in V$ then\n\\begin{equation}\n \\Phi_0+u > 0.\n\\end{equation}\n\n\n\\item\\textit{The map $G$ is well defined}\n\nTo prove that the map $G:\\mathbb{R}\\times V\\rightarrow L'^2_{w,-5\/2}$ is well defined we need to show that $||G(\\lambda,u)||_{L'^2_{w,-5\/2}}$ is bounded whenever $\\lambda \\in \\mathbb{R}$ and $u\\in V$. This is accomplished using the definition of the $H'^2_{w,-1\/2}$ norm, the triangle inequality, the asymptotic conditions on the background functions and the compact support of $q$, $\\omega$, $\\psi$ and $\\chi$, together with the inequalities presented in Appendix \\ref{useIneq}.\n\n\n\\item \\textit{Partial Fr\\'echet derivatives of $G$}\n\nTo obtain the partial Fr\\'echet derivatives of $G$ and to show that $G$ is $C^1$ we start by calculating the directional derivatives of $G$, namely\n\\begin{equation}\n \\left.\\frac{d}{dt}G(\\lambda+t\\gamma,u)\\right|_{t=0},\\qquad \\frac{d}{dt}G(\\lambda,u+tv)\\Bigg|_{t=0},\n\\end{equation}\nand propose them as the partial Fr\\'echet derivatives of $G$,\n\\begin{eqnarray}\n D_1G(\\lambda,u)[\\gamma] & = &\\Bigg[\\frac{1}{4}\\Delta_2q(\\Phi_0+u)+\\frac{\\partial\\omega(\\partial\\omega_0+\\lambda\\partial\\omega)}{8\\rho^4(\\Phi_0+u)^7} \\\\\n && +\\frac{\\partial\\psi(\\partial\\psi_0+\\lambda\\partial\\psi)}{2\\rho^2(\\Phi_0+u)^3}+\\frac{\\partial\\chi(\\partial\\chi_0+\\lambda\\partial\\chi)}{2\\rho^2(\\Phi_0+u)^3}\\Bigg]\\gamma,\\\\\n D_2G(\\lambda,u)[v] & = & \\Delta v + \\Bigg[\\frac{1}{4}\\Delta_2 q_0+\\frac{1}{4}\\lambda\\Delta_2q - \\frac{7(\\partial\\omega_0+\\lambda\\partial\\omega)^2}{16\\rho^4(\\Phi_0+u)^8} \\\\\n && -\\frac{3(\\partial\\psi_0+\\lambda\\partial\\psi)^2}{4\\rho^2(\\Phi_0+u)^4}-\\frac{3(\\partial\\chi_0+\\lambda\\partial\\chi)^2}{4\\rho^2(\\Phi_0+u)^4}\\Bigg]v.\n\\end{eqnarray}\nNow it can be shown that these operators are bounded using the properties mentioned in the point above, showing that they are linear operators between the following spaces\n\\begin{equation}\nD_1G(\\lambda,u):\\mathbb{R}\\rightarrow L'^2_{-5\/2}, \\qquad D_2G(\\lambda,u):H'^2_{-1\/2}\\rightarrow L'^2_{-5\/2}.\n\\end{equation}\nA lengthy but straightforward calculation shows that\n\\begin{equation}\n \\lim_{\\gamma\\ra0}\\frac{||G(\\lambda+\\gamma,u)-G(\\lambda,u)-D_1G(\\lambda,u)[\\gamma]||_{L'^2_{-5\/2}}}{|\\gamma|}=0,\n\\end{equation}\n\\begin{equation}\n \\lim_{v\\ra0}\\frac{||G(\\lambda,u+v)-G(\\lambda,u)-D_2G(\\lambda,u)[v]||_{L'^2_{-5\/2}}}{||v||_{H'^2_{-1\/2}}}=0\n\\end{equation}\nand therefore $D_1G$ and $D_2G$ are indeed the partial Fr\\'echet derivatives of $G$. It can also be checked that \nthe derivatives are continuous, and then $G$ is $C^1$.\n\n\n\\item \\textit{The map $D_2G(0,0):H'^2_{w,-1\/2}\\rightarrow L'^2_{w,-1\/2}$ is an isomorphism}\n\nAssociated with the map $D_2G(0,0)$ we define the operator $\\mathcal{L}$ through\n\\begin{equation}\n\\mathcal{L}v := -\\Delta v+\\alpha v,\n\\end{equation}\nwhere\n\\begin{equation}\\label{alpha0}\n \\alpha=-\\frac{\\Delta_2q_0}{4}+7\\frac{(\\partial\\omega_0)^2}{16\\rho^4\\Phi_0^8}+3\\frac{(\\partial\\psi_0)^2+(\\partial\\chi_0)^2}{4\\rho^2\\Phi_0^4}.\n\\end{equation}\nWe want to show that the equation\n\\begin{equation}\\label{ecL}\n\\mathcal Lu:=-\\Delta u+\\alpha u=f \\qquad \\mbox{in}\\; \\mathbb R^{3}\\setminus \\Pi\n\\end{equation}\nhas a unique solution $u\\in H'^2_{w,-1\/2}$ for each $f\\in L'^2_{w,-1\/2}$. The steps for this are the same as in \\cite{Acena:2015rxa},\nsee also the references therehin. The steps of the proof are as follows. First the Yamabe condition is used to show that for all $f\\in C_0^\\infty$, $f\\neq0$,\n\\begin{equation}\\label{yamflat} \n \\int_M|\\partial f|^2+\\alpha f^2 d\\mu>0,\n \\end{equation}\nwhere $\\alpha$ is given by \\eqref{alpha0} and the norm and volume\nelement in \\eqref{yamflat} are computed with respect to the flat metric. This is used to show that the bilinear form\n\\begin{equation}\\label{Bil}\nB[u,v]:=\\int_{\\mathbb{R}^3\\setminus\\Pi}\\partial u\\cdot \\partial v+\\alpha uv\\,d\\mu,\n\\end{equation}\nwith $u,v\\in H^{'1}_{w,-1\/2}$, which is well defined in virtue of Lemma \\ref{lemab1} and corresponds to \nthe linear operator $\\mathcal L$, is indeed an inner product. Using equation \\eqref{alpha0} and the standard Holder inequality for $L^2$ spaces (note that by Lemma \\ref{lemab1}, $\\partial u,\\;\\partial v,\\; uw^{-1},\\;v^{-1}\\in L^2$) it is shown that the linear functional $\\ell(\\cdot):=B[\\cdot,v]$ is \nbounded for all $v\\in H^{'1}_{w,-1\/2}$. With these conditions fulfilled, the Riesz Representation Theorem states that there exists a unique weak solution, $u\\in H^{'1}_{w,-1\/2}$, of $\\mathcal Lu=f$, for each $f\\in L^{'2}_{w,-5\/2}$. The last step is to show that the weak solution is indeed in $H^ {'2}_{w,-1\/2}$. This follows the same argument as in \\cite{gabach09}, the only new ingredient is the use of Theorem \\ref{lemaIsom}.\n\\end{itemize}\nThis completes the proof, as we have shown that the conditions for the Inverse Function theorem to hold are true in the case at hand.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}